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

Let the two positive numbers be a and b.

Given: a + b = 15 … 1

Also, a² + b² is minima

Assume, S = a² + b²

(from equation 1)

⇒ S = a² + (15 – a)²

⇒ S = a² + 225 + a² – 30a = 2a² – 30a + 225

⇒ = 4a - 30

⇒

Since, > 0 will give minimum value of S.

4a – 30 = 0

a = 7.5

Hence, two numbers will be 7.5 and 7.5.