How would I find the critical values of an absolute value function? $y=|x+1| + |x-1|$ over $[3, -2]$

Question

Would I just have to graph this to find the critical points? Because I tried the method where you rewrite the expressions in the absolute brackets as $\sqrt{(x+1)^2}$ and $\sqrt{(x-1)^2}$ and take the derivative of that, but I am still left with absolute expressions in the final equation when I set $y' = 0$. I don't know exactly what I should do if I have absolute value factors in the final equation.

Answer 1

Note that the graph of $|x+1|$ and $|x-1|$ are shifted versions of the graph of $|x|$ (1 unit to the left, respectively to the right)

• If $x \leq -1$, then $x+1 \leq 0$ and $x-1 \leq 0$, hence $f(x) = -(x+1) - (x-1) = -2x.$
• If $ -1 < x \leq 1$, then $x+1 \geq 0$, whereas $x-1 \leq 0$, hence $f(x) = x+1 - (x-1) = 2$.
• If $x > 1$, then $x+1 \geq 0$ and $x-1 \geq 0$, hence $f(x) = (x+1) + (x-1) = 2x.$

As a conclusion, $$f(x) = \begin{cases} -2x &\text{ if } x \leq -1\\ 2 &\text{ if } -1 < x \leq 1\\ 2x&\text{ if } x > 1 \end{cases}.$$

This should help you in finding the critical values.