💡Okay, so I'm trying to solve this algebra problem, and I want to make sure I understand it correctly before diving in. Let me read it again:Given ( A = 3a^2b - 2ab^2 + abc ), Xiao Ming mistakenly took "2A - B" as "2A + B", and calculated the result as ( C = 4a^2b - 3ab^2 + 4abc ).There are three parts to this problem:1. Calculate the expression for ( B ).2. Find the result of ( 2A - B ).3. Determine if the result from part (2) is independent of ( c ). If ( a = frac{1}{8} ) and ( b = frac{1}{5} ), find the value of the expression from part (2).Alright, let's tackle each part step by step.**Part 1: Calculating the expression for ( B ).**First, I know that Xiao Ming was supposed to compute ( 2A - B ) but instead computed ( 2A + B ) and got ( C = 4a^2b - 3ab^2 + 4abc ).So, if ( 2A + B = C ), then we can write:[ 2A + B = C ]We need to solve for ( B ). Let's rearrange the equation:[ B = C - 2A ]We have expressions for ( A ) and ( C ), so let's substitute them in.Given:[ A = 3a^2b - 2ab^2 + abc ][ C = 4a^2b - 3ab^2 + 4abc ]So, let's compute ( 2A ):[ 2A = 2(3a^2b - 2ab^2 + abc) = 6a^2b - 4ab^2 + 2abc ]Now, subtract ( 2A ) from ( C ):[ B = C - 2A = (4a^2b - 3ab^2 + 4abc) - (6a^2b - 4ab^2 + 2abc) ]Let's distribute the negative sign:[ B = 4a^2b - 3ab^2 + 4abc - 6a^2b + 4ab^2 - 2abc ]Now, combine like terms:- For ( a^2b ): ( 4a^2b - 6a^2b = -2a^2b )- For ( ab^2 ): ( -3ab^2 + 4ab^2 = ab^2 )- For ( abc ): ( 4abc - 2abc = 2abc )So, putting it all together:[ B = -2a^2b + ab^2 + 2abc ]Wait, let me double-check my calculations to make sure I didn't make a mistake.Starting with ( C - 2A ):[ (4a^2b - 3ab^2 + 4abc) - (6a^2b - 4ab^2 + 2abc) ]Subtracting term by term:- ( 4a^2b - 6a^2b = -2a^2b )- ( -3ab^2 - (-4ab^2) = -3ab^2 + 4ab^2 = ab^2 )- ( 4abc - 2abc = 2abc )Yes, that seems correct. So, ( B = -2a^2b + ab^2 + 2abc ).**Part 2: Finding the result of ( 2A - B ).**Now that we have ( B ), let's compute ( 2A - B ).We already have ( 2A ) from earlier:[ 2A = 6a^2b - 4ab^2 + 2abc ]And ( B = -2a^2b + ab^2 + 2abc )So, subtracting ( B ) from ( 2A ):[ 2A - B = (6a^2b - 4ab^2 + 2abc) - (-2a^2b + ab^2 + 2abc) ]Distribute the negative sign:[ 2A - B = 6a^2b - 4ab^2 + 2abc + 2a^2b - ab^2 - 2abc ]Now, combine like terms:- For ( a^2b ): ( 6a^2b + 2a^2b = 8a^2b )- For ( ab^2 ): ( -4ab^2 - ab^2 = -5ab^2 )- For ( abc ): ( 2abc - 2abc = 0 )So, the ( abc ) terms cancel out, and we're left with:[ 2A - B = 8a^2b - 5ab^2 ]Let me verify this again to be sure.Starting with ( 2A - B ):[ (6a^2b - 4ab^2 + 2abc) - (-2a^2b + ab^2 + 2abc) ]Which becomes:[ 6a^2b - 4ab^2 + 2abc + 2a^2b - ab^2 - 2abc ]Combine like terms:- ( 6a^2b + 2a^2b = 8a^2b )- ( -4ab^2 - ab^2 = -5ab^2 )- ( 2abc - 2abc = 0 )Yes, that's correct. So, ( 2A - B = 8a^2b - 5ab^2 ).**Part 3: Checking if the result is independent of ( c ) and calculating the value for specific ( a ) and ( b ).**Looking at the result from part (2), which is ( 8a^2b - 5ab^2 ), I notice that there is no ( c ) term in this expression. That means the value of ( c ) doesn't affect the result. So, Xiao Qiang is correct in saying that the result is independent of ( c ).Now, we need to find the value of ( 2A - B ) when ( a = frac{1}{8} ) and ( b = frac{1}{5} ).Let's substitute these values into the expression ( 8a^2b - 5ab^2 ).First, compute each term separately.Compute ( a^2 ):[ a^2 = left( frac{1}{8} right)^2 = frac{1}{64} ]Compute ( b ):[ b = frac{1}{5} ]Compute ( a^2b ):[ a^2b = frac{1}{64} times frac{1}{5} = frac{1}{320} ]Compute ( 8a^2b ):[ 8 times frac{1}{320} = frac{8}{320} = frac{1}{40} ]Now, compute ( ab^2 ):First, compute ( b^2 ):[ b^2 = left( frac{1}{5} right)^2 = frac{1}{25} ]Then, compute ( ab^2 ):[ ab^2 = frac{1}{8} times frac{1}{25} = frac{1}{200} ]Compute ( 5ab^2 ):[ 5 times frac{1}{200} = frac{5}{200} = frac{1}{40} ]Now, subtract ( 5ab^2 ) from ( 8a^2b ):[ 8a^2b - 5ab^2 = frac{1}{40} - frac{1}{40} = 0 ]Wait, that's interesting. The result is zero. Let me double-check my calculations to ensure I didn't make a mistake.Compute ( a^2b ):[ left( frac{1}{8} right)^2 times frac{1}{5} = frac{1}{64} times frac{1}{5} = frac{1}{320} ]Multiply by 8:[ 8 times frac{1}{320} = frac{8}{320} = frac{1}{40} ]Compute ( ab^2 ):[ frac{1}{8} times left( frac{1}{5} right)^2 = frac{1}{8} times frac{1}{25} = frac{1}{200} ]Multiply by 5:[ 5 times frac{1}{200} = frac{5}{200} = frac{1}{40} ]Subtract:[ frac{1}{40} - frac{1}{40} = 0 ]Yes, that's correct. So, the value of ( 2A - B ) when ( a = frac{1}{8} ) and ( b = frac{1}{5} ) is 0.**Summary of Findings:**1. The expression for ( B ) is ( -2a^2b + ab^2 + 2abc ).2. The result of ( 2A - B ) is ( 8a^2b - 5ab^2 ).3. The result from part (2) is indeed independent of ( c ), and when ( a = frac{1}{8} ) and ( b = frac{1}{5} ), the value is 0.I think I've covered all the steps thoroughly and checked my work to ensure accuracy. It's a bit surprising that the result is zero, but given the specific values of ( a ) and ( b ), it makes sense.

💡Alright, so I've got this problem here about a boat traveling between two piers, A and B. It goes downstream in 2 hours and then comes back upstream in 2.5 hours. The speed of the water flow is given as 3 km/h, and I need to find the average speed of the boat in still water. Hmm, okay, let's break this down.First, I remember that when dealing with boats or objects moving in water, their effective speed changes depending on whether they're going downstream or upstream. Downstream, the boat's speed increases because the current helps it, and upstream, its speed decreases because it's going against the current. So, if I let the boat's speed in still water be something like 'x' km/h, then downstream its speed would be 'x + 3' km/h, and upstream it would be 'x - 3' km/h.Now, the problem mentions that the boat takes 2 hours to go downstream and 2.5 hours to come back upstream. I also know that distance equals speed multiplied by time. Since the distance between piers A and B should be the same in both directions, I can set up an equation where the distance downstream equals the distance upstream.So, downstream distance would be (x + 3) * 2, and upstream distance would be (x - 3) * 2.5. Since these distances are equal, I can set them equal to each other:2(x + 3) = 2.5(x - 3)Okay, that seems like a good start. Let me check if this makes sense. If I solve for 'x,' I should get the average speed of the boat in still water. Let's try solving this equation:First, expand both sides:2x + 6 = 2.5x - 7.5Now, let's get all the 'x' terms on one side and constants on the other:2x - 2.5x = -7.5 - 6-0.5x = -13.5Now, divide both sides by -0.5:x = (-13.5) / (-0.5) = 27So, the average speed of the boat in still water is 27 km/h. That seems pretty fast for a boat, but maybe it's a speedboat or something.Wait, let me double-check my calculations to make sure I didn't make a mistake. Starting from:2(x + 3) = 2.5(x - 3)Expanding:2x + 6 = 2.5x - 7.5Subtract 2x from both sides:6 = 0.5x - 7.5Add 7.5 to both sides:13.5 = 0.5xMultiply both sides by 2:27 = xOkay, so that checks out. The boat's speed in still water is 27 km/h.Now, let's look at what the two students did. Bingbing wrote the equation 2(x + 3) = 2.5(x - 3), which is exactly what I did. So, Bingbing seems to have set it up correctly, with 'x' representing the boat's speed in still water. That makes sense because downstream speed is x + 3 and upstream is x - 3.Qianqian wrote a different equation: (x/2) - (x/2.5) = 3 * 2. Hmm, let's see what this means. If I interpret 'x' as the distance between the piers, then x/2 would be the downstream speed, and x/2.5 would be the upstream speed. The difference between these speeds is supposed to be twice the water flow speed, which is 6 km/h.So, setting up the equation:(x/2) - (x/2.5) = 6Let's solve this to see if it gives the same result for 'x' as the distance.First, find a common denominator for the fractions. 2 and 2.5 can be converted to 10 by multiplying numerator and denominator appropriately.So, (x/2) = (5x)/10 and (x/2.5) = (4x)/10.Subtracting these:(5x/10) - (4x/10) = x/10 = 6So, x = 60 km.Wait, that's the distance between the piers. If I use this distance, I can find the boat's speed in still water.Downstream speed = distance / time = 60 / 2 = 30 km/hUpstream speed = distance / time = 60 / 2.5 = 24 km/hSo, the boat's speed in still water would be the average of these two speeds:(30 + 24) / 2 = 27 km/hOkay, so Qianqian's equation also leads to the correct answer, but 'x' in her equation represents the distance between the piers, not the boat's speed in still water.So, going back to the original statements:① The variable x in Bingbing's equation represents the average speed of the boat in still water. – This is correct.② The variable x in Qianqian's equation represents the average speed of the boat in still water. – This is incorrect because x represents the distance.③ The variable x in Bingbing's equation represents the distance between piers A and B. – This is incorrect because x represents the speed.④ The variable x in Qianqian's equation represents the distance between piers A and B. – This is correct.So, the correct statements are ① and ④, which corresponds to option B.

💡Okay, so I have this problem where I need to find the value of m for point C(1, m) such that the sum of the distances AC + CB is minimized. Points A and B are given as A(-3, -4) and B(6, 3). Hmm, let me think about how to approach this.First, I remember that in geometry, the shortest path between two points is a straight line. So, if I want to minimize the sum of two distances, AC + CB, it should be when point C lies somewhere on the straight line connecting points A and B. That makes sense because if C is not on that line, the path would be longer.So, I need to find the equation of the line passing through points A and B. Let me calculate the slope first. The slope formula is (y2 - y1)/(x2 - x1). Plugging in the coordinates of A and B, that would be (3 - (-4))/(6 - (-3)) = (7)/(9). So, the slope is 7/9.Now, using the point-slope form of a line, which is y - y1 = m(x - x1), I can use point A(-3, -4) to write the equation. Substituting the values, I get y - (-4) = (7/9)(x - (-3)), which simplifies to y + 4 = (7/9)(x + 3).Let me simplify this equation further. Distribute the 7/9 on the right side: y + 4 = (7/9)x + (7/9)*3. Calculating (7/9)*3 gives 7/3. So, the equation becomes y + 4 = (7/9)x + 7/3.Now, subtract 4 from both sides to solve for y. Converting 4 to thirds, that's 12/3. So, y = (7/9)x + 7/3 - 12/3. Simplifying the constants, 7/3 - 12/3 is -5/3. Therefore, the equation of the line AB is y = (7/9)x - 5/3.Since point C(1, m) lies on this line, I can substitute x = 1 into the equation to find m. Plugging in x = 1, we get y = (7/9)(1) - 5/3. Calculating that, (7/9) - (5/3) needs a common denominator. Converting 5/3 to ninths, that's 15/9. So, 7/9 - 15/9 = -8/9.Therefore, the value of m is -8/9. Let me double-check my calculations to make sure I didn't make any mistakes. The slope was 7/9, which seems correct. The equation of the line was derived correctly using point A, and substituting x = 1 gave me m = -8/9. That seems right.I think I've got it. The value of m that minimizes AC + CB is -8/9.

💡Okay, so I have this problem about a cube with side length 8 cm. The vertices are colored alternately black and purple, and I need to find the volume of the tetrahedron formed by the purple vertices. Hmm, let me try to visualize this.First, I know a cube has 8 vertices. If they're alternately colored, that means half of them are purple and half are black. So, there should be 4 purple vertices and 4 black ones. The tetrahedron is formed by connecting these purple vertices. I remember that a tetrahedron has four triangular faces, so connecting four vertices in a cube should form such a shape.Wait, but how exactly are the purple vertices arranged? In a cube, each vertex is connected to three others. If they're alternately colored, each purple vertex should be connected only to black vertices, right? So, the purple vertices aren't connected to each other directly in the cube's edges. That means the tetrahedron isn't formed by the cube's edges but by connecting these purple vertices through the cube's space diagonals.I think the tetrahedron formed by connecting alternate vertices of a cube is called a regular tetrahedron. But is it regular? Let me check. In a cube, the distance between two opposite vertices (the space diagonal) is longer than the edges. So, if I connect the purple vertices, which are all opposite each other, the edges of the tetrahedron will all be space diagonals of the cube.Wait, no. Actually, in a cube, each space diagonal connects two vertices that are not adjacent. So, if I have four purple vertices, each pair of them is connected by a space diagonal. But hold on, in a cube, each vertex is connected to three others via edges, and the space diagonals connect to the opposite vertex. So, for a cube with side length 8 cm, the space diagonal length can be calculated using the formula for the space diagonal of a cube: ( ssqrt{3} ), where ( s ) is the side length. So, that would be ( 8sqrt{3} ) cm.But if all edges of the tetrahedron are space diagonals, then each edge of the tetrahedron is ( 8sqrt{3} ) cm. So, is this a regular tetrahedron? Because all edges are equal. Yes, I think so. A regular tetrahedron has all edges equal and all faces are equilateral triangles.Now, to find the volume of this regular tetrahedron. I remember the formula for the volume of a regular tetrahedron is ( frac{sqrt{2}}{12}a^3 ), where ( a ) is the edge length. So, plugging in ( a = 8sqrt{3} ), we get:( frac{sqrt{2}}{12} times (8sqrt{3})^3 ).Let me compute that step by step.First, compute ( (8sqrt{3})^3 ):( 8^3 = 512 ),( (sqrt{3})^3 = 3sqrt{3} ),so ( 512 times 3sqrt{3} = 1536sqrt{3} ).Now, multiply by ( frac{sqrt{2}}{12} ):( frac{sqrt{2}}{12} times 1536sqrt{3} = frac{1536sqrt{6}}{12} ).Simplify ( frac{1536}{12} ):1536 divided by 12 is 128.So, the volume is ( 128sqrt{6} ) cm³.Wait, but let me think again. Is the tetrahedron regular? Because in a cube, the space diagonals are equal, but when you connect four alternate vertices, the distances between them might not all be equal. Hmm, actually, in a cube, the distance between any two alternate vertices is the same, which is the space diagonal. So, yes, all edges of the tetrahedron are equal, making it a regular tetrahedron.Alternatively, maybe there's another way to compute the volume without using the regular tetrahedron formula. Maybe by using coordinates.Let me assign coordinates to the cube. Let's say the cube is axis-aligned with one corner at the origin (0,0,0) and extending to (8,8,8). Then, the purple vertices would be at (0,0,0), (8,8,0), (8,0,8), and (0,8,8). Wait, is that correct? Let me check: if we color the vertices alternately, starting from (0,0,0) as purple, then the adjacent vertices (along x, y, z) would be black, and their opposites would be purple. So, yes, the four purple vertices would be at (0,0,0), (8,8,0), (8,0,8), and (0,8,8).Now, to find the volume of the tetrahedron formed by these four points. The formula for the volume of a tetrahedron given four points ( A, B, C, D ) is:( V = frac{1}{6} | det(vec{AB}, vec{AC}, vec{AD}) | ).Let me compute vectors from point A (0,0,0) to the other three points:( vec{AB} = (8,8,0) - (0,0,0) = (8,8,0) ),( vec{AC} = (8,0,8) - (0,0,0) = (8,0,8) ),( vec{AD} = (0,8,8) - (0,0,0) = (0,8,8) ).Now, set up the determinant matrix with these vectors as columns:[begin{vmatrix}8 & 8 & 0 8 & 0 & 8 0 & 8 & 8 end{vmatrix}]Compute this determinant. Let's expand along the first row:( 8 times begin{vmatrix} 0 & 8 8 & 8 end{vmatrix} - 8 times begin{vmatrix} 8 & 8 0 & 8 end{vmatrix} + 0 times begin{vmatrix} 8 & 0 0 & 8 end{vmatrix} ).Compute each minor:First minor: ( 0 times 8 - 8 times 8 = 0 - 64 = -64 ).Second minor: ( 8 times 8 - 8 times 0 = 64 - 0 = 64 ).Third minor is multiplied by 0, so it doesn't contribute.So, determinant = ( 8 times (-64) - 8 times 64 + 0 = -512 - 512 = -1024 ).Taking absolute value, we get 1024.So, volume ( V = frac{1}{6} times 1024 = frac{1024}{6} = frac{512}{3} ) cm³.Wait, that's different from my earlier result of ( 128sqrt{6} ). Which one is correct?Let me check the determinant calculation again. Maybe I made a mistake there.The determinant was:[begin{vmatrix}8 & 8 & 0 8 & 0 & 8 0 & 8 & 8 end{vmatrix}]Expanding along the first row:( 8 times (0 times 8 - 8 times 8) - 8 times (8 times 8 - 8 times 0) + 0 times (8 times 8 - 0 times 0) ).So, ( 8 times (0 - 64) - 8 times (64 - 0) + 0 times (64 - 0) ).Which is ( 8 times (-64) - 8 times 64 + 0 = -512 - 512 = -1024 ). Absolute value is 1024. So, volume is ( frac{1024}{6} = frac{512}{3} ).Hmm, so that's 512/3 cm³.But earlier, using the regular tetrahedron formula, I got 128√6. Let me compute both numerically to see if they match.Compute ( 512/3 ): approximately 170.666...Compute ( 128sqrt{6} ): sqrt(6) is about 2.449, so 128 * 2.449 ≈ 313.5.These are different. So, one of the methods must be wrong.Wait, perhaps the tetrahedron isn't regular? Because in the cube, the edges of the tetrahedron are space diagonals, but the angles between them might not be 60 degrees, which is required for a regular tetrahedron.Wait, in a cube, the angle between two space diagonals can be found using the dot product. Let's take two vectors from the origin to two purple vertices: (8,8,0) and (8,0,8).Dot product: (8)(8) + (8)(0) + (0)(8) = 64 + 0 + 0 = 64.The magnitude of each vector: sqrt(8² + 8² + 0²) = sqrt(64 + 64) = sqrt(128) = 8√2.So, the cosine of the angle between them is 64 / (8√2 * 8√2) = 64 / (64 * 2) = 64 / 128 = 0.5.So, the angle is arccos(0.5) = 60 degrees. Wait, so the angle between two edges is 60 degrees. So, does that mean the tetrahedron is regular?Wait, but in a regular tetrahedron, all edges are equal and all angles between edges are 60 degrees. So, if all edges are equal and all angles are 60 degrees, it should be regular.But according to the determinant method, the volume is 512/3, while the regular tetrahedron formula gave 128√6. Let me compute both:512/3 ≈ 170.666...128√6 ≈ 128 * 2.449 ≈ 313.5.These are different. So, clearly, one approach is incorrect.Wait, perhaps I made a mistake in identifying the edge length of the tetrahedron. Earlier, I thought the edge length was 8√3, but actually, in the cube, the distance between two purple vertices is the space diagonal, which is indeed 8√3. But wait, in the coordinate system, the distance between (0,0,0) and (8,8,0) is sqrt((8)^2 + (8)^2 + (0)^2) = sqrt(128) = 8√2, not 8√3.Wait, hold on. That's a mistake. The space diagonal of a cube is from (0,0,0) to (8,8,8), which is sqrt(8² + 8² + 8²) = sqrt(192) = 8√3. But the distance between (0,0,0) and (8,8,0) is sqrt(8² + 8²) = 8√2. So, in the tetrahedron formed by the purple vertices, the edges are actually 8√2, not 8√3.So, that was my mistake earlier. I thought the edge length was 8√3, but it's actually 8√2.So, let's recalculate the volume using the regular tetrahedron formula with a = 8√2.Volume = ( frac{sqrt{2}}{12}a^3 ).Compute ( a^3 = (8√2)^3 = 512 * (2√2) = 1024√2 ).So, volume = ( frac{sqrt{2}}{12} * 1024√2 = frac{sqrt{2} * 1024√2}{12} ).Simplify: ( sqrt{2} * √2 = 2 ), so numerator is 1024 * 2 = 2048.Thus, volume = ( frac{2048}{12} = frac{512}{3} ) cm³.Ah, now it matches the determinant method. So, the volume is indeed 512/3 cm³.So, the confusion was because I initially thought the edge length was 8√3, but it's actually 8√2. The space diagonal of the cube is 8√3, but the distance between two purple vertices in the tetrahedron is the face diagonal, which is 8√2.Therefore, the correct volume is 512/3 cm³.

💡Okay, so I have this problem here where I need to prove that the sum of the reciprocals of the lengths of the unions of overlapping intervals is at least k squared. Let me try to unpack this step by step.First, let me restate the problem to make sure I understand it. We have k non-degenerate subintervals of [0,1], denoted as I₁, I₂, ..., I_k. A non-degenerate interval means that each interval has a positive length, right? So, each I_i has a length |I_i| > 0.The summation is over all pairs (i, j) such that the intersection of I_i and I_j is not empty. For each such pair, we take the reciprocal of the length of their union, which is |I_i ∪ I_j|, and sum all these reciprocals. The claim is that this sum is at least k².Hmm, okay. So, if I have k intervals, and I consider all pairs that overlap, then for each overlapping pair, I calculate 1 over the length of their union, and add all these up, and this total should be at least k squared.I wonder if there's a way to approach this using some sort of inequality or maybe an integral. Let me think about how to model this.Maybe I can assign some function to each interval and then use integration to capture the overlaps. Let me define a function f_i(x) for each interval I_i. Let's say f_i(x) is 1 outside of I_i and something else inside I_i. Maybe 1 minus the reciprocal of the length of I_i inside I_i? That might help in capturing the overlap.So, let's define f_i(x) as follows:- f_i(x) = 1 if x is not in I_i- f_i(x) = 1 - 1/|I_i| if x is in I_iThis way, f_i(x) is a step function that is 1 outside I_i and slightly less inside I_i. The idea is that when two intervals overlap, their functions will interact in a way that can be captured by integrating their product.Now, let's consider the integral of f_i(x) * f_j(x) over [0,1]. This integral will give us some measure of how much the two functions overlap. If I_i and I_j don't overlap, then their functions are both 1 in the region where they don't intersect, so the product would be 1. But if they do overlap, then in the overlapping region, the product would be (1 - 1/|I_i|)(1 - 1/|I_j|), which is less than 1.Wait, but actually, if they don't overlap, then f_i and f_j are both 1 in the entire interval except their own intervals. So, the product f_i * f_j would be 1 everywhere except inside I_i and I_j. Inside I_i, f_i is 1 - 1/|I_i| and f_j is 1. Similarly, inside I_j, f_j is 1 - 1/|I_j| and f_i is 1. So, the integral would be 1 minus the lengths where they are less than 1.Let me compute the integral for the case when I_i and I_j are disjoint. So, |I_i ∩ I_j| = 0. Then, the integral of f_i * f_j would be:Integral over [0,1] of f_i(x) * f_j(x) dx = Integral over [0,1] of 1 dx - Integral over I_i of (1 - (1 - 1/|I_i|)) dx - Integral over I_j of (1 - (1 - 1/|I_j|)) dx.Wait, that might not be the right way to compute it. Let me think again.If I_i and I_j are disjoint, then f_i(x) is 1 - 1/|I_i| on I_i, and 1 elsewhere. Similarly, f_j(x) is 1 - 1/|I_j| on I_j, and 1 elsewhere. So, their product f_i(x) * f_j(x) is:- 1 on [0,1] (I_i ∪ I_j)- (1 - 1/|I_i|) on I_i- (1 - 1/|I_j|) on I_jTherefore, the integral would be:| [0,1] (I_i ∪ I_j) | * 1 + |I_i| * (1 - 1/|I_i|) + |I_j| * (1 - 1/|I_j|)Simplify this:(1 - |I_i| - |I_j|) * 1 + |I_i| - 1 + |I_j| - 1Wait, that can't be right because |I_i|*(1 - 1/|I_i|) = |I_i| - 1, which would make the integral:(1 - |I_i| - |I_j|) + (|I_i| - 1) + (|I_j| - 1) = 1 - |I_i| - |I_j| + |I_i| - 1 + |I_j| - 1 = -1Hmm, so if I_i and I_j are disjoint, the integral of f_i * f_j is -1. Interesting.Now, what if I_i and I_j do overlap? Let's say their intersection has length c. Then, their union has length |I_i| + |I_j| - c. So, |I_i ∪ I_j| = |I_i| + |I_j| - c.In this case, f_i(x) * f_j(x) would be:- 1 on [0,1] (I_i ∪ I_j)- (1 - 1/|I_i|) on I_i I_j- (1 - 1/|I_j|) on I_j I_i- (1 - 1/|I_i|)(1 - 1/|I_j|) on I_i ∩ I_jSo, the integral becomes:(1 - |I_i| - |I_j| + c) * 1 + (|I_i| - c) * (1 - 1/|I_i|) + (|I_j| - c) * (1 - 1/|I_j|) + c * (1 - 1/|I_i|)(1 - 1/|I_j|)Let me compute each term:1. (1 - |I_i| - |I_j| + c) * 1 = 1 - |I_i| - |I_j| + c2. (|I_i| - c) * (1 - 1/|I_i|) = |I_i| - c - (|I_i| - c)/|I_i| = |I_i| - c - 1 + c/|I_i|3. Similarly, (|I_j| - c) * (1 - 1/|I_j|) = |I_j| - c - 1 + c/|I_j|4. c * (1 - 1/|I_i|)(1 - 1/|I_j|) = c - c/|I_i| - c/|I_j| + c/(|I_i||I_j|)Now, adding all these together:1. 1 - |I_i| - |I_j| + c2. + |I_i| - c - 1 + c/|I_i|3. + |I_j| - c - 1 + c/|I_j|4. + c - c/|I_i| - c/|I_j| + c/(|I_i||I_j|)Combine like terms:- Constants: 1 - 1 - 1 = -1- |I_i| terms: -|I_i| + |I_i| = 0- |I_j| terms: -|I_j| + |I_j| = 0- c terms: c - c - c + c = 0- c/|I_i| terms: c/|I_i| - c/|I_i| = 0- c/|I_j| terms: c/|I_j| - c/|I_j| = 0- Remaining: c/(|I_i||I_j|)So, the total integral is -1 + c/(|I_i||I_j|).But since |I_i| ≥ c and |I_j| ≥ c, we have |I_i||I_j| ≥ c(|I_i| + |I_j| - c). Therefore, c/(|I_i||I_j|) ≤ 1/(|I_i| + |I_j| - c) = 1/|I_i ∪ I_j|.So, the integral becomes ≤ -1 + 1/|I_i ∪ I_j|.Putting it all together, for overlapping intervals, the integral of f_i * f_j is ≤ -1 + 1/|I_i ∪ I_j|.Now, let's consider the sum over all i and j of the integrals of f_i * f_j. This is equal to the integral of (Σf_i(x))² dx, which is always non-negative because it's an integral of a square.So, 0 ≤ ∫₀¹ (Σf_i(x))² dx = Σ_{i,j} ∫₀¹ f_i(x)f_j(x) dx.From earlier, we have that for disjoint intervals, the integral is -1, and for overlapping intervals, it's ≤ -1 + 1/|I_i ∪ I_j|.Therefore, the sum Σ_{i,j} ∫₀¹ f_i(x)f_j(x) dx = Σ_{i,j disjoint} (-1) + Σ_{i,j overlapping} (-1 + 1/|I_i ∪ I_j|).Let me denote E(i,j) as 1 if I_i and I_j overlap, and 0 otherwise. Then, the sum becomes:Σ_{i,j} (-1) + Σ_{i,j overlapping} 1/|I_i ∪ I_j|.But Σ_{i,j} (-1) = -k², since there are k² terms.So, we have:0 ≤ -k² + Σ_{i,j overlapping} 1/|I_i ∪ I_j|.Rearranging, we get:Σ_{i,j overlapping} 1/|I_i ∪ I_j| ≥ k².Which is exactly what we needed to prove.I think this makes sense. By constructing these functions f_i and using the fact that the integral of their squares is non-negative, we can derive the required inequality. The key was to define the functions in such a way that their product captures the overlap and allows us to relate it to the reciprocal of the union lengths.

💡Okay, so I need to find the value of ( b ) that makes these two lines perpendicular. The lines are given by the equations:1. ( y = 3x + 5 )2. ( 4y + bx = 12 )First, I remember that for two lines to be perpendicular, the product of their slopes should be (-1). That is, if one line has a slope ( m_1 ) and the other has a slope ( m_2 ), then ( m_1 times m_2 = -1 ).Looking at the first equation, ( y = 3x + 5 ), it's already in slope-intercept form (( y = mx + c )), so the slope ( m_1 ) is 3. That part seems straightforward.Now, the second equation is ( 4y + bx = 12 ). I need to rewrite this equation in slope-intercept form to find its slope. Let me do that step by step.Starting with ( 4y + bx = 12 ), I want to solve for ( y ). So, I'll subtract ( bx ) from both sides:( 4y = -bx + 12 )Now, to isolate ( y ), I'll divide every term by 4:( y = -frac{b}{4}x + 3 )Okay, so now the equation is in the form ( y = m_2x + c ), where ( m_2 ) is the slope. Therefore, the slope ( m_2 ) is ( -frac{b}{4} ).Now, according to the condition for perpendicularity, the product of the slopes should be (-1). So, I'll set up the equation:( m_1 times m_2 = -1 )Substituting the known values:( 3 times left(-frac{b}{4}right) = -1 )Let me simplify this:( -frac{3b}{4} = -1 )To solve for ( b ), I'll first multiply both sides by 4 to eliminate the denominator:( -3b = -4 )Now, I'll divide both sides by -3:( b = frac{-4}{-3} )Simplifying the negatives:( b = frac{4}{3} )So, the value of ( b ) that makes the two lines perpendicular is ( frac{4}{3} ).Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting from the second equation:( 4y + bx = 12 )Solving for ( y ):( 4y = -bx + 12 )( y = -frac{b}{4}x + 3 )Yes, that looks correct. The slope is indeed ( -frac{b}{4} ).Then, setting the product of slopes equal to -1:( 3 times left(-frac{b}{4}right) = -1 )Multiplying:( -frac{3b}{4} = -1 )Multiplying both sides by 4:( -3b = -4 )Dividing by -3:( b = frac{4}{3} )Everything checks out. I think I did it right. So, ( b = frac{4}{3} ) is the correct answer.