Chapter 4.8, Problem 43ES

Use the Mean-Value Theorem to prove the following result: The graph of a function $f$ has a point of vertical tangency at $\left(x_0,f\left(x_0\right)\right)$ if $f$ is continuous at $x_0$ and $f^{\prime }\left(x\right)$ approaches either $+\infty$ or $-\infty$ as $x\to x_0^{+}$ and as $x\to x_0^{-}$ .