# What are the critical values, if any, of f(x)= abs(3x-4) ?

Dec 21, 2017

The only critical value is $\frac{4}{3}$

#### Explanation:

$c$ is a critical value fo a function, $f$ if and only if $c i s \in t h e \mathrm{do} m a \in o f$f$\mathmr{and}$f'(c) = 0$\mathmr{and}$f'(c) does not exist.

For $f \left(x\right) = \left\mid 3 x - 4 \right\mid$, the domain is $\left(- \infty , \infty\right)$

$f ' \left(x\right) = \left\{\begin{matrix}- 3 & x < \frac{4}{3} \\ 3 & x > \frac{4}{3}\end{matrix}\right.$

$f ' \left(x\right)$ is never $0$,

but $f ' \left(\frac{4}{3}\right)$ does not exist. (The left and right limits of the difference quotient are unequal.)

So, the only critical value is $\frac{4}{3}$