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\title{Math 120 Homework 5}
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\begin{itemize}
\item Due Friday October 21 at start of class.
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\item If your homework is longer than one page, {\bf staple} the pages together, and put your name on each sheet of paper. {\bf Homework that is not stapled will loose 1 point.}
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\item Each homework problem should be correct as stated. Occasionally, however, I might screw something up and give you an impossible homework problem. If you believe a problem is incorrect, please email me. If you are right, the first person to point out an error will get +1 on that homework, and I will post an updated version. 
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\subsection*{Newton iteration}

\noindent {\bf 1.} (5 points) Use Newton's method with an initial guess $x_1=1$ to approximate the (positive) fourth root of 2, and show the computations. It's enough to iterate 4 times, and feel free to use a calculator.

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\noindent {\bf 2.} Let $f(x) = x^3 - 2x + 2$.
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\noindent {\bf a.} (2 points) Prove that there is at least one point $x\in\RR$ for which $f(x)=0$ (i.e. $f$ has at least one root)
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\noindent {\bf b.} (2 points) Let $x_1=0$. Use Newton's method to compute the first few iterates $x_2,\ x_3,\ x_4$. 
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\noindent {\bf c.} (5 points) Prove that for each $n=1,2,3,\ldots$, $|f(x_n)|\geq 1$, where $x_0,x_1,$ etc are the points from part b. (you could do a proof by induction, but you don't have to). 

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\noindent Remark: problem 2c shows that if you choose a bad starting guess $x_1$, then Newton's method might not find a root of $f$.

\subsection*{Intermediate Value Theorem \& Rolle's theorem}

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\noindent {\bf 3.} (5 points) Let $f$ be a function that is continuous on the interval $[0,1],$ and suppose $0\leq f(x)\leq 1$ for all $x\in[0,1].$ Prove that there is at least one point $c\in [0,1]$ with $f(c)=c$.

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% \noindent {\bf 4.} (5 points) Let $f:[a,b]\to\RR$. Suppose that $f$ is continuous on $[a,b]$ and differentiable at every point of $(a,b)$. Suppose that $f(a)=f(b)=0$. Prove that for every $t\in\RR$, there is a point $c\in(a,b)$ with $f^\prime(c)=tf(c)$. 

% Hint: apply Rolle's theorem to some function $g(x)f(x)$ for some function $g$ depending on $t$.

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\noindent {\bf 4.} (5 points) Use Rolle's theorem to prove that for each value of $b$, if we define $f(x) = x^3-3x+b$ then there is at most one point $x\in [-1,1]$ with $f(x)=0$.

\subsection*{Mean Value theorem}
(This will be covered in lecture on Monday)
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\noindent {\bf 5.} (5 points) Let $f$ be a function that is continuous on the interval $[a,b]$ and differentiable on the interval $(a,b)$. Suppose that there exists $M>0$ so that for each $x\in (a,b)$, we have $|f^\prime(x)|\leq M$. Prove that $|f(b)-f(a)|\leq M|b-a|$.

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