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\noindent {\bf Question}

\noindent Let $f$ be a function which is continuous on the closed
interval $[a,b]$, where $a < b$.  Suppose that $f(b) < f(a)$.
Determine whether there exists a point $c$ in the open interval
$(a,b)$ so that $f(c) = c$.

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\noindent {\bf Answer}

\noindent Not necessarily: take $f(x) =100 -x$ on the interval
$[a,b] = [0,1]$.  Then, $f(1) =99 < f(0) =100$, but there are no
solutions to $x =100 -x$ in the interval $[0,1]$.  (The only
solution is at $x =50$.)



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