Problem Set - Mean value theorem

## Problem Set - Mean value theorem

 Use the mean value theorem to prove the following inequalities: (a)   $|\mathrm{sin}x-\mathrm{sin}y|\le |x-y|$ for all $x,y\in ℝ$. (b)   $|\mathrm{log}x-\mathrm{log}y|\le \frac{1}{2}|x-y|$ for all $x,y\in \left[2,\infty \right)$, (c)   $|{\left(x+1\right)}^{1/5}-{x}^{1/5}|\le {\left(5{x}^{4/5}\right)}^{-1}$ for all $x\in {ℝ}_{>0}$. Use the mean value theorem to show that if a function $f:\left(a,b\right)\to ℝ$ is differentiable with $f\prime \left(x\right)>0$ for all $x$ then $f$ is strictly increasing. Use the mean value theorem to show that if a function $f:\left(a,b\right)\to ℝ$ is twice differentiable with $f\prime \prime \left(x\right)>0$ then $f$ is strictly convex. ( $f$ is strictly convex if $f\left(tx+\left(1-t\right)y\right) for all $x,y\in \left(a,b\right)$ and $t,y\in \left(0,1\right)$.