Find two numbers whose difference is 96 and whose product is a minimum. = ? (smaller number) = ? (larger number)

Answer 1

Let x = larger number and y = smaller number

We are given that x - y = 96. So, y = x - 96.

Minimize P = xy = x(x - 96) = x² - 96x

dP/dx = 2x - 96

dP/dx = 0 when x = 48

When x < 48, dP/dx < 0. So, P is decreasing

When x > 48, dP/dx > 0. So, P is increasing.

P is minimized when x = 48and y = -48