💡First, I need to determine how many times the prime number 7 appears as a factor in the prime factorization of 100!.To do this, I'll start by dividing 100 by 7 to find the number of multiples of 7 within the range of 1 to 100.100 ÷ 7 = 14 with a remainder, which means there are 14 multiples of 7.Next, I'll divide 100 by 49 (which is 7 squared) to account for numbers that have more than one factor of 7.100 ÷ 49 = 2 with a remainder, indicating there are 2 multiples of 49.Finally, I'll add up all the multiples of 7 found in the previous steps to get the total number of times 7 appears as a factor in 100!.14 (from the first division) + 2 (from the second division) = 16.Therefore, the largest value of w such that 7^w is a factor of 100! is 16.

💡Okay, so I have this problem: I need to figure out into how many regions the x-axis and the graphs of ( y = 2 - x^2 ) and ( y = x^2 - 1 ) split the plane. Hmm, let me try to visualize this first.Alright, ( y = 2 - x^2 ) is a downward-opening parabola because the coefficient of ( x^2 ) is negative. Its vertex is at (0, 2). On the other hand, ( y = x^2 - 1 ) is an upward-opening parabola with its vertex at (0, -1). The x-axis is, of course, the line ( y = 0 ).I think the key here is to find all the intersection points between these curves because each intersection can potentially create a new region. So, first, I should find where these two parabolas intersect each other and where each parabola intersects the x-axis.Let me start by finding the points where ( y = 2 - x^2 ) and ( y = x^2 - 1 ) intersect. To do that, I'll set the two equations equal to each other:( 2 - x^2 = x^2 - 1 )Solving for x:( 2 + 1 = x^2 + x^2 )( 3 = 2x^2 )( x^2 = frac{3}{2} )So, ( x = sqrt{frac{3}{2}} ) or ( x = -sqrt{frac{3}{2}} ). Therefore, the two parabolas intersect at ( x = sqrt{frac{3}{2}} ) and ( x = -sqrt{frac{3}{2}} ). Plugging these back into either equation to find the y-coordinates:For ( x = sqrt{frac{3}{2}} ):( y = 2 - (sqrt{frac{3}{2}})^2 = 2 - frac{3}{2} = frac{1}{2} )Similarly, for ( x = -sqrt{frac{3}{2}} ):( y = 2 - (-sqrt{frac{3}{2}})^2 = 2 - frac{3}{2} = frac{1}{2} )So, the points of intersection are ( (sqrt{frac{3}{2}}, frac{1}{2}) ) and ( (-sqrt{frac{3}{2}}, frac{1}{2}) ).Next, I need to find where each parabola intersects the x-axis. For ( y = 2 - x^2 ), set ( y = 0 ):( 0 = 2 - x^2 )( x^2 = 2 )( x = sqrt{2} ) or ( x = -sqrt{2} )So, the points are ( (sqrt{2}, 0) ) and ( (-sqrt{2}, 0) ).For ( y = x^2 - 1 ), set ( y = 0 ):( 0 = x^2 - 1 )( x^2 = 1 )( x = 1 ) or ( x = -1 )So, the points are ( (1, 0) ) and ( (-1, 0) ).Alright, so now I have all the intersection points:- Between the two parabolas: ( (sqrt{frac{3}{2}}, frac{1}{2}) ) and ( (-sqrt{frac{3}{2}}, frac{1}{2}) )- Between ( y = 2 - x^2 ) and the x-axis: ( (sqrt{2}, 0) ) and ( (-sqrt{2}, 0) )- Between ( y = x^2 - 1 ) and the x-axis: ( (1, 0) ) and ( (-1, 0) )Now, I need to figure out how these curves divide the plane into regions. I think the best way is to sketch a rough graph mentally.Starting with the x-axis, which is a straight line. Then, the downward-opening parabola ( y = 2 - x^2 ) will intersect the x-axis at ( sqrt{2} ) and ( -sqrt{2} ), which are approximately 1.414 and -1.414. The upward-opening parabola ( y = x^2 - 1 ) intersects the x-axis at 1 and -1, which are inside the intersections of the first parabola.The two parabolas intersect each other at ( sqrt{frac{3}{2}} ) and ( -sqrt{frac{3}{2}} ), which are approximately 1.225 and -1.225. So, these intersection points are between -1.414 and -1, and between 1 and 1.414 on the x-axis.Now, let me try to visualize the regions:1. Above both parabolas: Since ( y = 2 - x^2 ) is above ( y = x^2 - 1 ) near the y-axis, there might be a region above both.2. Between the two parabolas: Since they intersect at two points, there should be regions between them on either side of the y-axis.3. Below the x-axis: The parabola ( y = x^2 - 1 ) dips below the x-axis, so there should be regions below the x-axis as well.Wait, actually, let me think more carefully. The x-axis is part of the plane, so all regions must be either above or below the x-axis, but since the x-axis itself is a boundary, it's part of the division.But actually, the x-axis is a line, so it divides the plane into two halves: upper half-plane and lower half-plane. Then, the two parabolas further divide these halves into regions.But since the two parabolas intersect each other and the x-axis, the number of regions will be more.I remember that in general, each new curve can intersect existing curves and create new regions. The formula for the maximum number of regions created by n curves is ( frac{n(n + 1)}{2} + 1 ), but that's when each pair of curves intersects in two points, which might not be the case here.Wait, but in this case, we have three curves: the x-axis, ( y = 2 - x^2 ), and ( y = x^2 - 1 ). So, n = 3.But actually, the x-axis is a straight line, and the other two are parabolas. Each parabola can intersect the x-axis at two points, and they can intersect each other at two points.So, the total number of intersection points is:- Between ( y = 2 - x^2 ) and x-axis: 2 points.- Between ( y = x^2 - 1 ) and x-axis: 2 points.- Between ( y = 2 - x^2 ) and ( y = x^2 - 1 ): 2 points.So, total 6 intersection points.But how does this translate to regions?I think each intersection can potentially create a new region, but it's not that straightforward.Wait, maybe I can use Euler's formula for planar graphs: ( V - E + F = 2 ), where V is the number of vertices (intersection points), E is the number of edges (segments of curves), and F is the number of faces (regions).But I need to be careful because the plane is infinite, so Euler's formula applies to planar graphs embedded on a sphere, but for the plane, it's similar except that we consider the unbounded face as well.So, let's try to apply it.First, count the number of vertices V. These are the intersection points:- 2 intersections between ( y = 2 - x^2 ) and x-axis.- 2 intersections between ( y = x^2 - 1 ) and x-axis.- 2 intersections between ( y = 2 - x^2 ) and ( y = x^2 - 1 ).So, V = 6.Next, count the number of edges E. Each curve is divided into segments by the intersection points.- The x-axis is divided into segments by the intersections with the parabolas. It intersects ( y = 2 - x^2 ) at ( sqrt{2} ) and ( -sqrt{2} ), and ( y = x^2 - 1 ) at 1 and -1. So, the x-axis is divided into 4 segments: from ( -infty ) to ( -sqrt{2} ), ( -sqrt{2} ) to -1, -1 to 1, 1 to ( sqrt{2} ), and ( sqrt{2} ) to ( infty ). Wait, but actually, each intersection splits the x-axis into segments. Since the x-axis is a single line, each intersection adds a new segment.Wait, no, actually, each intersection point on the x-axis splits it into more segments. Starting from the left, we have:- From ( -infty ) to ( -sqrt{2} )- ( -sqrt{2} ) to -1- -1 to 1- 1 to ( sqrt{2} )- ( sqrt{2} ) to ( infty )So, that's 5 segments on the x-axis.Similarly, each parabola is intersected by the other parabola and the x-axis.For ( y = 2 - x^2 ):- It intersects the x-axis at ( sqrt{2} ) and ( -sqrt{2} ), and intersects ( y = x^2 - 1 ) at ( sqrt{frac{3}{2}} ) and ( -sqrt{frac{3}{2}} ).So, the curve ( y = 2 - x^2 ) is divided into segments:- From ( -infty ) to ( -sqrt{2} )- ( -sqrt{2} ) to ( -sqrt{frac{3}{2}} )- ( -sqrt{frac{3}{2}} ) to ( sqrt{frac{3}{2}} )- ( sqrt{frac{3}{2}} ) to ( sqrt{2} )- ( sqrt{2} ) to ( infty )Wait, but actually, since it's a parabola opening downward, it's symmetric. So, it's divided into 5 segments as well.Similarly, for ( y = x^2 - 1 ):- It intersects the x-axis at 1 and -1, and intersects ( y = 2 - x^2 ) at ( sqrt{frac{3}{2}} ) and ( -sqrt{frac{3}{2}} ).So, the curve ( y = x^2 - 1 ) is divided into segments:- From ( -infty ) to ( -sqrt{frac{3}{2}} )- ( -sqrt{frac{3}{2}} ) to -1- -1 to 1- 1 to ( sqrt{frac{3}{2}} )- ( sqrt{frac{3}{2}} ) to ( infty )Again, 5 segments.So, in total, the number of edges E is:- 5 segments on the x-axis- 5 segments on ( y = 2 - x^2 )- 5 segments on ( y = x^2 - 1 )So, E = 5 + 5 + 5 = 15.But wait, each intersection point is shared by two curves, so each intersection is a vertex where two edges meet. So, in Euler's formula, each edge is a segment between two vertices.But actually, each intersection splits a curve into segments, so the number of edges should be calculated as the sum of all segments from all curves.But I think my initial count is correct: 15 edges.Now, applying Euler's formula:( V - E + F = 2 )We have V = 6, E = 15, so:( 6 - 15 + F = 2 )( -9 + F = 2 )( F = 11 )But wait, that can't be right because when I visualize the graph, I don't think there are 11 regions. Maybe I made a mistake in counting the edges.Wait, perhaps I overcounted the edges. Let me think again.Each intersection point is where two curves meet. So, each intersection is a vertex where two edges meet.But when I count the edges, I need to count each segment between two vertices as one edge.So, for the x-axis, which is a straight line, it's divided into segments by the intersection points at ( -sqrt{2} ), -1, 1, and ( sqrt{2} ). So, between ( -infty ) and ( -sqrt{2} ), that's one edge. Then ( -sqrt{2} ) to -1, another edge. Then -1 to 1, another. Then 1 to ( sqrt{2} ), and ( sqrt{2} ) to ( infty ). So, that's 5 edges on the x-axis.Similarly, for ( y = 2 - x^2 ), it's a parabola opening downward. It intersects the x-axis at ( sqrt{2} ) and ( -sqrt{2} ), and intersects ( y = x^2 - 1 ) at ( sqrt{frac{3}{2}} ) and ( -sqrt{frac{3}{2}} ). So, the segments are:- From ( -infty ) to ( -sqrt{2} )- ( -sqrt{2} ) to ( -sqrt{frac{3}{2}} )- ( -sqrt{frac{3}{2}} ) to ( sqrt{frac{3}{2}} )- ( sqrt{frac{3}{2}} ) to ( sqrt{2} )- ( sqrt{2} ) to ( infty )So, 5 edges.Similarly, for ( y = x^2 - 1 ), it's a parabola opening upward. It intersects the x-axis at -1 and 1, and intersects ( y = 2 - x^2 ) at ( -sqrt{frac{3}{2}} ) and ( sqrt{frac{3}{2}} ). So, the segments are:- From ( -infty ) to ( -sqrt{frac{3}{2}} )- ( -sqrt{frac{3}{2}} ) to -1- -1 to 1- 1 to ( sqrt{frac{3}{2}} )- ( sqrt{frac{3}{2}} ) to ( infty )Again, 5 edges.So, total edges E = 5 (x-axis) + 5 (( y = 2 - x^2 )) + 5 (( y = x^2 - 1 )) = 15.But wait, in planar graphs, each edge is shared by two faces, except for the unbounded face. But I'm not sure if that's directly applicable here.Wait, maybe I should consider that each intersection point is where two edges meet, so each vertex has degree 4? Because each intersection is where two curves cross, so each vertex is where four edges meet? Wait, no, each intersection is where two curves cross, so each vertex is where two edges meet, meaning degree 2.Wait, no, actually, each intersection is where two curves cross, so each vertex is where four edges meet? Because each curve contributes two edges at the intersection point.Wait, I'm getting confused. Let me clarify.Each intersection point is where two curves cross, so each curve is split into two segments at that point. So, each intersection adds two edges for each curve. But in terms of graph theory, each intersection is a vertex where four edges meet: two from each curve.But in our case, each intersection is a crossing of two curves, so each intersection is a vertex of degree 4.Wait, but in planar graphs, the degree of a vertex is the number of edges incident to it. So, if two curves cross at a point, each curve contributes two edges (one on each side of the intersection). So, each intersection is a vertex of degree 4.But in our case, the x-axis is a straight line, and the parabolas are curves. So, each intersection between the x-axis and a parabola is a vertex where two edges meet: one from the x-axis and one from the parabola.Similarly, the intersections between the two parabolas are vertices where two edges meet: one from each parabola.Wait, no, actually, when two curves cross, each curve is split into two segments at that point, so each intersection is a vertex where four edges meet: two from each curve.But in our case, the x-axis is a straight line, and the parabolas are curves. So, when the x-axis intersects a parabola, it's a vertex where two edges meet: one from the x-axis and one from the parabola.Similarly, when the two parabolas intersect, it's a vertex where two edges meet: one from each parabola.Wait, I think I'm overcomplicating this. Maybe it's better to think that each intersection is a vertex where two edges meet, regardless of how many curves are involved.But in reality, when two curves cross, they create four edges around the vertex, but in planar graph terms, each intersection is a single vertex where four edges meet.But I'm not sure. Maybe I should look for another approach.Alternatively, I can think about how many times the curves intersect and how that affects the number of regions.I remember that each new curve can intersect existing curves and create new regions. The formula for the maximum number of regions created by n curves is ( frac{n(n + 1)}{2} + 1 ), but that's when each pair of curves intersects in two points, which might not be the case here.Wait, but in our case, we have three curves: the x-axis, ( y = 2 - x^2 ), and ( y = x^2 - 1 ). So, n = 3.But the x-axis is a straight line, and the other two are parabolas. Each parabola can intersect the x-axis at two points, and they can intersect each other at two points.So, the total number of intersection points is:- Between ( y = 2 - x^2 ) and x-axis: 2 points.- Between ( y = x^2 - 1 ) and x-axis: 2 points.- Between ( y = 2 - x^2 ) and ( y = x^2 - 1 ): 2 points.So, total 6 intersection points.Now, the formula for the maximum number of regions created by n curves is ( frac{n(n + 1)}{2} + 1 ), but that's when each pair of curves intersects in two points, which is the case here.Wait, actually, no. The formula ( frac{n(n + 1)}{2} + 1 ) is for the maximum number of regions created by n lines. For curves, it's different because each pair can intersect more than once.But in our case, we have three curves: one line and two parabolas. Each parabola intersects the line at two points, and the two parabolas intersect each other at two points. So, total intersections: 2 + 2 + 2 = 6.Now, the general formula for the maximum number of regions created by n curves where each pair intersects in k points is ( R = n(n - 1)k / 2 + n + 1 ). Wait, I'm not sure if that's correct.Alternatively, I think the formula is similar to the one for lines, but adjusted for the number of intersections.Wait, actually, the formula for the maximum number of regions created by n curves where each pair intersects in k points is ( R = k cdot frac{n(n - 1)}{2} + n + 1 ).But I'm not sure. Maybe I should look for another approach.Alternatively, I can use the principle that each new curve can intersect all existing curves, and each intersection creates a new region.So, starting with the x-axis: it divides the plane into two regions: upper and lower.Then, adding the first parabola, ( y = 2 - x^2 ). It intersects the x-axis at two points, so it will cross from the upper half-plane to the lower half-plane twice. Each time it crosses, it can create a new region.Similarly, adding the second parabola, ( y = x^2 - 1 ), which also intersects the x-axis at two points and intersects the first parabola at two points.So, let's try to count step by step.1. Start with the plane: 1 region.2. Add the x-axis: divides the plane into 2 regions.3. Add ( y = 2 - x^2 ): it intersects the x-axis at two points, so it will cross from upper to lower half-plane twice. Each crossing can potentially create a new region. So, adding this parabola will add 2 regions. So, total regions: 2 + 2 = 4.Wait, but actually, when you add a curve that intersects the existing arrangement, the number of new regions created is equal to the number of times it crosses the existing curves plus 1.So, when adding ( y = 2 - x^2 ), it crosses the x-axis twice, so it creates 2 + 1 = 3 new regions. So, total regions: 2 + 3 = 5.Wait, that seems more accurate.Similarly, when adding the second parabola ( y = x^2 - 1 ), it will cross the x-axis twice and cross the first parabola twice. So, total intersections: 2 (with x-axis) + 2 (with first parabola) = 4 intersections.Therefore, the number of new regions created by adding this parabola is 4 + 1 = 5.So, total regions: 5 + 5 = 10.Wait, that seems to make sense.Let me verify:- Start with 1 region.- Add x-axis: 2 regions.- Add ( y = 2 - x^2 ): crosses x-axis twice, so adds 3 regions. Total: 5.- Add ( y = x^2 - 1 ): crosses x-axis twice and crosses ( y = 2 - x^2 ) twice, so 4 intersections, adds 5 regions. Total: 10.Yes, that seems to add up.Alternatively, using Euler's formula:V - E + F = 2We have V = 6 (intersection points), E = 15 (segments), so:6 - 15 + F = 2F = 11But wait, that contradicts the previous count of 10 regions.Hmm, maybe I made a mistake in counting edges.Wait, perhaps I overcounted the edges. Let me think again.Each intersection point is where two curves meet, so each intersection is a vertex where two edges meet.But when I counted edges, I considered each segment between two vertices as an edge. So, for the x-axis, which is divided into 5 segments, that's 5 edges.Similarly, each parabola is divided into 5 segments, so 5 edges each.So, total edges: 5 + 5 + 5 = 15.But in planar graph terms, each edge is shared by two faces, except for the unbounded face.Wait, but in our case, the plane is infinite, so the unbounded face is also counted.Wait, maybe I should consider that each edge is shared by two faces, so the number of edges E is related to the number of faces F by the formula:E = 3F / 2But that's for planar graphs where each face is bounded by at least three edges.Wait, no, that's not necessarily true here because some faces might be bounded by fewer edges.Alternatively, maybe I should use the formula V - E + F = 2.We have V = 6, E = 15, so:6 - 15 + F = 2F = 11But according to the step-by-step addition, we have 10 regions.So, there's a discrepancy here. Maybe my count of edges is incorrect.Wait, perhaps I'm counting the same edge multiple times.Wait, each intersection point is shared by two curves, so each edge is shared by two curves.Wait, no, each edge is a segment of a single curve between two intersection points.So, each edge is unique to a curve.Therefore, the total number of edges is indeed 15.But then, why does Euler's formula give F = 11, while the step-by-step count gives F = 10?Hmm, maybe because the plane is infinite, and Euler's formula counts the unbounded face as well, but in our case, the number of regions includes the unbounded face.Wait, actually, in planar graph terms, the formula V - E + F = 2 includes the unbounded face as one of the faces.So, if we have F = 11, that includes the unbounded face.But in our step-by-step count, we have 10 regions, which should include the unbounded face.Wait, maybe I made a mistake in the step-by-step count.Let me try again.1. Start with the plane: 1 region.2. Add the x-axis: divides the plane into 2 regions.3. Add ( y = 2 - x^2 ): it intersects the x-axis at two points, so it crosses from upper to lower half-plane twice. Each crossing creates a new region. So, the number of new regions created is equal to the number of times it crosses existing curves plus 1.So, it crosses the x-axis twice, so it creates 2 + 1 = 3 new regions. So, total regions: 2 + 3 = 5.4. Add ( y = x^2 - 1 ): it intersects the x-axis twice and intersects ( y = 2 - x^2 ) twice. So, total intersections: 4. Therefore, the number of new regions created is 4 + 1 = 5. So, total regions: 5 + 5 = 10.But according to Euler's formula, F = 11.So, there's a discrepancy of 1.Wait, maybe because when we add the second parabola, it also intersects the first parabola, which is already in the plane, so the formula might be slightly different.Alternatively, maybe I'm missing something in the edge count.Wait, let's recount the edges.Each curve is divided into segments by the intersection points.- The x-axis is divided into 5 segments: ( (-infty, -sqrt{2}) ), ( (-sqrt{2}, -1) ), ( (-1, 1) ), ( (1, sqrt{2}) ), ( (sqrt{2}, infty) ). So, 5 edges.- ( y = 2 - x^2 ) is divided into 5 segments: ( (-infty, -sqrt{2}) ), ( (-sqrt{2}, -sqrt{frac{3}{2}}) ), ( (-sqrt{frac{3}{2}}, sqrt{frac{3}{2}}) ), ( (sqrt{frac{3}{2}}, sqrt{2}) ), ( (sqrt{2}, infty) ). So, 5 edges.- ( y = x^2 - 1 ) is divided into 5 segments: ( (-infty, -sqrt{frac{3}{2}}) ), ( (-sqrt{frac{3}{2}}, -1) ), ( (-1, 1) ), ( (1, sqrt{frac{3}{2}}) ), ( (sqrt{frac{3}{2}}, infty) ). So, 5 edges.Total edges: 5 + 5 + 5 = 15.But in planar graph terms, each edge is shared by two faces, except for the unbounded face, which is only adjacent to one edge.Wait, no, actually, in planar graphs, each edge is shared by exactly two faces, including the unbounded face.So, if we have E edges, then the total number of face-edge incidences is 2E.But each face is bounded by at least three edges, so 2E ≥ 3F.But in our case, 2E = 30, and F = 11, so 30 ≥ 33, which is not true. So, that's a contradiction.Wait, that means my count of edges is incorrect.Alternatively, maybe I'm misapplying Euler's formula because the graph is not connected or because it's on the plane.Wait, actually, Euler's formula for planar graphs is V - E + F = 2, where F includes the unbounded face.But in our case, the graph is connected, right? Because all curves intersect each other, so the entire arrangement is connected.Wait, but if I have three curves, each intersecting the others, the graph should be connected.Wait, let me try to recount the edges properly.Each intersection point is where two curves meet, so each intersection is a vertex where two edges meet.But each curve is a single edge in the graph, right? Wait, no, each curve is divided into segments by the intersection points, so each segment is an edge.So, for the x-axis, which is a straight line, it's divided into 5 segments, so 5 edges.For ( y = 2 - x^2 ), it's divided into 5 segments, so 5 edges.For ( y = x^2 - 1 ), it's divided into 5 segments, so 5 edges.Total edges: 15.But each intersection is where two edges meet, so each vertex has degree 2.Wait, no, each intersection is where two curves cross, so each vertex is where four edges meet: two from each curve.Wait, that's not correct because each curve is a single edge, but when two curves cross, they create four edges around the vertex.Wait, no, in graph theory, each intersection is a vertex where two edges meet, one from each curve.Wait, I'm getting confused again.Let me think differently. Each intersection is a point where two curves cross, so each intersection is a vertex where two edges meet.But in reality, each curve is split into segments by the intersections, so each intersection is a vertex where two edges meet.So, for example, the x-axis is split into 5 segments, so it has 4 intersection points (at ( -sqrt{2} ), -1, 1, ( sqrt{2} )), but actually, it's split into 5 segments by 4 points.Wait, no, the x-axis is split into 5 segments by 4 intersection points.Similarly, each parabola is split into 5 segments by 4 intersection points.So, total number of edges:- x-axis: 5 edges- ( y = 2 - x^2 ): 5 edges- ( y = x^2 - 1 ): 5 edgesTotal: 15 edgesNumber of vertices V: each intersection point is a vertex. There are 6 intersection points.So, V = 6, E = 15.Now, applying Euler's formula:V - E + F = 26 - 15 + F = 2F = 11So, F = 11 regions.But according to the step-by-step count, we have 10 regions.So, which one is correct?Wait, maybe the step-by-step count is missing something.Let me try to visualize the regions.1. The upper half-plane is divided by the two parabolas and the x-axis.2. The lower half-plane is divided by the two parabolas and the x-axis.But actually, the x-axis is a single line, so it's just dividing the plane into upper and lower halves.Then, the two parabolas further divide these halves.Let me try to count the regions in the upper half-plane first.In the upper half-plane, we have:- The region above both parabolas.- The region between the two parabolas on the right side.- The region between the two parabolas on the left side.- The regions between the parabolas and the x-axis on both sides.Wait, actually, it's more complicated.Alternatively, let me think of the upper half-plane:- The parabola ( y = 2 - x^2 ) is above ( y = x^2 - 1 ) near the y-axis, and they intersect at ( sqrt{frac{3}{2}} ) and ( -sqrt{frac{3}{2}} ).So, in the upper half-plane:- Above ( y = 2 - x^2 ): 1 region.- Between ( y = 2 - x^2 ) and ( y = x^2 - 1 ): 2 regions (left and right of the y-axis).- Below ( y = x^2 - 1 ) but above the x-axis: 2 regions (left and right of the y-axis).Wait, that's 1 + 2 + 2 = 5 regions in the upper half-plane.In the lower half-plane:- Below the x-axis, we have the parabola ( y = x^2 - 1 ), which is below the x-axis between -1 and 1.So, the lower half-plane is divided into:- The region below ( y = x^2 - 1 ): 1 region.- The regions between ( y = x^2 - 1 ) and the x-axis on both sides: 2 regions.Wait, that's 1 + 2 = 3 regions in the lower half-plane.So, total regions: 5 (upper) + 3 (lower) = 8 regions.But according to Euler's formula, it's 11 regions.Hmm, that's a discrepancy.Wait, maybe I'm missing some regions.Let me try again.In the upper half-plane:1. Above both parabolas: 1 region.2. Between the two parabolas on the right side of the y-axis: 1 region.3. Between the two parabolas on the left side of the y-axis: 1 region.4. Between ( y = 2 - x^2 ) and the x-axis on the right side: 1 region.5. Between ( y = 2 - x^2 ) and the x-axis on the left side: 1 region.6. Between ( y = x^2 - 1 ) and the x-axis on the right side: 1 region.7. Between ( y = x^2 - 1 ) and the x-axis on the left side: 1 region.Wait, that's 7 regions in the upper half-plane.In the lower half-plane:1. Below ( y = x^2 - 1 ): 1 region.2. Between ( y = x^2 - 1 ) and the x-axis on the right side: 1 region.3. Between ( y = x^2 - 1 ) and the x-axis on the left side: 1 region.So, 3 regions in the lower half-plane.Total regions: 7 + 3 = 10.But Euler's formula says 11.Wait, maybe I'm still missing a region.Alternatively, maybe the unbounded region is counted differently.Wait, in the upper half-plane, above both parabolas is one region, but actually, it's unbounded on the top, so it's one region.Between the two parabolas on the right and left sides: two regions.Between ( y = 2 - x^2 ) and the x-axis on both sides: two regions.Between ( y = x^2 - 1 ) and the x-axis on both sides: two regions.So, that's 1 + 2 + 2 + 2 = 7 regions in the upper half-plane.In the lower half-plane:- Below ( y = x^2 - 1 ): 1 region.- Between ( y = x^2 - 1 ) and the x-axis on both sides: two regions.Total: 3 regions.So, total regions: 7 + 3 = 10.But Euler's formula gives F = 11.So, where is the 11th region?Wait, maybe I'm not considering the regions beyond the parabolas.Wait, in the upper half-plane, above both parabolas is one region, but beyond the parabolas, there are regions extending to infinity.Wait, actually, each parabola extends to infinity, so the regions above and below are unbounded.Wait, maybe I'm missing the regions beyond the parabolas.Wait, in the upper half-plane, above both parabolas is one region, but actually, beyond the points where the parabolas intersect the x-axis, there are regions extending to infinity.Wait, no, because the parabolas open downward and upward, so beyond their intersections with the x-axis, they don't bound any regions.Wait, I'm getting confused again.Alternatively, maybe the correct count is 10 regions, and Euler's formula is giving 11 because of an error in the edge count.Wait, perhaps I overcounted the edges.Let me recount the edges.Each curve is divided into segments by the intersection points.- The x-axis is divided into 5 segments: ( (-infty, -sqrt{2}) ), ( (-sqrt{2}, -1) ), ( (-1, 1) ), ( (1, sqrt{2}) ), ( (sqrt{2}, infty) ). So, 5 edges.- ( y = 2 - x^2 ) is divided into 5 segments: ( (-infty, -sqrt{2}) ), ( (-sqrt{2}, -sqrt{frac{3}{2}}) ), ( (-sqrt{frac{3}{2}}, sqrt{frac{3}{2}}) ), ( (sqrt{frac{3}{2}}, sqrt{2}) ), ( (sqrt{2}, infty) ). So, 5 edges.- ( y = x^2 - 1 ) is divided into 5 segments: ( (-infty, -sqrt{frac{3}{2}}) ), ( (-sqrt{frac{3}{2}}, -1) ), ( (-1, 1) ), ( (1, sqrt{frac{3}{2}}) ), ( (sqrt{frac{3}{2}}, infty) ). So, 5 edges.Total edges: 15.But in planar graph terms, each edge is shared by two faces, so the total number of face-edge incidences is 2E = 30.Each face is bounded by at least three edges, so 3F ≤ 2E.Given F = 11, 3*11 = 33 ≤ 30, which is not true. So, this is a contradiction.Therefore, my edge count must be wrong.Wait, perhaps I'm counting the same edge multiple times.Wait, no, each edge is a unique segment of a curve between two intersection points.Wait, maybe I'm miscounting the number of intersection points.Wait, we have:- ( y = 2 - x^2 ) intersects the x-axis at ( sqrt{2} ) and ( -sqrt{2} ).- ( y = x^2 - 1 ) intersects the x-axis at 1 and -1.- ( y = 2 - x^2 ) intersects ( y = x^2 - 1 ) at ( sqrt{frac{3}{2}} ) and ( -sqrt{frac{3}{2}} ).So, total intersection points:- On the x-axis: 4 points: ( -sqrt{2} ), -1, 1, ( sqrt{2} ).- Between the two parabolas: 2 points: ( -sqrt{frac{3}{2}} ), ( sqrt{frac{3}{2}} ).So, total V = 6.But wait, each intersection on the x-axis is a point where a parabola meets the x-axis, so each is a vertex where two edges meet: one from the x-axis and one from the parabola.Similarly, the intersections between the two parabolas are vertices where two edges meet: one from each parabola.So, in total, V = 6.Each edge is a segment between two vertices.So, for the x-axis, which is divided into 5 segments, that's 5 edges.For ( y = 2 - x^2 ), divided into 5 segments, 5 edges.For ( y = x^2 - 1 ), divided into 5 segments, 5 edges.Total edges: 15.But according to Euler's formula, V - E + F = 2, so 6 - 15 + F = 2, so F = 11.But when I count regions, I get 10.So, where is the 11th region?Wait, maybe I'm missing the region at infinity.Wait, in planar graphs, the unbounded face is counted as one region. So, if I have 10 regions, including the unbounded face, then F = 11 would mean there's another region, which doesn't make sense.Wait, no, actually, the unbounded face is one region, and the other regions are bounded.So, if I have 10 regions, including the unbounded face, then F = 11 would imply there's an extra region, which is not possible.Therefore, I must have made a mistake in counting the regions.Wait, let me try to visualize again.In the upper half-plane:1. Above both parabolas: 1 region.2. Between the two parabolas on the right side: 1 region.3. Between the two parabolas on the left side: 1 region.4. Between ( y = 2 - x^2 ) and the x-axis on the right side: 1 region.5. Between ( y = 2 - x^2 ) and the x-axis on the left side: 1 region.6. Between ( y = x^2 - 1 ) and the x-axis on the right side: 1 region.7. Between ( y = x^2 - 1 ) and the x-axis on the left side: 1 region.That's 7 regions in the upper half-plane.In the lower half-plane:1. Below ( y = x^2 - 1 ): 1 region.2. Between ( y = x^2 - 1 ) and the x-axis on the right side: 1 region.3. Between ( y = x^2 - 1 ) and the x-axis on the left side: 1 region.That's 3 regions in the lower half-plane.Total regions: 7 + 3 = 10.But according to Euler's formula, F = 11.So, where is the 11th region?Wait, maybe I'm not considering the regions beyond the parabolas.Wait, in the upper half-plane, above both parabolas is one region, but actually, beyond the points where the parabolas intersect the x-axis, there are regions extending to infinity.Wait, no, because the parabolas open downward and upward, so beyond their intersections with the x-axis, they don't bound any regions.Wait, I'm still confused.Alternatively, maybe the correct answer is 10 regions, and Euler's formula is not directly applicable here because the graph is not planar or because it's on the plane rather than the sphere.Wait, actually, Euler's formula applies to planar graphs embedded on a sphere, but for the plane, it's similar except that the unbounded face is considered as one face.So, if we have F = 11, that includes the unbounded face.But in our case, the unbounded face is just one region, so the total number of regions should be 11, but when I count, I get 10.Wait, maybe I'm missing a region.Wait, perhaps there's a region that's bounded by all three curves.Wait, let me think.In the upper half-plane, between the two parabolas, there are two regions: one on the right and one on the left.But actually, between the two parabolas, there's a lens-shaped region that's bounded by both parabolas.Similarly, in the lower half-plane, below ( y = x^2 - 1 ), there's a region that's unbounded below.Wait, but I already counted that.Wait, maybe I'm missing a region near the origin.Wait, no, the origin is inside the region between the two parabolas.Wait, I'm not sure.Alternatively, maybe the correct answer is 10 regions, and the discrepancy with Euler's formula is because the graph is not planar or because I'm miscounting something.Given that the step-by-step count gives 10 regions, and that seems to make sense, I think the answer is 10 regions.So, the final answer is 10.

💡First, I identify the relationships between the skipping speeds of each person based on the given information.1. **Tony's Speed (T):** Tony skips at twice Bruce's speed. [ T = 2B ] 2. **Brandon's Speed (Br):** Brandon skips at one-third the square of Tony's speed. [ Br = frac{1}{3}T^2 ] 3. **Colin's Speed (C):** Colin skips at six times Brandon's speed. [ C = 6Br ] 4. **Daniel's Speed (D):** Daniel skips at one-fourth Colin's speed. [ D = frac{1}{4}C ]Given that Bruce's speed (B) is 1 mile per hour, I substitute this value into the equations to find the speeds of the others.1. Calculate Tony's speed: [ T = 2 times 1 = 2 text{ mph} ] 2. Calculate Brandon's speed: [ Br = frac{1}{3} times (2)^2 = frac{4}{3} text{ mph} ] 3. Calculate Colin's speed: [ C = 6 times frac{4}{3} = 8 text{ mph} ] 4. Calculate Daniel's speed: [ D = frac{1}{4} times 8 = 2 text{ mph} ]Finally, Colin's skipping speed is determined to be 8 miles per hour.

💡Alright, so I'm trying to figure out which of the student's conclusions about the function ( f(x) = x^2e^x ) are correct. There are four conclusions, numbered ① through ④, and I need to check each one step by step. Let me start by understanding each conclusion and then verifying its correctness.First, let me recall some basic properties of the function ( f(x) = x^2e^x ). This is a product of ( x^2 ) and ( e^x ), both of which are familiar functions. ( x^2 ) is a parabola that opens upwards, and ( e^x ) is an exponential function that grows rapidly for positive ( x ) and approaches zero for negative ( x ). Combining these, ( f(x) ) will have behavior influenced by both factors.I think it would be helpful to compute the derivative of ( f(x) ) to analyze its increasing and decreasing intervals, as well as to find critical points which might be local maxima or minima. So, let's compute ( f'(x) ).Using the product rule, ( f'(x) = frac{d}{dx}(x^2) cdot e^x + x^2 cdot frac{d}{dx}(e^x) ). The derivative of ( x^2 ) is ( 2x ), and the derivative of ( e^x ) is ( e^x ). So,[f'(x) = 2x e^x + x^2 e^x]I can factor out ( x e^x ) from both terms:[f'(x) = x e^x (2 + x)]So, ( f'(x) = x e^x (x + 2) ). This expression will help me determine where the function is increasing or decreasing.Now, let's analyze the critical points by setting ( f'(x) = 0 ):[x e^x (x + 2) = 0]Since ( e^x ) is never zero, the critical points occur where ( x = 0 ) or ( x + 2 = 0 ), which gives ( x = -2 ). So, the critical points are at ( x = -2 ) and ( x = 0 ).Next, I need to determine the intervals where ( f'(x) ) is positive or negative to understand where ( f(x) ) is increasing or decreasing.Let me consider the sign of each factor in ( f'(x) = x e^x (x + 2) ):1. ( x ): Negative when ( x < 0 ), positive when ( x > 0 ).2. ( e^x ): Always positive for all real ( x ).3. ( x + 2 ): Negative when ( x < -2 ), positive when ( x > -2 ).So, combining these:- For ( x < -2 ): - ( x ) is negative. - ( x + 2 ) is negative. - So, ( x (x + 2) ) is positive (negative times negative). - Multiply by ( e^x ) (positive): ( f'(x) ) is positive. - Therefore, ( f(x) ) is increasing on ( (-infty, -2) ).- For ( -2 < x < 0 ): - ( x ) is negative. - ( x + 2 ) is positive. - So, ( x (x + 2) ) is negative (negative times positive). - Multiply by ( e^x ) (positive): ( f'(x) ) is negative. - Therefore, ( f(x) ) is decreasing on ( (-2, 0) ).- For ( x > 0 ): - ( x ) is positive. - ( x + 2 ) is positive. - So, ( x (x + 2) ) is positive. - Multiply by ( e^x ) (positive): ( f'(x) ) is positive. - Therefore, ( f(x) ) is increasing on ( (0, infty) ).So, summarizing:- Increasing on ( (-infty, -2) ) and ( (0, infty) )- Decreasing on ( (-2, 0) )This analysis should help me check conclusion ①.**Conclusion ①**: The interval where ( f(x) ) is monotonically decreasing is ( (-2, 0) ).From my analysis, yes, ( f(x) ) is decreasing on ( (-2, 0) ). So, conclusion ① is correct.**Conclusion ②**: ( f(x) ) has neither a minimum value nor a maximum value.Hmm, let's think about this. We found critical points at ( x = -2 ) and ( x = 0 ). Let's determine whether these are maxima or minima.Using the first derivative test:- At ( x = -2 ): - To the left of ( x = -2 ), ( f'(x) ) is positive (function increasing). - To the right of ( x = -2 ), ( f'(x) ) is negative (function decreasing). - Therefore, ( x = -2 ) is a local maximum.- At ( x = 0 ): - To the left of ( x = 0 ), ( f'(x) ) is negative (function decreasing). - To the right of ( x = 0 ), ( f'(x) ) is positive (function increasing). - Therefore, ( x = 0 ) is a local minimum.Now, let's check if these are global maxima or minima.For the local maximum at ( x = -2 ):- As ( x to -infty ), ( x^2 ) grows without bound, but ( e^x ) approaches zero. The product ( x^2 e^x ) approaches zero because exponential decay dominates polynomial growth. So, ( f(x) to 0 ) as ( x to -infty ).- At ( x = -2 ), ( f(-2) = (-2)^2 e^{-2} = 4 e^{-2} approx 4 times 0.135 = 0.54 ).- As ( x to infty ), ( x^2 e^x ) grows without bound because exponential growth dominates polynomial growth.So, the function has a local maximum at ( x = -2 ), and since the function tends to zero as ( x to -infty ) and tends to infinity as ( x to infty ), this local maximum is actually the global maximum.For the local minimum at ( x = 0 ):- ( f(0) = 0^2 e^0 = 0 ).- As ( x to infty ), ( f(x) ) tends to infinity, and as ( x to -infty ), ( f(x) ) tends to zero.So, the function has a local minimum at ( x = 0 ), and since the function approaches zero from the right and left of ( x = 0 ), this local minimum is also the global minimum.Therefore, the function does have both a minimum value (0) and a maximum value (approximately 0.54). So, conclusion ② is incorrect.**Conclusion ③**: The graph of ( f(x) ) intersects with its tangent line at ( (0, 0) ) at two points.First, let's find the tangent line to ( f(x) ) at ( (0, 0) ).We already computed ( f'(x) = x e^x (x + 2) ). So, ( f'(0) = 0 times e^0 times (0 + 2) = 0 ). Therefore, the slope of the tangent line at ( x = 0 ) is 0.The equation of the tangent line at ( (0, 0) ) is:[y - f(0) = f'(0)(x - 0) implies y - 0 = 0 times (x - 0) implies y = 0]So, the tangent line is the x-axis, ( y = 0 ).Now, we need to find the points where ( f(x) = 0 ). Let's solve ( x^2 e^x = 0 ).Since ( e^x ) is never zero, the equation ( x^2 e^x = 0 ) implies ( x^2 = 0 ), so ( x = 0 ).Therefore, the graph of ( f(x) ) intersects its tangent line at ( (0, 0) ) only at the point ( (0, 0) ). There are no other intersection points because ( f(x) ) is zero only at ( x = 0 ).Hence, conclusion ③ is incorrect.**Conclusion ④**: The graph of ( f(x) ) intersects with the line ( x - y + 2012 = 0 ) at two points.Let me rewrite the equation of the line:[x - y + 2012 = 0 implies y = x + 2012]We need to find the number of intersection points between ( f(x) = x^2 e^x ) and ( y = x + 2012 ). This requires solving the equation:[x^2 e^x = x + 2012]Let me define a new function ( g(x) = x^2 e^x - x - 2012 ). The number of real roots of ( g(x) = 0 ) will give the number of intersection points.To find the number of real roots, I can analyze the behavior of ( g(x) ) as ( x ) approaches infinity and negative infinity, and check for sign changes.First, let's consider the limits:- As ( x to infty ): - ( x^2 e^x ) dominates because exponential growth is much faster than polynomial growth. - So, ( g(x) to infty ).- As ( x to -infty ): - ( x^2 e^x ) approaches zero because ( e^x ) decays to zero faster than ( x^2 ) grows. - So, ( g(x) approx -x - 2012 ). - As ( x to -infty ), ( -x ) becomes very large positive, so ( g(x) to infty ).Wait, that seems contradictory. Let me double-check:Wait, as ( x to -infty ), ( e^x ) approaches zero, so ( x^2 e^x ) approaches zero. Therefore, ( g(x) = x^2 e^x - x - 2012 approx -x - 2012 ).But as ( x to -infty ), ( -x ) becomes very large positive, so ( g(x) approx text{large positive} - 2012 ), which is still very large positive. So, ( g(x) to infty ) as ( x to -infty ).Wait, but that can't be right because if ( x ) is very large negative, ( -x ) is positive, but ( x^2 e^x ) is approaching zero. So, ( g(x) ) is approximately ( -x - 2012 ), which for ( x to -infty ), ( -x ) is positive and large, so ( g(x) ) tends to positive infinity.But then, as ( x ) increases from ( -infty ) to ( infty ), ( g(x) ) starts at ( infty ), goes somewhere, and then tends back to ( infty ). So, to find the number of roots, I need to check if ( g(x) ) dips below zero somewhere in between.Let me compute ( g(x) ) at some specific points:1. At ( x = 0 ): - ( g(0) = 0^2 e^0 - 0 - 2012 = 0 - 0 - 2012 = -2012 ).2. At ( x = 5 ): - ( g(5) = 25 e^5 - 5 - 2012 ). - ( e^5 approx 148.413 ), so ( 25 times 148.413 approx 3710.325 ). - ( 3710.325 - 5 - 2012 approx 3710.325 - 2017 = 1693.325 ), which is positive.3. At ( x = -10 ): - ( g(-10) = 100 e^{-10} - (-10) - 2012 ). - ( e^{-10} approx 0.0000454 ), so ( 100 times 0.0000454 approx 0.00454 ). - ( 0.00454 + 10 - 2012 approx 10.00454 - 2012 approx -2001.995 ), which is negative.Wait, so at ( x = -10 ), ( g(-10) approx -2001.995 ), which is negative, and as ( x to -infty ), ( g(x) to infty ). Therefore, between ( x = -infty ) and ( x = -10 ), ( g(x) ) goes from ( infty ) to approximately -2002, so it must cross zero at least once in that interval.Similarly, at ( x = 0 ), ( g(0) = -2012 ), which is negative, and at ( x = 5 ), ( g(5) approx 1693.325 ), which is positive. Therefore, between ( x = 0 ) and ( x = 5 ), ( g(x) ) goes from -2012 to 1693.325, so it must cross zero at least once in that interval.Therefore, there are at least two real roots: one in ( (-infty, -10) ) and another in ( (0, 5) ). But wait, let me check if there could be more roots.Wait, let's analyze the behavior of ( g(x) ) in more detail.First, let's consider the derivative ( g'(x) ) to understand the critical points of ( g(x) ):[g'(x) = frac{d}{dx}(x^2 e^x) - frac{d}{dx}(x) - frac{d}{dx}(2012)][g'(x) = (2x e^x + x^2 e^x) - 1 - 0][g'(x) = x e^x (x + 2) - 1]This derivative is a bit complicated, but let's see if we can analyze it.We can note that ( x e^x (x + 2) ) is the same as ( f'(x) ), which we already analyzed earlier. So, ( g'(x) = f'(x) - 1 ).From our earlier analysis, ( f'(x) ) is positive on ( (-infty, -2) ) and ( (0, infty) ), and negative on ( (-2, 0) ).Therefore, ( g'(x) = f'(x) - 1 ):- On ( (-infty, -2) ), ( f'(x) > 0 ), so ( g'(x) = f'(x) - 1 ). Since ( f'(x) ) can be large positive here, ( g'(x) ) is positive.- On ( (-2, 0) ), ( f'(x) < 0 ), so ( g'(x) = f'(x) - 1 ) is negative.- On ( (0, infty) ), ( f'(x) > 0 ), so ( g'(x) = f'(x) - 1 ). Depending on the value of ( f'(x) ), ( g'(x) ) could be positive or negative.Wait, let's see:At ( x = 0 ), ( f'(0) = 0 ), so ( g'(0) = 0 - 1 = -1 ).At ( x = 5 ), ( f'(5) = 5 e^5 (5 + 2) = 5 e^5 times 7 approx 5 times 148.413 times 7 approx 5200 ), so ( g'(5) approx 5200 - 1 = 5199 ), which is positive.Therefore, ( g'(x) ) is negative at ( x = 0 ) and positive at ( x = 5 ), so by the Intermediate Value Theorem, there must be some ( c ) in ( (0, 5) ) where ( g'(c) = 0 ). That is, ( g(x) ) has a critical point in ( (0, 5) ).Similarly, on ( (-infty, -2) ), ( g'(x) ) is positive, and on ( (-2, 0) ), ( g'(x) ) is negative. So, ( g(x) ) has a critical point at ( x = -2 ) as well.Wait, but let me check ( g'(x) ) at ( x = -2 ):( f'(-2) = (-2) e^{-2} (-2 + 2) = (-2) e^{-2} (0) = 0 ). So, ( g'(-2) = 0 - 1 = -1 ).Wait, that contradicts my earlier statement. Let me correct that.Wait, ( f'(x) = x e^x (x + 2) ). So, at ( x = -2 ), ( f'(-2) = (-2) e^{-2} (0) = 0 ). Therefore, ( g'(-2) = 0 - 1 = -1 ).So, actually, ( g'(x) ) is negative at ( x = -2 ). So, perhaps my earlier analysis was incorrect.Wait, let's re-examine ( g'(x) = f'(x) - 1 ).- On ( (-infty, -2) ), ( f'(x) > 0 ), so ( g'(x) = f'(x) - 1 ). Depending on how large ( f'(x) ) is, ( g'(x) ) could be positive or negative.Wait, let's compute ( f'(x) ) at some point in ( (-infty, -2) ), say ( x = -3 ):( f'(-3) = (-3) e^{-3} (-3 + 2) = (-3) e^{-3} (-1) = 3 e^{-3} approx 3 times 0.05 = 0.15 ).So, ( g'(-3) = 0.15 - 1 = -0.85 ), which is negative.Wait, but earlier I thought ( f'(x) ) is positive on ( (-infty, -2) ), which it is, but it's not necessarily large. So, ( g'(x) = f'(x) - 1 ) could be negative even in ( (-infty, -2) ) if ( f'(x) < 1 ).Wait, let's see:( f'(x) = x e^x (x + 2) ).For ( x < -2 ), ( x ) is negative, ( x + 2 ) is negative, so ( f'(x) = x e^x (x + 2) ) is positive because negative times negative is positive, and ( e^x ) is positive.But how large is ( f'(x) ) in ( (-infty, -2) )?At ( x = -2 ), ( f'(-2) = 0 ).As ( x to -infty ), ( x e^x (x + 2) ) behaves like ( x^2 e^x ), which tends to zero because exponential decay dominates polynomial growth. So, ( f'(x) ) approaches zero from the positive side as ( x to -infty ).Therefore, ( f'(x) ) is positive but less than 1 for all ( x < -2 ), because as ( x ) approaches -2 from the left, ( f'(x) ) approaches zero, and as ( x to -infty ), ( f'(x) to 0 ). So, ( f'(x) ) is always less than 1 in ( (-infty, -2) ).Therefore, ( g'(x) = f'(x) - 1 ) is negative in ( (-infty, -2) ).So, putting it all together:- On ( (-infty, -2) ), ( g'(x) < 0 ).- At ( x = -2 ), ( g'(-2) = -1 ).- On ( (-2, 0) ), ( g'(x) < 0 ) because ( f'(x) < 0 ), so ( g'(x) = f'(x) - 1 < -1 ).- At ( x = 0 ), ( g'(0) = -1 ).- On ( (0, infty) ), ( g'(x) ) starts negative at ( x = 0 ) and becomes positive as ( x ) increases because ( f'(x) ) grows without bound.Therefore, ( g(x) ) has critical points:- A local maximum somewhere in ( (0, infty) ) where ( g'(x) = 0 ).- No local minima in ( (-infty, -2) ) because ( g'(x) ) is negative throughout.Wait, actually, since ( g'(x) ) is negative on ( (-infty, -2) ) and negative on ( (-2, 0) ), and then becomes positive on ( (0, infty) ), the function ( g(x) ) is decreasing on ( (-infty, 0) ) and increasing on ( (0, infty) ).But wait, that can't be because ( g'(x) ) is negative on ( (-infty, -2) ) and ( (-2, 0) ), so ( g(x) ) is decreasing on ( (-infty, 0) ), and increasing on ( (0, infty) ).Wait, but earlier, I saw that ( g(x) ) at ( x = -10 ) is approximately -2002, and as ( x to -infty ), ( g(x) to infty ). So, if ( g(x) ) is decreasing on ( (-infty, 0) ), starting from ( infty ) at ( x = -infty ), decreasing to ( g(-10) approx -2002 ), and then continuing to decrease to ( g(0) = -2012 ). Wait, but that would mean ( g(x) ) is decreasing throughout ( (-infty, 0) ), but at ( x = -10 ), it's already at -2002, and at ( x = 0 ), it's -2012, which is lower. So, the function is decreasing from ( infty ) to -2012 as ( x ) goes from ( -infty ) to 0.Then, on ( (0, infty) ), ( g(x) ) starts at -2012 and increases to ( infty ) as ( x to infty ).Therefore, the function ( g(x) ) has only one crossing from positive to negative as ( x ) approaches 0 from the left, but wait, no, because ( g(x) ) is decreasing on ( (-infty, 0) ) from ( infty ) to -2012, so it must cross zero exactly once in ( (-infty, 0) ).Similarly, on ( (0, infty) ), ( g(x) ) starts at -2012 and increases to ( infty ), so it must cross zero exactly once in ( (0, infty) ).Therefore, ( g(x) = 0 ) has exactly two real roots: one in ( (-infty, 0) ) and one in ( (0, infty) ).Therefore, the graph of ( f(x) ) intersects the line ( y = x + 2012 ) at two points. So, conclusion ④ is correct.Wait, but earlier, I thought that ( g(x) ) might have more roots, but upon further analysis, it seems that ( g(x) ) is decreasing on ( (-infty, 0) ) and increasing on ( (0, infty) ), crossing zero once in each interval. Therefore, exactly two intersection points.Hence, conclusion ④ is correct.So, summarizing my findings:- Conclusion ①: Correct.- Conclusion ②: Incorrect.- Conclusion ③: Incorrect.- Conclusion ④: Correct.Therefore, the correct conclusions are numbered ① and ④.

💡Alright, so I've got this vector problem here, and I need to find the angle between vectors p and r. Let me try to break it down step by step.First, the problem states that p, q, and r are unit vectors. That means each of them has a magnitude of 1. Also, they form a linearly independent set, which is important because it tells me that none of them can be expressed as a combination of the others. So, they're all pointing in different directions, not just scaled versions of each other.The equation given is a vector triple product: p cross (q cross r) equals (q + 2r)/2. I remember that there's a vector triple product identity which might help here. Let me recall it. I think it's something like a cross (b cross c) equals (a dot c)b minus (a dot b)c. Yeah, that sounds right. So applying that identity to the left side of the equation, I can rewrite it.So, p cross (q cross r) becomes (p dot r)q minus (p dot q)r. That's the expansion. Now, the right side of the equation is (q + 2r)/2, which I can write as (1/2)q + r. So, putting it all together, I have:(p dot r)q - (p dot q)r = (1/2)q + r.Hmm, okay. Now, since q and r are vectors, and they're part of a linearly independent set, they can't be scalar multiples of each other. That means the coefficients of q and r on both sides of the equation must be equal. Otherwise, we'd have a linear dependence, which contradicts the given condition.So, looking at the coefficients of q on both sides: on the left side, it's (p dot r), and on the right side, it's 1/2. Similarly, for r, the coefficient on the left is -(p dot q), and on the right, it's 1.So, setting these equal, I get two equations:1. (p dot r) = 1/22. -(p dot q) = 1From the second equation, multiplying both sides by -1, I find that (p dot q) = -1.Now, since p and q are unit vectors, their dot product is equal to the cosine of the angle between them. So, p dot q = cos(theta) where theta is the angle between p and q. But here, p dot q is -1, which means cos(theta) = -1. The angle whose cosine is -1 is 180 degrees. So, p and q are pointing in exactly opposite directions. That makes sense because their dot product is -1.But wait, the problem is asking for the angle between p and r, not p and q. So, let's focus on the first equation: p dot r = 1/2. Again, since p and r are unit vectors, their dot product is the cosine of the angle between them. So, cos(phi) = 1/2, where phi is the angle between p and r.Now, what angle has a cosine of 1/2? I remember that cos(60 degrees) is 0.5, which is 1/2. So, phi must be 60 degrees. That seems straightforward.But just to make sure I didn't make any mistakes, let me go through the steps again.1. I used the vector triple product identity correctly: a cross (b cross c) = (a dot c)b - (a dot b)c. Applied to p, q, r, it becomes (p dot r)q - (p dot q)r.2. Set this equal to (q + 2r)/2, which simplifies to (1/2)q + r.3. Equated the coefficients of q and r on both sides because q and r are linearly independent, so their coefficients must match.4. Solved for p dot r and p dot q, getting 1/2 and -1 respectively.5. Converted the dot products to angles using the definition of the dot product for unit vectors: p dot r = cos(phi) = 1/2, so phi = 60 degrees.Everything seems to check out. I don't see any errors in the reasoning. The key was recognizing the vector triple product identity and then using the linear independence of the vectors to set the coefficients equal. That allowed me to solve for the necessary dot products and then find the angle.I was a bit concerned initially about whether the linear independence was necessary, but it makes sense because if the vectors weren't linearly independent, we couldn't be sure that the coefficients had to match. They could have been scalar multiples or something else, which would complicate things. But since they're linearly independent, the only way the equation holds is if the coefficients of q and r are equal on both sides.Another thing I thought about was whether there could be other angles that satisfy cos(phi) = 1/2. For example, 300 degrees also has a cosine of 1/2, but since we're talking about the angle between two vectors, it's always the smallest angle between them, which is 60 degrees. So, that's why we take 60 degrees and not 300.Also, considering that p and q have a dot product of -1, which is the minimum possible value for the dot product of two unit vectors, that tells me they are antiparallel, meaning they point in exactly opposite directions. That's a useful piece of information, but it's not directly needed to solve for the angle between p and r. However, it does reinforce that the set {p, q, r} is linearly independent because p and q are not just scalar multiples of each other—they're pointing in opposite directions, but r is another vector that's not in the same line as p or q.I wonder if there's another way to approach this problem without using the vector triple product identity. Maybe by expressing the vectors in components or using geometric interpretations. But honestly, the triple product identity seems like the most straightforward method here because it directly relates the cross product to the dot products, which are scalar quantities that are easier to handle.If I were to try another approach, perhaps I could consider the properties of cross products and dot products. For example, knowing that the cross product of two vectors is perpendicular to both of them, so p cross (q cross r) would be perpendicular to p and to q cross r. But I'm not sure if that line of thinking would lead me anywhere useful without bringing in the triple product identity.Another thought: since p, q, and r are unit vectors, their magnitudes are all 1. So, when I compute p dot r, it's just the cosine of the angle between them, which is exactly what I need. Similarly, p dot q is the cosine of the angle between p and q, which we found to be -1, indicating they're antiparallel.I also considered whether the order of the cross products matters, but in this case, the triple product identity takes care of that, so I don't have to worry about the order beyond what's given in the problem.In summary, the key steps were:1. Apply the vector triple product identity to expand the left side.2. Equate the coefficients of q and r on both sides due to linear independence.3. Solve the resulting equations for the dot products.4. Convert the dot product of p and r into the angle using the definition of the dot product.This process gave me the angle between p and r as 60 degrees, which seems correct based on the calculations and the properties of unit vectors and dot products.I think I'm confident with this answer. It makes sense mathematically, and all the steps follow logically without any contradictions or errors. The use of the triple product identity was crucial here, and recognizing the implications of linear independence allowed me to set up the equations properly. Without that, I might have been stuck trying to find another way to relate the vectors, which could have been more complicated or even impossible.So, yeah, I think 60 degrees is the right answer. It's always good to double-check, but in this case, everything lines up.**Final Answer**The angle between (mathbf{p}) and (mathbf{r}) is (boxed{60}) degrees.

💡First, let's denote Christina's current age as ( C ).In five years, Christina's age will be ( C + 5 ).We are told that in 15 years, Oscar's age will be ( frac{3}{5} ) times Christina's current age. Since Oscar is currently 6 years old, in 15 years, he will be ( 6 + 15 = 21 ) years old.Setting up the equation based on the given information:[frac{3}{5}C = 21]Solving for ( C ):[C = 21 times frac{5}{3} = 35]So, Christina is currently 35 years old. In five years, she will be:[35 + 5 = 40 text{ years old}]Finally, the fraction of Christina's age in five years to 80 years is:[frac{40}{80} = frac{1}{2}]