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\centerline{MATH 133 -- Calculus 1 with Fundamentals}
\centerline{Second Derivative and Concavity}
\centerline{November 27, 2017}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
We say $f$ (or the graph $y = f(x)$) is \emph{concave up} on an interval if $f'$
is increasing on that interval, and similarly, $f$ or its graph is \emph{concave down}
of $f'$ is decreasing on that interval.  Combined with our results from before Thanksgiving, this
says:
\begin{itemize}
\item If $f''(x) > 0$ on an interval, then $f$ or its graph is concave up on that
interval
\item If $f''(x) < 0$ on an interval, then $f$ or its graph is concave down on 
that interval
\item A point $(c,f(c))$ on the graph of $f$ where the concavity changes is 
called a \emph{point of inflection} of $f$.
\end{itemize}
The notion of concavity can also be used to state a second method for determining
whether critical points are local maxima or local minima, called the Second Derivative 
Test:
\begin{theorem} (Second Derivative Test)  Let $f$ be differentiable on some open interval
containing a critical point $c$.  In addition, assume $f''(c)$ exists.
\begin{itemize}
\item[(a)]  If $f''(c) > 0$, then $f(c)$ is a local minimum 
\item[(b)]  If $f''(c) < 0$, then $f(c)$ is a local maximum 
\item[(c)]  If $f''(c) = 0$, there is no conclusion.
\end{itemize}
\end{theorem}
In the last case here, $f$ could have either a local maximum or a local minimum, or neither, 
so no conclusion is possible.  \emph{Technical Comment}:
In the other cases, the intuition is that $f'$ should be 
increasing or decreasing on an interval containing $c$ depending on the sign of 
$f''(c) = (f')'(c)$, 
so that (a) corresponds to a case where the graph is concave up at $c$ and (b) corresponds
to a case where the graph is concave down at $c$.
This would follow, for instance, if we knew (in addition) that $f''$ was continuous on some
interval containing $c$.  
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\noindent
{\it Questions}
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\noindent
\begin{enumerate}
\item[1.]  Consider the graph $f(x) = x^3 - 3x^2 + 2x$ on the interval $[-1,3]$ from 
last time (the plot is also on the back of this sheet).  
Find the intervals where $f$ is concave up and the intervals where $f$ is 
concave down.  How many points of inflection are there on this graph and where
are they located?
\item[2.]  Consider $f(x) = x^2 e^{-x}$.
\begin{enumerate}
\item[(a)]  Compute $f'(x)$ and find all critical points.
\item[(b)]  Determine the sign of $f'(x)$ on each interval between successive critical points, and
use that to classify the critical points as local maxima or local minima by the First Derivative
Test.
\item[(c)]  Now compute $f''(x)$ and check your answers in (b) by using the Second
Derivative Test.
\item[(d)]  Determine all points of inflection of $f$.
\end{enumerate} 
\item[3.]  Repeat question 2 for  $f(x) =  2x^4 - 3x^2 + 2$. 
\end{enumerate}
\begin{figure}
\begin{center}
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\caption{Plot for question 1}
\end{center}
\end{figure}
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