💡First, I need to find the unknown number, which I'll call ( x ).The given equation is:[(28 + frac{48}{x}) times x = 1980]I'll start by distributing ( x ) to both terms inside the parentheses:[28x + 48 = 1980]Next, I'll subtract 48 from both sides to isolate the term with ( x ):[28x = 1980 - 48][28x = 1932]Finally, I'll divide both sides by 28 to solve for ( x ):[x = frac{1932}{28}][x = 69]So, the unknown number is 69.

💡Okay, so I need to find the coefficient of ( x^2 ) in the expansion of ( left( frac{3}{x} + x right) left( 2 - sqrt{x} right)^6 ). Hmm, let me think about how to approach this.First, I remember that when you have a product of two expressions, you can expand them term by term. So, in this case, I have ( frac{3}{x} ) and ( x ) multiplied by the expansion of ( left( 2 - sqrt{x} right)^6 ). Maybe I should first expand ( left( 2 - sqrt{x} right)^6 ) using the binomial theorem.The binomial theorem says that ( (a + b)^n = sum_{k=0}^{n} binom{n}{k} a^{n - k} b^k ). So, applying this to ( left( 2 - sqrt{x} right)^6 ), I can write it as:[sum_{k=0}^{6} binom{6}{k} 2^{6 - k} (-sqrt{x})^k]Simplifying that, each term becomes:[binom{6}{k} 2^{6 - k} (-1)^k x^{k/2}]So, the expansion of ( left( 2 - sqrt{x} right)^6 ) is:[sum_{k=0}^{6} binom{6}{k} 2^{6 - k} (-1)^k x^{k/2}]Now, I need to multiply this expansion by ( left( frac{3}{x} + x right) ). Let me distribute this multiplication:[left( frac{3}{x} + x right) times left( sum_{k=0}^{6} binom{6}{k} 2^{6 - k} (-1)^k x^{k/2} right)]This can be split into two separate sums:[frac{3}{x} times sum_{k=0}^{6} binom{6}{k} 2^{6 - k} (-1)^k x^{k/2} + x times sum_{k=0}^{6} binom{6}{k} 2^{6 - k} (-1)^k x^{k/2}]Simplifying each term:1. For the first sum, ( frac{3}{x} times x^{k/2} = 3 x^{(k/2) - 1} )2. For the second sum, ( x times x^{k/2} = x^{(k/2) + 1} )So, the entire expression becomes:[3 sum_{k=0}^{6} binom{6}{k} 2^{6 - k} (-1)^k x^{(k/2) - 1} + sum_{k=0}^{6} binom{6}{k} 2^{6 - k} (-1)^k x^{(k/2) + 1}]Now, I need to find the coefficient of ( x^2 ) in this combined expression. That means I need to find all terms where the exponent of ( x ) is 2.Let me handle each sum separately.**First Sum: ( 3 sum_{k=0}^{6} binom{6}{k} 2^{6 - k} (-1)^k x^{(k/2) - 1} )**We need the exponent ( (k/2) - 1 = 2 ). Let's solve for ( k ):[frac{k}{2} - 1 = 2 frac{k}{2} = 3 k = 6]So, in the first sum, the term corresponding to ( k = 6 ) will contribute to ( x^2 ).Calculating the coefficient for ( k = 6 ):[3 times binom{6}{6} times 2^{6 - 6} times (-1)^6 = 3 times 1 times 1 times 1 = 3]**Second Sum: ( sum_{k=0}^{6} binom{6}{k} 2^{6 - k} (-1)^k x^{(k/2) + 1} )**We need the exponent ( (k/2) + 1 = 2 ). Let's solve for ( k ):[frac{k}{2} + 1 = 2 frac{k}{2} = 1 k = 2]So, in the second sum, the term corresponding to ( k = 2 ) will contribute to ( x^2 ).Calculating the coefficient for ( k = 2 ):[binom{6}{2} times 2^{6 - 2} times (-1)^2 = 15 times 16 times 1 = 240]Wait, hold on. Let me double-check that calculation. ( binom{6}{2} ) is 15, ( 2^{4} ) is 16, and ( (-1)^2 ) is 1. So, 15 * 16 = 240. That seems correct.Now, adding the contributions from both sums:- First sum: 3- Second sum: 240Total coefficient of ( x^2 ) is ( 3 + 240 = 243 ).Hmm, that seems straightforward. Let me just make sure I didn't miss any other terms that could contribute to ( x^2 ).Looking back at the first sum: the exponent was ( (k/2) - 1 ). We set that equal to 2 and found ( k = 6 ). Since ( k ) can only go up to 6, that's the only term contributing from the first sum.In the second sum, the exponent was ( (k/2) + 1 ). Setting that equal to 2 gives ( k = 2 ), which is within the range of 0 to 6. So, that's the only term contributing from the second sum.Therefore, adding those two contributions together gives the total coefficient of ( x^2 ) as 243.I think that's it. I don't see any mistakes in my reasoning. So, the coefficient should be 243.**Final Answer**The coefficient of ( x^2 ) is boxed{243}.

💡Okay, so I have this function f(x) defined piecewise. It's (x - a)^2 when x is less than or equal to 0, and it's x + 1/x - a when x is greater than 0. The question says that f(0) is the minimum value of f(x), and I need to find the range of the real number a. Hmm, okay, let me try to break this down.First, let me recall what it means for f(0) to be the minimum value. That means that for all x, f(x) is greater than or equal to f(0). So, I need to make sure that both pieces of the function are always greater than or equal to f(0). That makes sense.Let me compute f(0) first. Since 0 is less than or equal to 0, I use the first piece of the function: f(0) = (0 - a)^2 = a^2. So, f(0) is a squared. Got that.Now, I need to ensure that for all x ≤ 0, f(x) = (x - a)^2 is greater than or equal to a^2. Similarly, for all x > 0, f(x) = x + 1/x - a is greater than or equal to a^2. So, I have two conditions to check.Starting with the first piece: f(x) = (x - a)^2 for x ≤ 0. I want this to be greater than or equal to a^2 for all x ≤ 0. Let me write that inequality:(x - a)^2 ≥ a^2.Expanding the left side: x^2 - 2a x + a^2 ≥ a^2.Subtracting a^2 from both sides: x^2 - 2a x ≥ 0.Factor out an x: x(x - 2a) ≥ 0.So, the inequality x(x - 2a) ≥ 0 must hold for all x ≤ 0.Now, let's analyze this inequality. The product of x and (x - 2a) is non-negative. Since x is less than or equal to 0, x is non-positive. So, for the product to be non-negative, (x - 2a) must also be non-positive because a negative times a negative is positive.So, (x - 2a) ≤ 0 for all x ≤ 0.Which simplifies to x ≤ 2a for all x ≤ 0.But wait, x can be any number less than or equal to 0, so the smallest x can be is negative infinity, but we need x ≤ 2a for all x ≤ 0. That means that 2a must be greater than or equal to the maximum value of x in the domain x ≤ 0. The maximum value of x in x ≤ 0 is 0. So, 2a ≥ 0, which implies a ≥ 0.Okay, so from the first piece, we get that a must be greater than or equal to 0.Now, moving on to the second piece: f(x) = x + 1/x - a for x > 0. We need this to be greater than or equal to a^2 for all x > 0.So, the inequality is:x + 1/x - a ≥ a^2.Let me rearrange this:x + 1/x ≥ a + a^2.So, x + 1/x is a function of x, and we need its minimum value to be greater than or equal to a + a^2.I remember that for x > 0, the function x + 1/x has a minimum value. Let me recall: the derivative of x + 1/x is 1 - 1/x^2. Setting that equal to zero gives 1 - 1/x^2 = 0, so x^2 = 1, so x = 1 (since x > 0). So, the minimum occurs at x = 1.Calculating the minimum value: f(1) = 1 + 1/1 - a = 2 - a.Wait, hold on. Wait, in the inequality, we have x + 1/x ≥ a + a^2. So, the minimum of x + 1/x is 2, so 2 ≥ a + a^2.So, 2 ≥ a + a^2.Let me write that as a quadratic inequality:a^2 + a - 2 ≤ 0.Let me solve this quadratic inequality. First, find the roots of a^2 + a - 2 = 0.Using the quadratic formula: a = [-1 ± sqrt(1 + 8)] / 2 = [-1 ± 3] / 2.So, the roots are ( -1 + 3 ) / 2 = 1, and ( -1 - 3 ) / 2 = -2.So, the quadratic a^2 + a - 2 factors as (a - 1)(a + 2).So, the inequality (a - 1)(a + 2) ≤ 0.To solve this, we can consider the intervals determined by the roots: a < -2, -2 < a < 1, and a > 1.Testing intervals:1. For a < -2, say a = -3: (-3 - 1)(-3 + 2) = (-4)(-1) = 4 > 0. So, inequality not satisfied.2. For -2 < a < 1, say a = 0: (0 - 1)(0 + 2) = (-1)(2) = -2 < 0. So, inequality satisfied.3. For a > 1, say a = 2: (2 - 1)(2 + 2) = (1)(4) = 4 > 0. So, inequality not satisfied.At the endpoints:a = -2: ( -2 - 1)( -2 + 2 ) = (-3)(0) = 0. So, equality holds.a = 1: (1 - 1)(1 + 2) = (0)(3) = 0. So, equality holds.So, the solution to the inequality is -2 ≤ a ≤ 1.But wait, earlier from the first piece, we had a ≥ 0. So, combining both conditions, a must satisfy both a ≥ 0 and -2 ≤ a ≤ 1.Therefore, the range of a is the intersection of [0, ∞) and [-2, 1], which is [0, 1].So, a must be between 0 and 1, inclusive.Let me double-check to make sure I didn't make a mistake.First, for the first piece: f(x) = (x - a)^2. We found that a must be ≥ 0. That makes sense because if a were negative, then for x ≤ 0, (x - a)^2 would be minimized at x = a, but since a is negative, x can't reach a because x is ≤ 0. So, the minimum would actually be at x = 0, which is a^2. So, if a were negative, then for x approaching negative infinity, (x - a)^2 would go to infinity, but we need f(0) to be the minimum. So, if a is negative, maybe f(0) isn't the minimum? Wait, actually, if a is negative, then (x - a)^2 is minimized at x = a, which is less than 0, but x can't be less than a if a is negative because x is ≤ 0. Wait, no, x can be less than a if a is negative because x can be any number ≤ 0, which includes numbers less than a if a is negative. So, if a is negative, then the minimum of (x - a)^2 occurs at x = a, which is less than 0, and f(a) = 0. But f(0) is a^2, which is positive. So, f(a) = 0 < f(0) = a^2. That would contradict f(0) being the minimum. Therefore, a cannot be negative. So, a must be ≥ 0. That seems correct.For the second piece: x + 1/x - a. We found that its minimum is 2 - a, and we set 2 - a ≥ a^2, leading to a^2 + a - 2 ≤ 0, which gives -2 ≤ a ≤ 1. But since a must be ≥ 0, combining gives 0 ≤ a ≤ 1.So, that seems consistent.Therefore, the range of a is [0, 1].**Final Answer**The range of the real number ( a ) is boxed{[0,1]}.

💡Alright, so I've got this problem about knights and liars sitting on chairs, and I need to figure out how many knights are now sitting on red chairs after they've all changed seats. Let me try to break this down step by step.First, let me understand the setup. There are 20 chairs, half blue and half red, so that's 10 blue chairs and 10 red chairs. Each chair has either a knight or a liar sitting on it. Knights always tell the truth, and liars always lie. Initially, everyone said they were sitting on a blue chair. That means the knights were actually sitting on blue chairs because they tell the truth, and the liars were sitting on red chairs because they lied about being on blue chairs.So, initially, all the knights were on blue chairs, and all the liars were on red chairs. Let's denote the number of knights as K and the number of liars as L. Since there are 20 chairs, K + L = 20.Now, after they changed seats, half of them (which is 10 people) claim to be sitting on blue chairs, and the other half claim to be sitting on red chairs. I need to figure out how many knights are now on red chairs.Let me think about what this means. After the switch, some knights might now be on red chairs, and some liars might now be on blue chairs. The ones who are telling the truth (knights) will correctly state their current chair color, and the liars will lie about it.So, among the 10 people who now say they're on blue chairs, some are knights who are actually on blue chairs and some are liars who are on red chairs. Similarly, among the 10 people who say they're on red chairs, some are knights who are actually on red chairs and some are liars who are on blue chairs.Let me define some variables to make this clearer:- Let K_b be the number of knights now on blue chairs.- Let K_r be the number of knights now on red chairs.- Let L_b be the number of liars now on blue chairs.- Let L_r be the number of liars now on red chairs.We know that:1. The total number of knights is K = K_b + K_r.2. The total number of liars is L = L_b + L_r.3. The total number of people is 20, so K + L = 20.4. After the switch, 10 people say they're on blue chairs and 10 say they're on red chairs.Now, the people who say they're on blue chairs consist of the knights actually on blue chairs (K_b) and the liars on red chairs (L_r), because liars will lie about their chair color. Similarly, the people who say they're on red chairs consist of the knights actually on red chairs (K_r) and the liars on blue chairs (L_b).So, we have:5. K_b + L_r = 10 (those who say blue)6. K_r + L_b = 10 (those who say red)We also know that the total number of blue chairs is 10 and red chairs is 10. So:7. K_b + L_b = 10 (total blue chairs)8. K_r + L_r = 10 (total red chairs)Now, let's see if we can solve these equations.From equation 7: K_b + L_b = 10From equation 6: K_r + L_b = 10If we subtract equation 6 from equation 7, we get:(K_b + L_b) - (K_r + L_b) = 10 - 10K_b - K_r = 0So, K_b = K_rThat's interesting. The number of knights on blue chairs equals the number of knights on red chairs.Similarly, from equation 5: K_b + L_r = 10From equation 8: K_r + L_r = 10Since K_b = K_r, we can say that both K_b and K_r are equal, and both plus L_r equal 10. So, L_r must be the same in both cases, which makes sense.Now, let's recall that initially, all knights were on blue chairs, and all liars were on red chairs. So, initially, K_b_initial = K and L_r_initial = L.After the switch, some knights have moved to red chairs, and some liars have moved to blue chairs. Let's denote the number of knights who moved to red chairs as x, and the number of liars who moved to blue chairs as y.So, K_b = K - xK_r = xL_b = yL_r = L - yFrom equation 7: K_b + L_b = 10Substituting, (K - x) + y = 10From equation 8: K_r + L_r = 10Substituting, x + (L - y) = 10We also know that K + L = 20Let me write down these equations:1. (K - x) + y = 102. x + (L - y) = 103. K + L = 20Let me try to solve these equations.From equation 1: K - x + y = 10From equation 2: x + L - y = 10Let me add equations 1 and 2:(K - x + y) + (x + L - y) = 10 + 10K + L = 20Which is consistent with equation 3, so no new information there.Let me try to express y from equation 1:From equation 1: y = 10 - K + xSubstitute y into equation 2:x + L - (10 - K + x) = 10x + L -10 + K - x = 10L + K -10 = 10L + K = 20Again, consistent with equation 3.Hmm, seems like I'm going in circles. Maybe I need another approach.Let me think about the number of knights and liars.Initially, all knights were on blue chairs, and all liars were on red chairs.After the switch, some knights moved to red chairs, and some liars moved to blue chairs.The number of knights on red chairs is x, and the number of liars on blue chairs is y.From the initial setup:K_b_initial = KL_r_initial = LAfter the switch:K_b = K - xK_r = xL_b = yL_r = L - yFrom the total blue chairs:K_b + L_b = 10(K - x) + y = 10From the total red chairs:K_r + L_r = 10x + (L - y) = 10We have two equations:1. K - x + y = 102. x + L - y = 10Let me add these two equations:(K - x + y) + (x + L - y) = 10 + 10K + L = 20Which is consistent with K + L = 20.Let me subtract equation 2 from equation 1:(K - x + y) - (x + L - y) = 10 - 10K - x + y - x - L + y = 0K - 2x - L + 2y = 0But since K + L = 20, we can write L = 20 - KSubstitute L into the equation:K - 2x - (20 - K) + 2y = 0K - 2x -20 + K + 2y = 02K - 2x + 2y -20 = 0Divide by 2:K - x + y -10 = 0But from equation 1: K - x + y = 10So, K - x + y = 10Which is the same as equation 1. So, no new information.Hmm, seems like I need another way to approach this.Let me think about the number of people who changed chairs.Initially, all knights were on blue chairs, and all liars were on red chairs.After the switch, some knights are on red chairs, and some liars are on blue chairs.Let me denote:- x = number of knights who moved to red chairs- y = number of liars who moved to blue chairsSo, the number of knights on red chairs is x, and the number of liars on blue chairs is y.From the total blue chairs:Initially, there were K knights on blue chairs and L liars on red chairs.After the switch:- Blue chairs have (K - x) knights and y liars- Red chairs have x knights and (L - y) liarsWe know that the total number of blue chairs is 10, so:(K - x) + y = 10Similarly, the total number of red chairs is 10, so:x + (L - y) = 10We also know that K + L = 20Let me write these equations:1. K - x + y = 102. x + L - y = 103. K + L = 20From equation 3, L = 20 - KSubstitute L into equation 2:x + (20 - K) - y = 10x + 20 - K - y = 10x - K - y = -10From equation 1: K - x + y = 10Let me write these two equations:a. K - x + y = 10b. x - K - y = -10Let me add equations a and b:(K - x + y) + (x - K - y) = 10 + (-10)0 = 0No new information.Let me try to solve for one variable.From equation a: K - x + y = 10Let me solve for y:y = 10 - K + xSubstitute y into equation b:x - K - (10 - K + x) = -10x - K -10 + K - x = -10-10 = -10Again, no new information.Hmm, seems like I need to find another relationship.Wait, let's think about the number of people who changed chairs.Initially, all knights were on blue chairs, and all liars were on red chairs.After the switch, x knights moved to red chairs, and y liars moved to blue chairs.So, the number of people who moved is x + y.But we don't know x + y.Wait, but the total number of chairs is 20, and the number of blue and red chairs remains the same, 10 each.So, the number of knights on blue chairs decreased by x, and the number of liars on red chairs decreased by y.But I'm not sure if that helps.Wait, let's think about the number of knights and liars.Initially, K knights on blue chairs, L liars on red chairs.After the switch:Knights: K - x on blue, x on redLiars: y on blue, L - y on redWe know that:(K - x) + y = 10 (blue chairs)x + (L - y) = 10 (red chairs)And K + L = 20Let me try to express everything in terms of K.From K + L = 20, L = 20 - KSubstitute L into the blue chairs equation:(K - x) + y = 10And into the red chairs equation:x + (20 - K - y) = 10Simplify the red chairs equation:x + 20 - K - y = 10x - K - y = -10Now, we have:1. K - x + y = 102. x - K - y = -10Let me add these two equations:(K - x + y) + (x - K - y) = 10 + (-10)0 = 0No new information.Let me subtract equation 2 from equation 1:(K - x + y) - (x - K - y) = 10 - (-10)K - x + y - x + K + y = 202K - 2x + 2y = 20Divide by 2:K - x + y = 10Which is equation 1 again.Hmm, seems like I'm stuck in a loop.Maybe I need to make an assumption or find another relationship.Wait, let's think about the number of knights and liars.Initially, all knights were on blue chairs, so K = number of blue chairs initially occupied by knights.But blue chairs are 10, so K ≤ 10.Similarly, liars were on red chairs, so L = number of red chairs initially occupied by liars, which is also 10.Wait, no, initially, all knights were on blue chairs, and all liars were on red chairs, but the number of blue and red chairs is 10 each.So, K ≤ 10 and L ≤ 10.But K + L = 20, so K = 10 and L = 10.Wait, that makes sense because there are 10 blue chairs and 10 red chairs.So, initially, all 10 blue chairs had knights, and all 10 red chairs had liars.So, K = 10 and L = 10.Wait, that's a key point I missed earlier.So, K = 10 and L = 10.That simplifies things.So, initially:- 10 knights on blue chairs- 10 liars on red chairsAfter the switch:- Some knights moved to red chairs- Some liars moved to blue chairsLet me denote:- x = number of knights who moved to red chairs- y = number of liars who moved to blue chairsSo, after the switch:- Knights on blue chairs: 10 - x- Knights on red chairs: x- Liars on blue chairs: y- Liars on red chairs: 10 - yFrom the total blue chairs:(10 - x) + y = 10From the total red chairs:x + (10 - y) = 10Let me solve these equations.From blue chairs:10 - x + y = 10Subtract 10 from both sides:-x + y = 0So, y = xFrom red chairs:x + 10 - y = 10Subtract 10 from both sides:x - y = 0So, x = yWhich is consistent with y = x.So, x = yNow, let's recall that after the switch, 10 people say they're on blue chairs and 10 say they're on red chairs.The people who say they're on blue chairs are:- Knights on blue chairs: 10 - x- Liars on red chairs: 10 - y = 10 - x (since y = x)So, total saying blue: (10 - x) + (10 - x) = 20 - 2xBut we know that only 10 people say blue, so:20 - 2x = 10Subtract 20:-2x = -10Divide by -2:x = 5So, x = 5Therefore, the number of knights on red chairs is x = 5So, the answer is 5 knights are now sitting on red chairs.

💡Okay, so I have this inequality problem from the 2000 IMO selection test. It says that for positive real numbers (a, b, c), the following expression is greater than or equal to 3:[left(a+frac{1}{b}-cright)left(b+frac{1}{c}-aright) + left(b+frac{1}{c}-aright)left(c+frac{1}{a}-bright) + left(c+frac{1}{a}-bright)left(a+frac{1}{b}-cright) geq 3.]Hmm, this looks a bit complicated with all these terms. Let me try to break it down step by step.First, I notice that each term in the sum is a product of two expressions. Each of these expressions has a similar structure: one variable, the reciprocal of another variable, and minus the third variable. So, for example, the first term is ((a + frac{1}{b} - c)) multiplied by ((b + frac{1}{c} - a)).I wonder if there's a substitution that can simplify this expression. Maybe if I let (a = frac{x}{y}), (b = frac{y}{z}), and (c = frac{z}{x}), it might make things clearer. Let me try that.Substituting these into the expression, each term becomes:[left(frac{x}{y} + frac{z}{y} - 1right)left(frac{y}{z} + frac{x}{z} - 1right)]So the entire sum becomes:[sum left(frac{x + z - y}{y}right)left(frac{y + x - z}{z}right)]Which simplifies to:[sum frac{(x + z - y)(x + y - z)}{yz}]Hmm, that seems a bit more manageable. Let me write this out explicitly:[frac{(x + z - y)(x + y - z)}{yz} + frac{(y + x - z)(y + z - x)}{zx} + frac{(z + y - x)(z + x - y)}{xy}]Now, if I multiply each term by (xyz), I get:[x(x + z - y)(x + y - z) + y(y + x - z)(y + z - x) + z(z + y - x)(z + x - y)]This should be greater than or equal to (3xyz). So, the inequality becomes:[x(x + z - y)(x + y - z) + y(y + x - z)(y + z - x) + z(z + y - x)(z + x - y) geq 3xyz]Let me expand one of these terms to see what happens. Taking the first term:[x(x + z - y)(x + y - z)]Expanding the product inside:[x left[(x + z - y)(x + y - z)right]]Let me denote (A = x + z - y) and (B = x + y - z). Then, (A cdot B = (x + z - y)(x + y - z)). Let me expand this:[(x + z - y)(x + y - z) = x^2 + xy - xz + xz + zy - z^2 - xy - y^2 + yz]Wait, that seems messy. Let me do it step by step.Multiply (x) with each term in the second bracket:- (x cdot x = x^2)- (x cdot y = xy)- (x cdot (-z) = -xz)Then, multiply (z) with each term in the second bracket:- (z cdot x = xz)- (z cdot y = zy)- (z cdot (-z) = -z^2)Then, multiply (-y) with each term in the second bracket:- (-y cdot x = -xy)- (-y cdot y = -y^2)- (-y cdot (-z) = yz)Now, combine all these terms:[x^2 + xy - xz + xz + zy - z^2 - xy - y^2 + yz]Simplify by combining like terms:- (x^2)- (xy - xy = 0)- (-xz + xz = 0)- (zy + yz = 2yz)- (-z^2)- (-y^2)So, the product simplifies to:[x^2 + 2yz - y^2 - z^2]Therefore, the first term becomes:[x(x^2 + 2yz - y^2 - z^2) = x^3 + 2xyz - xy^2 - xz^2]Similarly, the second term would be:[y(y^2 + 2xz - x^2 - z^2) = y^3 + 2xyz - yx^2 - yz^2]And the third term:[z(z^2 + 2xy - x^2 - y^2) = z^3 + 2xyz - zx^2 - zy^2]Now, adding all three terms together:[x^3 + 2xyz - xy^2 - xz^2 + y^3 + 2xyz - yx^2 - yz^2 + z^3 + 2xyz - zx^2 - zy^2]Combine like terms:- (x^3 + y^3 + z^3)- (2xyz + 2xyz + 2xyz = 6xyz)- (-xy^2 - xz^2 - yx^2 - yz^2 - zx^2 - zy^2)So, the entire expression becomes:[x^3 + y^3 + z^3 + 6xyz - (xy^2 + xz^2 + yx^2 + yz^2 + zx^2 + zy^2)]Hmm, I need to show that this is greater than or equal to (3xyz). So, subtracting (3xyz) from both sides, the inequality becomes:[x^3 + y^3 + z^3 + 3xyz - (xy^2 + xz^2 + yx^2 + yz^2 + zx^2 + zy^2) geq 0]Wait, this looks familiar. Isn't this related to the Schur's inequality? Let me recall Schur's inequality.Schur's inequality states that for any real numbers (x, y, z) and (r geq 0):[x^r(x - y)(x - z) + y^r(y - z)(y - x) + z^r(z - x)(z - y) geq 0]For (r = 1), it becomes:[x(x - y)(x - z) + y(y - z)(y - x) + z(z - x)(z - y) geq 0]Expanding this, we get:[x^3 + y^3 + z^3 + 3xyz geq x^2y + x^2z + y^2x + y^2z + z^2x + z^2y]Which is exactly the inequality I have here! So, the expression simplifies to Schur's inequality for (r = 1), which is known to hold for all positive real numbers (x, y, z).Therefore, the original inequality holds true, and the minimum value is indeed 3.**Final Answer**boxed{3}

💡Alright, so I've got this problem about a function and its derivative. Let me try to work through it step by step.First, the function is given as f(x) = -2x ln x + x² - 2a x + a². They mention that g(x) is the derivative of f(x). So, I need to find f'(x) to get g(x).Starting with f(x) = -2x ln x + x² - 2a x + a². To find f'(x), I'll differentiate term by term.The derivative of -2x ln x: I remember that the derivative of x ln x is 1 + ln x, so using the constant multiple rule, the derivative of -2x ln x should be -2*(1 + ln x). Let me double-check that. Yes, using the product rule: d/dx [x ln x] = ln x + x*(1/x) = ln x + 1. So, multiplying by -2 gives -2 ln x - 2.Next term: x². The derivative is straightforward, that's 2x.Then, -2a x. The derivative of that is -2a.Lastly, a² is a constant, so its derivative is 0.Putting it all together, f'(x) = -2 ln x - 2 + 2x - 2a. So, g(x) = f'(x) = -2 ln x - 2 + 2x - 2a.Wait, that's a bit messy. Let me write it more neatly: g(x) = 2x - 2 ln x - 2 - 2a. Maybe factor out the 2? So, g(x) = 2(x - ln x - 1 - a). Hmm, not sure if that helps yet.Moving on to part (I): The tangent line to the curve y = f(x) at the point (1, f(1)) is perpendicular to the line x + y + 3 = 0. I need to find the value of a.First, let's recall that two lines are perpendicular if the product of their slopes is -1. So, I need to find the slope of the given line and then find the slope of the tangent line that is perpendicular to it.The given line is x + y + 3 = 0. Let me rewrite it in slope-intercept form (y = mx + b). Subtract x and 3 from both sides: y = -x - 3. So, the slope (m) of this line is -1.Since the tangent line is perpendicular to this, its slope should be the negative reciprocal of -1. The negative reciprocal of -1 is 1. So, the slope of the tangent line at (1, f(1)) should be 1.But wait, the slope of the tangent line at a point is given by the derivative at that point. So, f'(1) should equal 1.From earlier, f'(x) = 2x - 2 ln x - 2 - 2a. Let's compute f'(1):f'(1) = 2*(1) - 2 ln(1) - 2 - 2a.Simplify each term:2*(1) = 2ln(1) = 0, so -2 ln(1) = 0-2 remains as is.So, f'(1) = 2 + 0 - 2 - 2a = 0 - 2a = -2a.But we know that f'(1) should be 1 because the tangent line is perpendicular to the given line. So, set -2a = 1.Solving for a: a = -1/2.Wait, that seems straightforward. Let me just verify:If a = -1/2, then f'(1) = -2*(-1/2) = 1, which is correct. So, yes, a = -1/2.Okay, that was part (I). Now, moving on to part (II): Discuss the number of solutions to g(x) = 0.So, g(x) = f'(x) = 2x - 2 ln x - 2 - 2a = 0.Let me write that equation again: 2x - 2 ln x - 2 - 2a = 0.I can simplify this equation by dividing both sides by 2: x - ln x - 1 - a = 0.So, x - ln x - 1 = a.Let me denote h(x) = x - ln x - 1. Then, the equation becomes h(x) = a.So, the number of solutions to g(x) = 0 is the same as the number of intersections between h(x) and the horizontal line y = a.Therefore, to discuss the number of solutions, I need to analyze the function h(x) = x - ln x - 1.First, let's find the domain of h(x). Since ln x is defined for x > 0, the domain is x > 0.Next, let's find the critical points of h(x) by taking its derivative.h'(x) = derivative of x is 1, derivative of -ln x is -1/x, derivative of -1 is 0. So, h'(x) = 1 - 1/x.Set h'(x) = 0 to find critical points:1 - 1/x = 01 = 1/xx = 1.So, x = 1 is the critical point.Now, let's analyze the behavior of h(x) around x = 1.For x < 1, say x = 0.5:h'(0.5) = 1 - 1/0.5 = 1 - 2 = -1 < 0. So, h(x) is decreasing on (0,1).For x > 1, say x = 2:h'(2) = 1 - 1/2 = 1 - 0.5 = 0.5 > 0. So, h(x) is increasing on (1, ∞).Therefore, h(x) has a minimum at x = 1.Let's compute h(1):h(1) = 1 - ln(1) - 1 = 1 - 0 - 1 = 0.So, the minimum value of h(x) is 0 at x = 1.Now, let's see the behavior as x approaches 0+ and as x approaches ∞.As x approaches 0+:ln x approaches -∞, so -ln x approaches +∞. Therefore, h(x) = x - ln x - 1 approaches +∞.As x approaches ∞:x grows to infinity, ln x grows to infinity but much slower than x. So, x - ln x - 1 approaches infinity.Therefore, the graph of h(x) is a U-shaped curve with a minimum at (1, 0), approaching infinity as x approaches 0+ and as x approaches ∞.So, the function h(x) = x - ln x - 1 has a minimum value of 0 at x = 1, and it's decreasing on (0,1) and increasing on (1, ∞).Now, to find the number of solutions to h(x) = a, we can consider different cases based on the value of a.Case 1: a < 0.Since the minimum value of h(x) is 0, and h(x) is always greater than or equal to 0, there are no solutions when a < 0.Case 2: a = 0.Here, h(x) = 0. The equation h(x) = 0 has exactly one solution at x = 1, since that's the point where h(x) attains its minimum.Case 3: a > 0.In this case, since h(x) approaches infinity as x approaches 0+ and as x approaches ∞, and it has a minimum of 0 at x = 1, the horizontal line y = a will intersect the graph of h(x) at two points: one in the interval (0,1) and another in the interval (1, ∞). Therefore, there are two solutions.So, summarizing:- If a < 0: No solutions.- If a = 0: Exactly one solution.- If a > 0: Exactly two solutions.Therefore, the number of solutions to g(x) = 0 depends on the value of a as described above.Wait, let me just make sure I didn't miss anything. For a > 0, since h(x) is decreasing before x=1 and increasing after, and h(1) = 0, then for any a > 0, the line y = a will cross h(x) once on (0,1) and once on (1, ∞). That seems correct.For a = 0, it only touches at x=1.For a < 0, since h(x) never goes below 0, there are no solutions.Yes, that makes sense.So, I think that's the complete analysis for part (II).**Final Answer**Ⅰ. boxed{-dfrac{1}{2}} Ⅱ. When ( a < 0 ), there are no solutions; when ( a = 0 ), there is one solution; and when ( a > 0 ), there are two solutions.end{think>**Final Answer**Ⅰ. boxed{-dfrac{1}{2}} Ⅱ. When ( a < 0 ), there are no solutions; when ( a = 0 ), there is one solution; and when ( a > 0 ), there are two solutions.