Answer
$f$ is not differentiable at $x=1$ and $x=5$
Work Step by Step
There are 3 cases at which a graph is not differentiable at a point:
- There is a corner (a pointy shape) at a point in the graph (a pointy point cannot have any tangent lines there)
- The graph is not continuous at that point (differentiable means continuous)
- There is a vertical tangent line at that point in the graph (since $f'(x)=\infty$)
In this graph, there are 2 points at which $f$ is not differentiable there:
- At $x=5$, the line of the graph is continuous, but quite vertical. So the tangent line there would be vertical, which means the derivative of $f$ there would be $\infty$. So $f$ is not differentiable there.
- At $x=1$, the graph is not continuous. Instead, the graph approaches infinity. Therefore, $f$ is not differentiable there.
