Question

Find two numbers whose difference is 176 and whose product is a minimum.

Answer 1

Let the two required numbers be x and y, then x - y = 176, thus y = x - 176.

Let the product of the two numbers be represented by

$P = xy x(x - 176) = x^2 - 176x$

For the product to be minimum, $\frac{dP}{dx} =0$

$\frac{dP}{dx} =2x - 176=0 \\ \\ \Rightarrow2x=176 \\ \\ \Rightarrow x=88$

Thus, y = 88 - 176 = -88

Therefore, the two numbers are 88 and -88.