Chapter 4.8, Problem 28ES

(a) Use the Mean-Value Theorem to show that if $f$ is differentiable on an open interval, and if $\left|f^{\prime }\left(x\right)\right|\ge M$ for all values of $x$ in the interval, then $\left|f\left(x\right)-f\left(y\right)\right|\ge M\left|x-y\right|$ for all values of $x$ and $y$ in the interval.

(b) Use the result in part (a) to show that $\left|\tan x-\tan y\right|\ge \left|x-y\right|$ for all values of $x$ and $y$ in the interval $\left(-\pi /2,\pi /2\right)$ .

(c) Use the result in part (b) to show that $\left|\tan x+\tan y\right|\ge \left|x+y\right|$ for all values of $x$ and $y$ in the interval $\left(-\pi /2,\pi /2\right)$ .