Find two positive numbers whose sum is 14 and the sum of whose square

1.	Find two positive numbers whose sum is 14 and the sum of whose square is minimum.
Answer» Let the two numbers be x and y Given x + y = 14 & S = x² + y² where y = 14 – x ∴ S = x² + (14 – x)² = x² + 14² + x² – 28x = 2x² – 28x + 14² \(\frac{ds}{dx}\) = 4x – 28 → (1) \(\frac{ds}{dx}\) = 0 ⇒ 4x – 28 = 0 ⇒ x = 7 \(\frac{d^2s}{dx^2}\) = 4 > 0, sum is minimum. ∴ y = 14 – x = 14 – 7 = 7 ∴ the two positive number are 7 & 7.

Find two positive numbers whose sum is 14 and the sum of whose square is minimum.

Answer»

Let the two numbers be x and y

Given x + y = 14 & S = x² + y²

where y = 14 – x

∴ S = x² + (14 – x)² = x² + 14² + x² – 28x = 2x² – 28x + 14²

\(\frac{ds}{dx}\) = 4x – 28 → (1) \(\frac{ds}{dx}\) = 0

⇒ 4x – 28 = 0 ⇒ x = 7

\(\frac{d^2s}{dx^2}\) = 4 > 0, sum is minimum.

∴ y = 14 – x = 14 – 7 = 7

∴ the two positive number are 7 & 7.