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This problem shows an interesting application of the mean value theorem.

\begin{enumerate}
\item Use algebra to verify that 
$(1+(x-1)^2)(1+(x+1)^2) = 4 + x^4$.

\item Differentiate $\displaystyle \arctan(x+1) - \arctan(x-1)
+ \arctan\!\left(\!{{x^2}\over 2}\!\right)$. Use the result of a) and
a {\bf m}ostly {\bf v}ital {\bf t}hing to show that $\displaystyle
\arctan (x+1) - \arctan(x-1) + \arctan\!\left(\!{{x^2}\over
2}\!\right)$ is always $\displaystyle{\pi\over 2}$.
\end{enumerate}

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