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%  Extreme Values: The Second Derivative Test           %
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\problemgrouptitle{Extreme Values: The Second Derivative Test}

\begin{problemblock}{Determine the \relative maxima and minima of the following functions and their locations. Use the Second Derivative Test.}

\item $f(x)=x^3-12x+1$
\answer{\Relative maximum: $f(-2)=17$; \Relative mininum: $f(2)=-15$}

\item $f(x)=\cos(2x)-4\sin(x)$ on the interval $(-\pi,\pi)$
\answer{ \Relative maximum at $\displaystyle f\left(-\frac{\pi}{2}\right)=3$; \Relative minimum: $\displaystyle f\left(\frac{\pi}{2}\right)=-5$;}

\end{problemblock}

\item{Can you use the Second Derivative Test to categorize the critical number $x=0$ of the function $f(x)=\sin^4x$\,?  Explain why or why not.}
\answer{No since $f''(x)=12\cos^2x\sin^2x-4\sin^4x$ vanishes at $x=0$ so the test is inconclusive.  The first derivative test shows that $x=0$ is a the location of a local minimum of $f$\,.}
