Optimization

The sum of two nonnegative numbers is 36. Find the numbers if

a. the difference of their square roots is to be as large as possible.

b. the sum of their square roots is to be as large as possible.

Step 1 of 6:

a)In this problem, we need to find the numbers if the difference of their square roots is to be as large as possible.

Step 2 of 6:

Given:The sum of two nonnegative numbers is 36.

Let us consider, x and y are two nonnegative numbers.

Given:

Difference of their square roots is maximum means

, where

Now, differentiate on both sides we get.

Therefore, does not exist at and .

Step 3 of 6:

Clearly, f(x) has critical points at 0,18, and 36.

At ,

At ,

At ,

Hence, the maximum value attains at

Another number is:

Therefore, the numbers are “0 and 36”.