Question

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

Hint:

For optimization problems, express one variable in terms of another, differentiate, and find critical points.

Answer 1

Let the two numbers be \( x \) and \( y \). Given: \[ x + y = 15. \] We need to minimize: \[ S = x^2 + y^2. \] Using \( y = 15 - x \), we rewrite: \[ S = x^2 + (15-x)^2. \] Expanding: \[ S = x^2 + 225 - 30x + x^2 = 2x^2 - 30x + 225. \] Differentiating: \[ \frac{dS}{dx} = 4x - 30. \] Setting \( \frac{dS}{dx} = 0 \): \[ 4x - 30 = 0 \Rightarrow x = \frac{30}{4} = 7.5. \] Since \( y = 15 - x = 7.5 \), the numbers are \( 7.5 \) and \( 7.5 \).