Question

What are the critical values, if any, of `f(x)=x^3 + 6x^2 − 135x`?

Answer 1

`-9` and `5`.

Explanation:

A critical number for a function `f`, is a number in the domain of `f` at which either the derivative of `f` is `0` or it fails to exist.

For `f(x)=x^3 + 6x^2 − 135x`, the domain is `(-oo,oo)` and

`f'(x) = 3x^2+12x-136`.

This `f'(x)` exists for all numbers.

`f'(x)=0` at `-9` and at `5`, both of which are in the domain of `f`, so both are critical numbers for `f`.

To solve `f'(x)=0`, use the quadratic formula or factor:

`3x^2+12x-136 = 3(x^2+4x-45) = 3(x+9)(x-5)`