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\textbf{Math 165: Honors Calculus I}

\textbf{Assignment 21}\quad \emph{Nov. 8, 1999}

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\begin{enumerate}
\item Let $f$ be continuous and positive on $[a,b]$. Prove that there
exists a constant $c>0$ such that $f(x)\ge c$ for all $x\in[a,b]$.

\answerspace

\problemspace
\item Give an example of a function $g$ on $[0,1]$ such that $g(x) > 0$ for
all $x\in[0,1]$, but there is no constant $c > 0$ such that $g(x)\ge c$ for
all $x\in[0,1]$.

\answerspace

\problemspace
\item Let $f$ be continuous on $[a,b]$ and let $(p,q)$ be any point in the
plane. Prove that there exists a point on the graph of $f$ that is closest
to $(p,q)$. (Hint: What is the distance from $(p,q)$ to $(x,f(x))$?)

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