# What are the critical values, if any, of f(x)= -2 x^3 + 27 x^2 - 48 x + 7?

Dec 23, 2015

$c = 1 , 8$

#### Explanation:

A critical value is the $x$-value at which $f ' \left(x\right) = 0$ or does not exist.

Find $f ' \left(x\right)$.

$f ' \left(x\right) = - 6 {x}^{2} + 54 x - 48$

Find when $f ' \left(x\right) = 0$.

$- 6 {x}^{2} + 54 x - 48 = 0$
$\implies - 6 \left({x}^{2} - 9 x + 8\right) = 0$
$\implies - 6 \left(x - 1\right) \left(x - 8\right) = 0$
color(red)(x=1,8

Find when $f ' \left(x\right)$ does not exist.

$f ' \left(x\right)$ is defined for all $x$.

Thus, the only critical values are $c = 1 , 8$.

graph{-2 x^3 + 27 x^2 - 48 x + 7 [-5, 15, -289.3, 405]}