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

Let the two positive numbers be x and 15 – x

We have to find value of x such that x² + (15 – x)² is minimum

Let f(x) = x² + (15 – x)²

To find at what value of x f(x) is minimum we first find the critical points where f’(x) = 0

Differentiate f(x) and equate to 0

⇒ f’(x) = 2x + 2(15 – x)(-1)

⇒ 0 = 2x – 2(15 – x)

⇒ 0 = x – (15 – x)

⇒ x = 15 – x

⇒ 2x = 15

Now this can be point of minima or maxima but observe the function f(x) both the terms are positive due to squares and hence the maximum value will be infinity and we will get f(x) as infinity only when x is infinity hence has to be a point of minima

Hence the two numbers are and

Hence the numbers are and