Problem Set - Mean value theorem

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 7 December 2009

Mean value theorem

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

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)