# How do you find the critical numbers for f(x) = |x + 3| - 1 to determine the maximum and minimum?

Sep 19, 2016

The only critical number is $- 3$. (And f(-3)=1 is a minimum.)

#### Explanation:

The derivative of $f \left(x\right) = \left\mid x \right\mid$ is

$f ' \left(x\right) = \left\{\begin{matrix}- 1 & \text{ if" & x<0 \\ 1 & " if} & x > 0\end{matrix}\right.$
(and does not exist at $x = 0$).

$f \left(x\right) = \left\mid x + 3 \right\mid - 1$ is $\left\mid x \right\mid$ translated $3$ left and down $1$.

$f ' \left(x\right)$ is $\left\{\begin{matrix}- 1 & \text{ if" & x " is left of the vertex" \\ 1 & " if" & x " is right of the vertex}\end{matrix}\right.$
and is undefined at the vertex.
The vertex is at $x = - 3$.

I must admit, this seems like an awful lot of work to find the minimum and maximum.
We know that $\left\mid x \right\mid$ has a "V" shaped graph, so a translation also has a "V" shaped graph with minimum at the vertex and no maximum.