Problem Set - Rolle's Theorem and the 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

Rolle's Theorem and the Mean Value Theorem

	State Rolle's theorem and draw a picture which illustrates the statement of the theorem.
	State the mean value theorem and draw a picture which illustrates the statement of the theorem.
	Explain why Rolle's theorem is a special case of the mean value theorem.
	Verify Rolle's theorem for the function $f\left(x\right)=\left(x-1\right)\left(x-2\right)\left(x-3\right)$ on the interval $\left[1,3\right]$.
	Verify Rolle's theorem for the function $f\left(x\right)={\left(x-2\right)}^2{\left(x-3\right)}^6$ on the interval $\left[2,3\right]$.
	Verify Rolle's theorem for the function $f\left(x\right)=\sin x-1$ on the interval $\left[\pi /2,5\pi /2\right]$.
	Verify Rolle's theorem for the function $f\left(x\right)=e^{-x}\sin x$ on the interval $\left[0,\pi \right]$.
	Verify Rolle's theorem for the function $f\left(x\right)=x^3-6x^2+11x-6$.
	Let $f\left(x\right)=1-x^{2/3}$. Show that $f\left(-1\right)=f\left(1\right)$ but there is no number $c$ in the interval $\left[-1,1\right]$ such that ${\displaystyle {\left.\frac{df}{dx}\right|}_{x=c}=0}$. Why does this not contradict Rolle's theorem?
	Let $f\left(x\right)={\left(x-1\right)}^{-2}$. Show that $f\left(0\right)=f\left(2\right)$ but there is no number $c$ in the interval $\left[0,2\right]$ such that ${\displaystyle {\left.\frac{df}{dx}\right|}_{x=c}=0}$. Why does this not contradict Rolle's theorem?
	Discuss the applicability of Rolle's theorem when $f\left(x\right)=\left(x-1\right)\left(2x-3\right)$ on the interval $1\le x\le 3$.
	Discuss the applicability of Rolle's theorem when $f\left(x\right)=2+{\left(x-1\right)}^{2/3}$ on the interval $0\le x\le 2$.
	Discuss the applicability of Rolle's theorem when $f\left(x\right)=⌊x⌋$ on the interval $-1\le x\le 1$.
	At what point on the curve $y=6-{\left(x-3\right)}^2$ on the interval $\left[0,6\right]$ is the tangent to the curve parallel to the $x$-axis?
	Show that the equation $x^5+10x+3=0$ has exactly one real solution.
	Show that a polynomial of degree three has at most three real roots.
	Verify the mean value theorem for the function $f\left(x\right)=x^{2/3}$ on the interval $\left[0,1\right]$.
	Verify the mean value theorem for the function $f\left(x\right)=\ln x$ on the interval $\left[1,e\right]$.
	Verify the mean value theorem for the function $f\left(x\right)=x$ on the interval $\left[\mathrm{a},\mathrm{b}\right]$, where $a$ and $b$ are constants.
	Verify the mean value theorem for the function $f\left(x\right)=l{x}^2+mx+n$ on the interval $\left[\mathrm{a},\mathrm{b}\right]$, where $l,m,n,a$ and $b$ are constants.
	Show that the mean value theorem is not applicable to the function $f\left(x\right)=\left|x\right|$ in the interval $\left[-1,1\right]$.
	Show that the mean value theorem is not applicable to the function $f\left(x\right)=1/x$ in the interval $\left[-1,1\right]$.
	Find the points on the curve $y=x^3-3x$ where the tangent is parallel to the chord joining $\left(1,-2\right)$ and $\left(2,2\right)$.
	If $f\left(x\right)=x\left(1-\ln x\right)$, $x>0$, show that $\left(a-b\right)\ln c=b\left(1-\ln b\right)-a\left(1-\ln a\right)$, for some $c\in \left[a,b\right]$ where $0<a<b$.

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)