Question

What are the critical points for `f(x) = (x^2-10x)^4`?

Answer 1

For critical numbers for `f`, we need the values of `x` that are in the domain of `f` and at which `f'(x) = 0` or `f'(x)` does not exist.

For `f(x) = (x^2-10x)^4`, we use the power rule and the chain rule to get:

`f'(x) = 4(x^2-10x)^3(2x-10)`.

This derivative always exists, so we only need the zeros.

Solve: `f'(x) = 4(x^2-10x)^3(2x-10) = 0`.

`4(x^2-10x)^3(2x-10) = 4[x(x-10)]^3 2(x-5) = 8x^3(x-10)^3(x-5)`

The zeros of `f'` are: `0, 5, and 10`

All three are in the domain of `f`, so the critical numbers are:

`0, 5, and 10`

Alternative Terminology
I am used to calling these points on the line, "critical points". Alternative terminology may say that critical points, are points in the plane. In this terminology, we need to find the `y` values. And the critical points will be:

`(0, 0)`, `(5, f(5))` and `(10, f(10))`