Question

Two non negative numbers x and y exist such that the sum of the numbers if 12 and that the product of one number and the square of the other number is a maximum. What is the maximum product?

Answer 1

Maximum product: `256`

Explanation:

Suppose that in forming the product `x` is the variable being squared.

Since `x+y=12`
`color(white)("XXX")y=12-x`
Note that for both `x` and `y` to be non-negative, `x in [0,12]`

The product function can be written as
`f(x)=y * x^2 = (12-x) * x^2 = 12x^2-x^3`

The maximum will occur at one of this function's critical points or at the end points for the `x` Domain.

Critical points will happen where the slope of this function (that is the derivative) is equal to zero.

`f'(x)=24x-3x^2`

So we will have critical points where
`color(white)("XXX")24x-3x^2=0`

`color(white)("XXX")rarr x(24-3x)=0`

`color(white)("XXX")rarr x=0color(white)("xx")orcolor(white)("xx")24-3x=0`
`color(white)("xxxxxxxxxxxxxxxxxxx")rarr x=8`

at `x=0`, `f(x=0)=(12-0) * 0^2 = 0`
and
at `x=8`, `f(x=8)=(12-8) * 8^2 =256`

The end points `x=0` and `x=12` of the Domain give values for the product function. both give function values of `0`

Therefore, the maximum product function value is `256` (at critical point `x=8`).