💡Alright, so I have this function f(x) that's defined piecewise. It's equal to (1/2)x + m when x is less than 1, and it's equal to x minus the natural logarithm of x when x is greater than or equal to 1. The problem is asking for the range of the real number m such that the function f(x) is monotonically increasing on all real numbers.Okay, first, I need to recall what it means for a function to be monotonically increasing. A function is monotonically increasing if, as x increases, the value of the function doesn't decrease. In other words, for any two points x1 and x2 where x1 < x2, f(x1) ≤ f(x2). So, the function either stays the same or increases as x increases.Since this function is piecewise, I need to check the monotonicity in both pieces and also ensure that there's no decrease at the point where the function changes from one piece to the other, which is at x = 1.Let me start by analyzing each piece separately.First, for x < 1, the function is f(x) = (1/2)x + m. This is a linear function with a slope of 1/2. Since the slope is positive (1/2 is greater than 0), this part of the function is increasing. So, as x increases, f(x) increases. That's good because it's part of the requirement for the entire function to be monotonically increasing.Next, for x ≥ 1, the function is f(x) = x - ln(x). To determine if this part is increasing, I can take its derivative. The derivative of f(x) with respect to x is f'(x) = 1 - (1/x). Now, let's analyze this derivative. For x > 1, 1/x is less than 1, so 1 - (1/x) is positive. That means the derivative is positive for all x > 1, so the function is increasing on that interval as well. At x = 1, the derivative is 1 - 1/1 = 0. So, the slope at x = 1 is zero, which means the function has a horizontal tangent there.Wait, that might be a problem. If the derivative is zero at x = 1, does that mean the function isn't increasing at that exact point? Hmm, but for a function to be monotonically increasing, it just needs to not decrease. So, having a derivative of zero at a single point doesn't necessarily violate the condition, as long as the function doesn't decrease anywhere else.However, I also need to ensure that the function doesn't have a decrease at the point where the two pieces meet, which is at x = 1. So, I need to check the continuity and the behavior around x = 1.First, let's check the continuity at x = 1. For the function to be continuous at x = 1, the left-hand limit as x approaches 1 from below should equal the right-hand limit as x approaches 1 from above, and both should equal f(1).Calculating the left-hand limit: as x approaches 1 from below, f(x) approaches (1/2)(1) + m = 1/2 + m.Calculating the right-hand limit: as x approaches 1 from above, f(x) approaches 1 - ln(1) = 1 - 0 = 1.So, for the function to be continuous at x = 1, we need 1/2 + m = 1. Solving for m, we get m = 1 - 1/2 = 1/2.But wait, the problem doesn't specify that the function needs to be continuous, just that it needs to be monotonically increasing. So, maybe continuity isn't required? Hmm, but if the function isn't continuous at x = 1, there could be a jump discontinuity, which might cause the function to decrease there.Let me think about that. If the left-hand limit at x = 1 is 1/2 + m and the right-hand limit is 1, then if 1/2 + m > 1, there would be a jump downward at x = 1, which would make the function decreasing at that point, violating the monotonicity. On the other hand, if 1/2 + m ≤ 1, then the function either continues smoothly or has a jump upward, which doesn't violate the monotonicity.So, to ensure that the function doesn't decrease at x = 1, we need 1/2 + m ≤ 1. Solving for m, we subtract 1/2 from both sides: m ≤ 1/2.Therefore, the range of m is all real numbers less than or equal to 1/2.Wait, but what if m is less than 1/2? Then, the left-hand limit at x = 1 would be less than 1, meaning the function would have a jump upward at x = 1. That's fine because the function is still increasing overall.If m is exactly 1/2, then the function is continuous at x = 1, and there's no jump. That's also acceptable.But if m were greater than 1/2, then the left-hand limit would be greater than 1, causing a downward jump at x = 1, which would make the function decrease there, violating the monotonicity.So, putting it all together, m must be less than or equal to 1/2 for the function to be monotonically increasing on all real numbers.I think that's the solution. Let me just recap:1. For x < 1, f(x) is linear with a positive slope, so it's increasing.2. For x ≥ 1, f(x) has a derivative that's positive for x > 1, so it's increasing there as well.3. At x = 1, to prevent a decrease, the value from the left must be less than or equal to the value from the right. This gives the condition 1/2 + m ≤ 1, leading to m ≤ 1/2.Therefore, the range of m is (-∞, 1/2].**Final Answer**The range of the real number ( m ) is boxed{(-infty, frac{1}{2}]}.

💡Alright, so I have this inequality to prove: for all positive real numbers ( a, b, ) and ( c ), it's supposed to be true that ( a^2 + b^2 + c^2 geq ab + bc + ca ). And then, using that result, I need to deduce that ( (a + b + c)^2 geq 3(ab + bc + ca) ).Okay, let's start with the first part. I remember that inequalities involving squares often relate to things like the AM-GM inequality or maybe even the Cauchy-Schwarz inequality. But I'm not sure which one applies here. Maybe I can try expanding or rearranging terms to see if something pops out.So, if I take ( a^2 + b^2 + c^2 ) and subtract ( ab + bc + ca ) from both sides, I get:[ a^2 + b^2 + c^2 - ab - bc - ca geq 0 ]Hmm, that looks a bit complicated. Maybe I can factor this expression somehow. I recall that sometimes expressions like this can be written as sums of squares because squares are always non-negative. Let me try that.If I consider ( (a - b)^2 ), that's ( a^2 - 2ab + b^2 ). Similarly, ( (b - c)^2 = b^2 - 2bc + c^2 ) and ( (c - a)^2 = c^2 - 2ca + a^2 ). If I add all these up:[ (a - b)^2 + (b - c)^2 + (c - a)^2 ]Expanding each term:[ (a^2 - 2ab + b^2) + (b^2 - 2bc + c^2) + (c^2 - 2ca + a^2) ]Combine like terms:[ 2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca ]Oh, that's exactly twice the expression I had earlier! So,[ (a - b)^2 + (b - c)^2 + (c - a)^2 = 2(a^2 + b^2 + c^2 - ab - bc - ca) ]Since squares are always non-negative, the left side is always greater than or equal to zero. Therefore,[ 2(a^2 + b^2 + c^2 - ab - bc - ca) geq 0 ]Dividing both sides by 2:[ a^2 + b^2 + c^2 - ab - bc - ca geq 0 ]Which means:[ a^2 + b^2 + c^2 geq ab + bc + ca ]Alright, that works out nicely. So, the first part is proven by expressing the difference as a sum of squares, which is always non-negative.Now, moving on to the second part: deducing that ( (a + b + c)^2 geq 3(ab + bc + ca) ).I know that expanding ( (a + b + c)^2 ) gives:[ a^2 + b^2 + c^2 + 2ab + 2bc + 2ca ]So,[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) ]From the first part, we have ( a^2 + b^2 + c^2 geq ab + bc + ca ). Maybe I can use this to bound the expression.Let me substitute ( a^2 + b^2 + c^2 ) with something greater than or equal to ( ab + bc + ca ). So,[ (a + b + c)^2 geq (ab + bc + ca) + 2(ab + bc + ca) ]Because ( a^2 + b^2 + c^2 geq ab + bc + ca ), replacing it with the smaller value gives a lower bound.Simplifying the right side:[ (a + b + c)^2 geq 3(ab + bc + ca) ]And that's exactly what I needed to deduce! So, by using the result from the first part, I was able to establish the second inequality.Wait, let me double-check that substitution step. If ( a^2 + b^2 + c^2 geq ab + bc + ca ), then replacing ( a^2 + b^2 + c^2 ) with ( ab + bc + ca ) in the expression ( (a + b + c)^2 ) would indeed give a smaller value, right? So, since ( (a + b + c)^2 ) is equal to ( a^2 + b^2 + c^2 + 2(ab + bc + ca) ), and ( a^2 + b^2 + c^2 ) is at least ( ab + bc + ca ), then the whole expression is at least ( ab + bc + ca + 2(ab + bc + ca) = 3(ab + bc + ca) ). Yeah, that makes sense.I think that's solid. So, both inequalities are proven using the sum of squares technique and substitution.**Final Answer**The inequality is proven, and the deduction is valid. Thus,[boxed{(a + b + c)^2 geq 3(ab + bc + ca)}]

💡First, I need to translate the given percentage equation into a mathematical expression. The equation states that 30% of (x - y) is equal to 20% of (x + y). I'll convert the percentages to decimals to make the calculations easier.Next, I'll set up the equation by multiplying 0.30 by (x - y) and 0.20 by (x + y). This gives me 0.30(x - y) = 0.20(x + y).After setting up the equation, I'll distribute the decimals inside the parentheses to simplify the expression. This results in 0.30x - 0.30y = 0.20x + 0.20y.Now, I'll move all terms containing x to one side of the equation and all terms containing y to the other side. This helps in isolating the variables for easier comparison. Subtracting 0.20x from both sides and adding 0.30y to both sides gives me 0.10x = 0.50y.With the variables isolated, I'll solve for y in terms of x by dividing both sides of the equation by 0.50. This simplifies to y = 0.20x.Finally, I'll convert the decimal 0.20 back to a percentage to determine what percent of x is y. This means y is 20% of x.

💡First, I need to identify the unknown number in the problem. Let's call this number "x".According to the problem, the product of this number and 3 is increased by 4, resulting in 19. This can be written as the equation:3x + 4 = 19To solve for x, I'll start by isolating the term with x. I'll subtract 4 from both sides of the equation:3x + 4 - 4 = 19 - 43x = 15Next, I'll divide both sides of the equation by 3 to find the value of x:3x / 3 = 15 / 3x = 5Therefore, the certain number is 5.

💡Alright, so I have this problem about a circle given in parametric equations, and I need to find its polar equation and then find the length of a chord where a line intersects the circle. Hmm, okay, let me break this down step by step.First, part (1) asks to establish a polar coordinate system and derive the polar equation of circle C. The parametric equations given are:[begin{cases}x = 1 + 2cos theta y = 2sin thetaend{cases}]where (theta) is the parameter.I remember that parametric equations for a circle usually have the form (x = h + rcos theta) and (y = k + rsin theta), where ((h, k)) is the center and (r) is the radius. Comparing that to what I have here, it looks like the center is at ((1, 0)) and the radius is 2. So, the standard Cartesian equation of this circle should be:[(x - 1)^2 + y^2 = 4]Let me expand that to make sure:[x^2 - 2x + 1 + y^2 = 4]Simplify:[x^2 + y^2 - 2x + 1 = 4]Subtract 4 from both sides:[x^2 + y^2 - 2x - 3 = 0]So, (x^2 + y^2 = 2x + 3). That seems right.Now, to convert this into polar coordinates. I remember the conversion formulas:[x = rho cos theta y = rho sin theta x^2 + y^2 = rho^2]So, substituting these into the equation (x^2 + y^2 = 2x + 3), we get:[rho^2 = 2rho cos theta + 3]That should be the polar equation of the circle. Let me just check if that makes sense. If I rearrange it:[rho^2 - 2rho cos theta - 3 = 0]This is a quadratic in terms of (rho), which is typical for circles in polar coordinates. Okay, that seems good.Moving on to part (2). The polar equation of line (l) is given as (theta = frac{pi}{3}). I need to find where this line intersects the circle (C) and then calculate the length of chord (AB).First, let me recall that in polar coordinates, (theta = frac{pi}{3}) represents a straight line at an angle of (frac{pi}{3}) from the polar axis (which is the positive x-axis). In Cartesian coordinates, this line would have a slope of (tan frac{pi}{3} = sqrt{3}), so its equation is (y = sqrt{3}x).Now, I need to find the points of intersection between this line and the circle (C). The circle has the equation ((x - 1)^2 + y^2 = 4), and the line is (y = sqrt{3}x). Let me substitute (y) from the line into the circle's equation.Substituting (y = sqrt{3}x) into ((x - 1)^2 + y^2 = 4):[(x - 1)^2 + (sqrt{3}x)^2 = 4]Let me expand and simplify this:First, expand ((x - 1)^2):[x^2 - 2x + 1]Then, expand ((sqrt{3}x)^2):[3x^2]So, putting it all together:[x^2 - 2x + 1 + 3x^2 = 4]Combine like terms:[4x^2 - 2x + 1 = 4]Subtract 4 from both sides:[4x^2 - 2x - 3 = 0]Now, I have a quadratic equation in terms of (x). Let me solve for (x) using the quadratic formula:[x = frac{2 pm sqrt{(-2)^2 - 4 cdot 4 cdot (-3)}}{2 cdot 4}]Calculate the discriminant:[(-2)^2 = 4 4 cdot 4 cdot (-3) = -48 So, discriminant = 4 - (-48) = 4 + 48 = 52]Therefore:[x = frac{2 pm sqrt{52}}{8}]Simplify (sqrt{52}):[sqrt{52} = sqrt{4 cdot 13} = 2sqrt{13}]So, substituting back:[x = frac{2 pm 2sqrt{13}}{8} = frac{2(1 pm sqrt{13})}{8} = frac{1 pm sqrt{13}}{4}]Therefore, the x-coordinates of points (A) and (B) are (frac{1 + sqrt{13}}{4}) and (frac{1 - sqrt{13}}{4}).Now, let me find the corresponding y-coordinates using (y = sqrt{3}x):For (x = frac{1 + sqrt{13}}{4}):[y = sqrt{3} cdot frac{1 + sqrt{13}}{4} = frac{sqrt{3} + sqrt{39}}{4}]For (x = frac{1 - sqrt{13}}{4}):[y = sqrt{3} cdot frac{1 - sqrt{13}}{4} = frac{sqrt{3} - sqrt{39}}{4}]So, the coordinates of points (A) and (B) are:[Aleft( frac{1 + sqrt{13}}{4}, frac{sqrt{3} + sqrt{39}}{4} right) Bleft( frac{1 - sqrt{13}}{4}, frac{sqrt{3} - sqrt{39}}{4} right)]Now, to find the length of chord (AB), I can use the distance formula between points (A) and (B):[|AB| = sqrt{ left( frac{1 + sqrt{13}}{4} - frac{1 - sqrt{13}}{4} right)^2 + left( frac{sqrt{3} + sqrt{39}}{4} - frac{sqrt{3} - sqrt{39}}{4} right)^2 }]Simplify the differences inside the square roots:For the x-coordinates:[frac{1 + sqrt{13}}{4} - frac{1 - sqrt{13}}{4} = frac{1 + sqrt{13} - 1 + sqrt{13}}{4} = frac{2sqrt{13}}{4} = frac{sqrt{13}}{2}]For the y-coordinates:[frac{sqrt{3} + sqrt{39}}{4} - frac{sqrt{3} - sqrt{39}}{4} = frac{sqrt{3} + sqrt{39} - sqrt{3} + sqrt{39}}{4} = frac{2sqrt{39}}{4} = frac{sqrt{39}}{2}]So, plugging these back into the distance formula:[|AB| = sqrt{ left( frac{sqrt{13}}{2} right)^2 + left( frac{sqrt{39}}{2} right)^2 } = sqrt{ frac{13}{4} + frac{39}{4} } = sqrt{ frac{52}{4} } = sqrt{13 }]Wait, that seems a bit too straightforward. Let me double-check my steps.Starting from the distance formula:[|AB| = sqrt{ left( frac{sqrt{13}}{2} right)^2 + left( frac{sqrt{39}}{2} right)^2 } = sqrt{ frac{13}{4} + frac{39}{4} } = sqrt{ frac{52}{4} } = sqrt{13 }]Yes, that's correct. So, the length of chord (AB) is (sqrt{13}).Alternatively, I remember there's a formula for the length of a chord given the radius and the distance from the center to the chord. The formula is:[|AB| = 2 sqrt{r^2 - d^2}]where (r) is the radius and (d) is the distance from the center to the chord.In this case, the radius (r = 2). I need to find the distance from the center of the circle to the line (l). The center of the circle is at ((1, 0)), and the line (l) is (y = sqrt{3}x).The distance (d) from a point ((x_0, y_0)) to the line (ax + by + c = 0) is given by:[d = frac{ |ax_0 + by_0 + c| }{ sqrt{a^2 + b^2} }]First, let me write the equation of line (l) in standard form:[y = sqrt{3}x implies sqrt{3}x - y = 0]So, (a = sqrt{3}), (b = -1), and (c = 0).The center of the circle is ((1, 0)), so plugging into the distance formula:[d = frac{ | sqrt{3} cdot 1 + (-1) cdot 0 + 0 | }{ sqrt{ (sqrt{3})^2 + (-1)^2 } } = frac{ | sqrt{3} | }{ sqrt{ 3 + 1 } } = frac{ sqrt{3} }{ 2 }]So, (d = frac{sqrt{3}}{2}).Now, using the chord length formula:[|AB| = 2 sqrt{r^2 - d^2} = 2 sqrt{ 2^2 - left( frac{sqrt{3}}{2} right)^2 } = 2 sqrt{ 4 - frac{3}{4} } = 2 sqrt{ frac{16}{4} - frac{3}{4} } = 2 sqrt{ frac{13}{4} } = 2 cdot frac{sqrt{13}}{2} = sqrt{13}]So, both methods give the same result, which is reassuring.Therefore, the length of chord (AB) is (sqrt{13}).**Final Answer**The length of chord (AB) is boxed{sqrt{13}}.

💡First, I'll assign a variable to represent the revenue in December. Let's denote December's revenue as ( D ).According to the problem, November's revenue is (frac{2}{5}) of December's revenue. So, November's revenue is:[text{November's revenue} = frac{2}{5}D]Next, January's revenue is (frac{1}{5}) of November's revenue. Therefore, January's revenue is:[text{January's revenue} = frac{1}{5} times frac{2}{5}D = frac{2}{25}D]Now, I'll calculate the average of November's and January's revenues:[text{Average} = frac{text{November's revenue} + text{January's revenue}}{2} = frac{frac{2}{5}D + frac{2}{25}D}{2}]To add the fractions, I'll find a common denominator:[frac{2}{5}D = frac{10}{25}D][frac{10}{25}D + frac{2}{25}D = frac{12}{25}D][text{Average} = frac{frac{12}{25}D}{2} = frac{12}{50}D = frac{6}{25}D]Finally, to determine how many times December's revenue is of this average:[frac{D}{frac{6}{25}D} = frac{25}{6}]Therefore, the store's revenue in December was (frac{25}{6}) times the average of its revenues in November and January.