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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Problem Set 8, {\it Due:} 11/18}
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I.  From Abbott: 4.5.2, 4.5.3 (of course, you are supposed
to {\it prove} your assertion!), 4.5.7, 5.2.3, 5.2.4.
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{\it Additional Problems}
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\noindent
II.  
\item{A)}  Show that if $f(x)$ is any polynomial of {\it odd}
degree, then $f$ has a real root (that is a real number
$c$ where $f(c) = 0$).
\item{B)}  Suppose that $f(x)$ is a polynomial of {\it even}
degree and that there exist $a \ne b$ in ${\bf R}$ such
that $f(a) < 0 < f(b)$.  Show that $f$ has at least two 
distinct real roots.
\item{C)}  True or False:  Any polynomial of even degree 
with a real root has at least two distinct real roots.
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\noindent
III.  Show that the equation $x^2 = 3^x$ has at least one real
solution.
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\noindent
IV.  
\item{A)}  Suppose $f$ is continuous on $[0,1]$ with 
$f(0) < 0$ and $f(1) > 1$.  Prove that there is at least
one point $c \in (0,1)$ where $f(c) = c^2$.
\item{B)}  Generalize your reasoning from part A to 
show that if $f$ is as before and $g$ is any continuous
function on $[0,1]$ with $g(0) \ge 0$, $g(1) \le 1$, 
the there is at least one point $c \in (0,1)$ where $f(c) = g(c)$.
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