Problem Set - Rolle's Theorem and the Mean Value Theorem

## Problem Set - 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 $f\left(x\right)=\mathrm{sin}x-1$ on the interval $\left[\pi /2,5\pi /2\right]$. Verify Rolle's theorem for the function $f\left(x\right)={e}^{-x}\mathrm{sin}x$ on the interval $\left[0,\pi \right]$. Verify Rolle's theorem for the function $f\left(x\right)={x}^{3}-6{x}^{2}+11x-6$. 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 ${\frac{df}{dx}|}_{x=c}=0$. 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 ${\frac{df}{dx}|}_{x=c}=0$. 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 $f\left(x\right)=\mathrm{ln}x$ on the interval $\left[1,e\right]$. Verify the mean value theorem for the function $f\left(x\right)=x$ on the interval $\left[a,b\right]$, 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 $\left[a,b\right]$, 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 $f\left(x\right)=x\left(1-\mathrm{ln}x\right)$, $x>0$, show that $\left(a-b\right)\mathrm{ln}c=b\left(1-\mathrm{ln}b\right)-a\left(1-\mathrm{ln}a\right)$, for some $c\in \left[a,b\right]$ where $0.