Find two positive numbers whose sum is 15 and | vedclass

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Question

(A) Let the two positive numbers be $x$ and $y$. Given that $x + y = 15$,so $y = 15 - x$.Let $S$ be the sum of the squares of the numbers,so $S = x^2

Vedclass · Accepted Answer

$7.5, 7.5$

Vedclass · Answer

(A) Let the two positive numbers be $x$ and $y$. Given that $x + y = 15$,so $y = 15 - x$.Let $S$ be the sum of the squares of the numbers,so $S = x^2 + y^2$.Substituting $y = 15 - x$,we get $S(x) = x^2 + (15 - x)^2 = x^2 + 225 - 30x + x^2 = 2x^2 - 30x + 225$.To find the minimum,we find the first derivative $S'(x) = 4x - 30$.Setting $S'(x) = 0$,we get $4x = 30$,which implies $x = \frac{30}{4} = 7.5$.Now,find the second derivative $S''(x) = 4$.Since $S''(7.5) = 4 > 0$,the function $S(x)$ has a local minimum at $x = 7.5$.Thus,the two numbers are $x = 7.5$ and $y = 15 - 7.5 = 7.5$.

Find two positive numbers whose sum is $15$ and the sum of whose squares is minimum.

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