What are the critical points for f(x) = (x^2-10x)^4f(x)=(x2−10x)4?

1 Answer
Jim H
May 28, 2015

For critical numbers for ff, we need the values of xx that are in the domain of ff 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))

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