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\title{Homework 7 for 18.100B/C, Fall 2010\\
Due Thursday 4 November}
%lasteqno BASE@  2
%\author{Richard Melrose}
%\address{Department of Mathematics, Massachusetts Institute of Technology}
%\email{rbm@math.mit.edu}
%\dedicatory{\sethouse\datverp}
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As usual due in 2-108 or lecture by 11AM or by email before 5PM.

\begin{enumerate}
\item Let $K_1,$ $K_2\subset M$ be two compact subsets of a
metric space $(M,d).$ Show that there exist points $p\in K_1$ and $q\in
K_2$ such that
$$
d(p,q)=\sup_{y\in K_2}\inf_{x\in K_1}d(x,y).
$$
Define
$$
D(K_1,K_2)=\max\left(\sup_{y\in K_2}\inf_{x\in K_1}d(x,y),
\sup_{x\in K_1}\inf_{y\in K_2}d(x,y)\right).
$$
Show that $D$ defines a metric on the collection of (non-empty) compact
subsets of $M.$

\item If $f:[a,b]\longrightarrow \mathbb{R}$ is
differentiable (where $a<b)$ and $f'(x)\not=0$ for all $x\in(a,b)$
show that $f(b)\not=f(a).$

\item Rudin Chap 5 No 4. If $C_i$ for $0\le i\le n$ are real constants such that
$$
C_0+\frac{C_1}2+\frac{C_2}3+\cdots+\frac{C_{n-1}}n+\frac{C_n}{n+1}=0
$$
show that the equation
$$
C_0+C_1x+C_2x^2+\cdots+C_nx^n=0
$$
has at least one real solution $x$ in the interval $(0,1).$

\item Suppose $f:\mathbb{R}\longrightarrow \mathbb{R}$ is
  differentiable and that $f'(x)\not=1$ for all $x\in\mathbb{R}$ show that
  there can be \emph{at most one} $x\in\mathbb{R}$ such that $f(x)=x$ (`a
  fixed point of $f$').

\item Rudin Chap 5 No 15. Suppose $a\in\mathbb{R},$ $f$ is a
  twice-differentiable real function on $(a,\infty)$ and $M_0,$ $M_1$ and
  $M_2$ are the suprema of $|f(x)|,$ $|f'(x)|$ and $|f''(x)|$ on
  $(a,\infty)$ (so all are assumed to be finite). Prove that 
%
\begin{equation*}
M_1^2\le 4M_0M_2.
\label{BASE.1}\end{equation*}
%
[There is a hint in Rudin, namely Taylor's theorem shows that given any
  $h>0$ and $x\in(a,\infty)$ there is $\xi\in(x,x+2h)$ such that 
%
\begin{equation*}
f'(x)=\frac1{2h}(f(x+2h)-f(x))-hf''(\xi).
\label{BASE.2}\end{equation*}
%
Use this to show that $|f'(x)|\le hM_2+\frac{M_0}h.$ For what value of $h$
is the RHS smallest?]

\end{enumerate}

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