Calculus

Find two numbers whose sum is 10 for which the sum of their squares is a minimum.

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  1. two numbers : x and 10-x

    S = x^2 + (10-x)^2
    dS/dx = 2x - 2(10-x)
    = 0 for max/min
    x = 5

    so the numbers are both 5

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  2. x and (10-x)
    s = x^2 + (10-x)^2
    s = x^2 + 100 -20x + x^2
    s = 2 x^2 - 20 x + 100
    s/2 = x^2 -10 x + 100 we can minimize half the sum easier than the whole sum
    That is a parabola and you could find the vertex but since you said this was "calculus" we will take the derivative and set to zero.
    0 = 2 x - 10
    x = 5
    10-x = 5

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  3. Interesting the answer is halfway between.
    Exploring that
    Say a sum of two numbers is s
    We want to minimize the sum of squares of x^2 and (s-x)^2
    sum = 2 x^2 -2sx
    d sum/dx = 0 = 4 x -2s
    x = s/2
    so it works for any old sum, not just 10

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