Question

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

Hint:

To minimize the sum of squares for a constant sum, distribute the total equally among variables.

Answer 1

Step 1: Define variables. Let the two numbers be \( x \) and \( y \), so \[ x + y = 15 \] The function to minimize is: \[ S = x^2 + y^2 \] Step 2: Express in terms of one variable. Using \( y = 15 - x \), we get: \[ S = x^2 + (15 - x)^2 \] Step 3: Differentiate and find the minimum. \[ \frac{dS}{dx} = 2x + 2(15 - x)(-1) = 2x - 2(15 - x) \] Setting \( \frac{dS}{dx} = 0 \): \[ 2x - 2(15 - x) = 0 \] \[ 4x = 30 \Rightarrow x = 7.5, \quad y = 7.5 \] Thus, the minimum sum of squares occurs when \( x = y = 7.5 \).