What is the smallest possible product you can form from two non-negative numbers whose sum is 82? The minimum product is

Question

What is the smallest possible product you can form from two non-negative numbers whose sum is 82?

The minimum product is?

Answer 1

x + y = 82

You want to minimize x*y. To do so, you want to solve for y, y = 82-x and substitute this into x*y:

x(82-x) = 82x-x^2

Taking a derivative of 82x - x² : 82 - 2x

Set the derivative = 0 and solve for x: 82 - 2x = 0

2x = 82

x = 41

The minimum product should be 41*41 = 1681

We can quasi check with other sums that add up to 82 (aka, 38 and 44 - give you a product of 1672).

Answer 2

f(x) = x(82 - x)

f(x) = -x² + 82x

This is a down-opening parabola. It has a maximum, yet no minimum.

The minimum with positive values would be at the zeros.

0 = -x² + 82x

0 = -x(x - 82)

{82}

Answer 3

The problem with this exercise is that it is asking for the smallest possible product and the domain is assumed to be any non-negative real number. This exercise would make more sense if it was asking for the largest possible product since the function is concave down.