Question

Find two positive numbers whose sum is 18 and the product of the first number and the square of the other is a maximum. Help!?

Answer 1

The numbers are `9` and `9`. See explanation.

Explanation:

Let `x` and `y` be the numbers.

Then the task is to find such `x` and `y` that:

`{(x+y=18),(x*y=max):}`

From these equations we have:

`y=18-x`

Then the function to maximize is:

`f(x)=x*(18-x)=-x^2+18x`

The derivative is

`f^'(x)=-2x+18`

`f^'(x)=0 iff -2x+18=0 iff x=9`

The dervative decreases at `x=9`, so it is a maximum:

The maximum values are `x=9` and `y=9`