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

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Answer 1 · 2015-11-19T17:12:36

$-9$ and $5$ .

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:

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

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