Single Variable Calculus, Chapter 3, 3.3, Section 3.3, Problem 92

On what values of $x$ does the function $f(x) = |x-1| + |x+2|$ differentiable? Find a formula for $f'$ and sketch its graph.




By referring to the graph and by using the definition of absolute value, we can deduce $f(x)$ as


$
f(x) = \left\{
\begin{array}{c}
2x +1 & \text{for} & x \geq 1\\
3 & \text{for} & -2 < x < 1\\
-2x-1 & \text{for} & x \leq -2
\end{array}\right.
$


Now, we can find the formula $f'(x)$ by taking the derivative of the Piecewise Function $f(x)$


$
f'(x) = \left\{
\begin{array}{c}
2 & \text{for} & x \geq 1\\
0 & \text{for} & -2 < x < 1\\
-2 & \text{for} & x \leq -2
\end{array}\right.
$





By referring to the graph, we can conclude that $f(x)$ is differentiable everywhere except
at $x=-2$ and $x=1$ because of jump discontinuity that makes its limit from left and right unequal.