Question

Find two positive numbers whose product is 25 and whose sum is a minimum. (if both values are the same number, enter it into both blanks.)

Answer 1

Let xy = 25, x and y > 0.

$y = \frac{25}{x}$
$\text{Denote S as: } x + y$
$S(x) = x + \frac{25}{x}$

$\text{Minimum: }S'(x) = 0, S''(x) > 0$
$S'(x) = 1 - \frac{25}{x^{2}}$
$1 - \frac{25}{x^{2}} = 0$
$x^{2} = 25$
$x = 5 (x > 0)$

$S''(x) = \frac{50}{x^{3}} > 0, (x > 0)$

Thus, at x = 5, there lies a minimum summation, and this forms a square with side length of 5.