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\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
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\newtheorem{proposition}[theorem]{Proposition}
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\lhead{\color{blue} \large Lecture Notes}
\chead{\color{black} \Large Concavity Behavior}
\rhead{ page \thepage}
\cfoot{}
\rfoot{\small   Last revised: November 29, 2016}
\lfoot{\small   \copyright \;  Hidegkuti, 2014}
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\textbf{Definition:} \ A function $f$ is \textbf{concave up} on an interval $%
I$ if for all $a$,$b$ in $I,$ the secant line segment connecting $\left(
a,f\left( a\right) \right) $ and $\left( b,f\left( b\right) \right) $ is
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$\ \ \ \ \ \ \ \ \ \ \ \ \ \ f$ is concave up increasing \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ $f$ is concave up decrasing\bigskip \bigskip

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\textbf{Definition:} \ A function $f$ is \textbf{concave down} on an
interval $I$ if for all $a$,$b$ in $I,$ the secant line segment connecting $%
\left( a,f\left( a\right) \right) $ and $\left( b,f\left( b\right) \right) $
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$\ \ \ \ \ \ \ \ \ \ \ \ \ \ f$ is concave down increasing \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ $f$ is concave down decrasing\bigskip \bigskip 
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Let us precisely express what it means for a function $f$ to be concave up.
\ Suppose that $a<b$ and $x$ is any number with $a<x<b$. \ \ If we denote
the line connecting $\left( a,f\left( a\right) \right) $ and $\left(
b,f\left( b\right) \right) $ by $g,$ then we have that $f\left( x\right)
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\ \ 

Let us write the equation of the line $g\left( x\right) $. \ Since it is
connecting the points $\left( a,f\left( a\right) \right) $ and $\left(
b,f\left( b\right) \right) $, its slope is $m=\dfrac{f\left( b\right)
-f\left( a\right) }{b-a}$. \ Using the point-slope form with $\left(
a,f\left( a\right) \right) $, we get%
\begin{eqnarray*}
y-f\left( a\right) &=&m\left( x-a\right) \\
y &=&m\left( x-a\right) +f\left( a\right) =\dfrac{f\left( b\right) -f\left(
a\right) }{b-a}\left( x-a\right) +f\left( a\right)
\end{eqnarray*}%
and so we have that $g\left( x\right) =\dfrac{f\left( b\right) -f\left(
a\right) }{b-a}\left( x-a\right) +f\left( a\right) $.%
\begin{eqnarray*}
f\left( x\right) &\leq &g\left( x\right) \\
f\left( x\right) &\leq &\dfrac{f\left( b\right) -f\left( a\right) }{b-a}%
\left( x-a\right) +f\left( a\right) \text{ \ \ \ \ \ \ \ \ \ add }f\left(
a\right) \\
f\left( x\right) -f\left( a\right) &\leq &\dfrac{f\left( b\right) -f\left(
a\right) }{b-a}\left( x-a\right) \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
divide by }x-a \\
\dfrac{f\left( x\right) -f\left( a\right) }{x-a} &\leq &\dfrac{f\left(
b\right) -f\left( a\right) }{b-a}
\end{eqnarray*}%
The last inequality compares two slopes, that of line segment between $%
\left( a,f\left( a\right) \right) $ and $\left( x,f\left( x\right) \right) $%
, and that of the line segment between $\left( a,f\left( a\right) \right) $
and $\left( b,f\left( b\right) \right) $. \ This is not a proof, but it
suggests that in case of a concave up function, if we step to the right, the
slope of the secant line increases. \ The limit of the slopes of these same
secant lines is the derivative $f^{\prime }$.\bigskip

So this computation suggests that in case of a concave up function $f$, the
derivative, $f^{\prime }$ is increasing.\bigskip

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\textbf{Theorem:} \ A function $f$ is concave up if its derivative, $%
f^{\prime }$ is increasing. \ $f$ is concave down if $f^{\prime }$ is
decreasing.%
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\medskip \bigskip

It easily follows that an increasing $f^{\prime }$ means that the second
derivative, $f^{\prime \prime }$ is positive.\bigskip \bigskip

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\begin{eqnarray*}
f\text{ concave up \ \ } &\Longleftrightarrow &\text{ \ \ \ }f^{\prime }%
\text{ \ increasing \ \ }\Longleftrightarrow \text{ \ \ }f^{\prime \prime }%
\text{ \ positive} \\
f\text{ concave down \ \ } &\Longleftrightarrow &\text{ \ \ \ }f^{\prime }%
\text{ \ decreasing \ \ }\Longleftrightarrow \text{ \ \ }f^{\prime \prime }%
\text{ \ negative}
\end{eqnarray*}%
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\medskip

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\textbf{Definition:} \ A point $\left( x,f\left( x\right) \right) $ is a 
\textbf{point of inflection} if $x$ separates two intervals on which $f$
behaves differently with respect to concavity.%
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\pagebreak

\begin{center}
{\normalsize {\LARGE Practice Problems\bigskip } }
\end{center}

In case of each of the following functions given, determine the intervals
upon which the function is concave up and concave down. \ State the $x-$%
coordinate of all points of inflection.\bigskip

\bigskip

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\begin{enumerate}
\item $f\left( x\right) =x^{4}-6x^{2}+x-3$

\item $f\left( x\right) =x^{3}+6x^{2}-3x-1$

\item $f\left( x\right) =x^{4}-10x^{3}+8x+1$

\item $f\left( x\right) =2x^{4}-8x^{3}-36x^{2}-120x+80$

\item $f\left( x\right) =\sin x$
\end{enumerate}

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{\normalsize {\LARGE \bigskip }}\pagebreak

\begin{center}
{\normalsize {\LARGE Answers\bigskip } }
\end{center}

\begin{enumerate}
\item 
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$f^{\prime \prime }\left( x\right) =12\left( x^{2}-1\right) =12\left(
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$f$ is concave up on $\left( -\infty ,-1\right) $ and on $\left( 1,\infty
\right) $

concave down on $\left( -1,1\right) $

points of inflection at $x=-1$ and $1$%
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\item 
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$f^{\prime \prime }\left( x\right) =6\left( x+2\right) $\FRAME{dtbpFX}{%
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$f$ is concave down on $\left( -\infty ,-2\right) $

and concave up on $\left( -2,\infty \right) $

point of inflection at $x=-2$%
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\item 
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$f^{\prime \prime }\left( x\right) =12x\left( x-5\right) $\FRAME{dtbpFX}{%
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$f$ is concave up on $\left( -\infty ,0\right) $ and on $\left( 5,\infty
\right) $

concave down on $\left( 0,5\right) $

points of inflection at $x=0$ and $5$%
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\item 
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$f^{\prime \prime }\left( x\right) =24x^{2}-48x-72=24\left( x+1\right)
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$f$ is concave up on $\left( -\infty ,-1\right) $ and on $\left( 3,\infty
\right) $

concave down on $\left( -1,3\right) $

points of inflection at $x=-1$ and $3$%
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\item $f^{\prime \prime }\left( x\right) =-\sin x=-f\left( x\right) $

So $\sin x$ is concave up where it is negative and concave down where it is
positive. \ All of its zeroes are points of inflection.

Concave up: \ \ when $\pi +2k\pi <x<2\pi +2k\pi $ where $k$ is an integer

Concave down: \ when \ $2k\pi <x<\pi +2k\pi $ \ where $k$ is an integer

points of inflection: at $x=k\pi $ where $k$ is an integer
\end{enumerate}

{\small 
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For more documents like this, visit our page at\
https://teaching.martahidegkuti.com and click on Lecture Notes. \ E-mail
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\end{document}