# Chapter 2 - Review - Exercises: 47

$f$ is not differentiable at the points where $x=-4$, $x=-1$, $x=2$ and $x=5$.

#### Work Step by Step

There are 3 cases at which a function fails to be differentiable at a point: 1) Its graph has a corner or a kink there, since there is no tangent line at a corner or a kink. 2) The graph of the function is not continuous at that point. Not continuous means not differentiable. 3) The graph has a vertical tangent line at that point. Because a vertical tangent line would lead to $\lim\limits_{x\to a}|f'(x)|=\infty$ In this graph, there are 4 points at which function $f$ is not differentiable: 1) The point where $x=-4$, since the graph is not continuous there. 2) The point where $x=-1$, since the graph has a corner / a kink there. 3) The point where $x=2$, since the graph is again not continuous there. 4) The point where $x=5$, since at that point, only a vertical tangent line can be drawn.

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