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%MODULE 1: Optimization
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Extrema
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A common problem is one of find 
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hilite
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 extrema
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}
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, which are 
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 minimum
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 and 
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 maximum
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 values of a function.
 Methods of approximating an exremum are possible, but generally unsatisfactory.
 Derivatives give us an way to compute extrema in a very efficienty manner.
 
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Sample problems
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Find the maximum and minimum values of 
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 on the interval 
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Where does the graph of a function change direction from increasing to decreasin
g or vice versa? 
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A manufacturer's profit depends on the volume of sales: if not enough units
 are sold, income is insufficient to cover the costs, but if too many units
 are sold, the costs of making too much of a product grow faster than income.
 The profit function for the manufacturer Widgets-R-Us is known to be 
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, where 
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 is the number of units sold, in thousands.
 What number of units sold should the manufacturer target in order to maximize
 profit? 
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%1
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SAGE worksheets
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Extrema?
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An 
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 extremum
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 is either a minimum or a maximum value of a function.
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 at 
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Extrema: Approximation
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 values of 
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<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size scriptsize
7676.43 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size scriptsize
23847.66 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size scriptsize
72927.76 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size scriptsize
220264.66 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\size default

\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
\begin_inset VSpace -2ex
\end_inset


\end_layout

\begin_layout Standard
\noindent
It 
\emph on
looks
\emph default
 as if the minimum and maximum values are approximately 
\begin_inset Formula $\ec-0.37$
\end_inset

 and 
\begin_inset Formula $\ec220264.66$
\end_inset

.
 They appear at 
\begin_inset Formula $\ec x=-1$
\end_inset

 and 
\begin_inset Formula $\ec x=10$
\end_inset

.
 How can we be 
\emph on
certain
\emph default
 about this, and how can we find the 
\emph on
precise
\emph default
 minimum and maximum?
\end_layout

\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Extrema: Certainty of computation
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Think about what must happen to the 
\begin_inset Formula $y$
\end_inset

 values around a minimum 
\begin_inset Formula $\ec m=f(c)$
\end_inset

 at 
\begin_inset Formula $\ec x=a$
\end_inset

.
 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/x_ex_left_decreasing.png
	lyxscale 20
	scale 20

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Itemize
The 
\begin_inset Formula $y$
\end_inset

 values should be larger than 
\begin_inset Formula $\ec m$
\end_inset

, because 
\begin_inset Formula $\ec m$
\end_inset

 is a minimum.
 
\end_layout

\begin_layout Itemize
If 
\begin_inset Formula $\ec x<c$
\end_inset

 but close to 
\begin_inset Formula $\ec c$
\end_inset

, the slopes of the secant lines will be negative:
\begin_inset Formula 
\[
\ec\frac{f(x)-f(c)}{x-c}=\frac{\mathrm{positive}}{\mathrm{negative}}
\]

\end_inset

so the slopes of the tangent lines should also be negative.
 Thus the derivative should be negative for 
\begin_inset Formula $\ec x<c$
\end_inset

.
 
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Extrema: Certainty of computation
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Think about what must happen to the 
\begin_inset Formula $y$
\end_inset

 values around a minimum 
\begin_inset Formula $\ec m=f(c)$
\end_inset

 at 
\begin_inset Formula $\ec x=a$
\end_inset

.
 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/x_ex_right_decreasing.png
	lyxscale 20
	scale 20

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Itemize
The 
\begin_inset Formula $y$
\end_inset

 values should be larger than 
\begin_inset Formula $\ec m$
\end_inset

, because 
\begin_inset Formula $\ec m$
\end_inset

 is a minimum.
 
\end_layout

\begin_layout Itemize
If 
\begin_inset Formula $\ec x>c$
\end_inset

 but close to 
\begin_inset Formula $\ec c$
\end_inset

, the slopes of the secant lines will be positive:
\begin_inset Formula 
\[
\ec\frac{f(x)-f(c)}{x-c}=\frac{\mathrm{positive}}{\mathrm{positive}}
\]

\end_inset

so the slopes of the tangent lines should also be positive.
 Thus the derivative should be positive for 
\begin_inset Formula $\ec x>c$
\end_inset

.
 
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Extrema: Certainty of computation
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The preceding argument implies the following result.
\end_layout

\begin_layout Theorem
If the function 
\begin_inset Formula $\ec f(x)$
\end_inset

 is continuous on the interval 
\begin_inset Formula $\ec[a,b]$
\end_inset

, it has a 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 minimum
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 at 
\begin_inset Formula $\ec x=c$
\end_inset

 when 
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\ec f'(x)<0$
\end_inset

 for all 
\begin_inset Formula $\ec x\in[a,c)$
\end_inset

, and 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec f'(x)>0$
\end_inset

 for all 
\begin_inset Formula $\ec x\in(c,b]$
\end_inset

.
 
\end_layout

\end_deeper
\begin_layout Standard
\begin_inset VSpace bigskip
\end_inset

A similar theorem exists for a 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 maximum
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

, but reversing the inequalities: 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec f'(x)>0$
\end_inset

 for all 
\begin_inset Formula $\ec x\in[a,c)$
\end_inset

, and 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec f'(x)<0$
\end_inset

 for all 
\begin_inset Formula $\ec x\in(c,b]$
\end_inset

.
 
\end_layout

\begin_layout BeginFrame
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

How to find extrema
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
If 
\begin_inset Formula $\ec f(x)$
\end_inset

 is continuous on the interval 
\begin_inset Formula $\ec[a,b]$
\end_inset

, you can find extrema in the following way: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{method}
\end_layout

\end_inset

 
\end_layout

\begin_layout Itemize
Find all places where 
\begin_inset Formula $\ec f'(x)=0$
\end_inset

 or 
\begin_inset Formula $\ec f'(x)$
\end_inset

 does not exist.
 These 
\begin_inset Formula $x$
\end_inset

 values are called 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 critical points
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
 
\end_layout

\begin_layout Itemize
Find the 
\begin_inset Formula $\ec y$
\end_inset

 values at each critical points 
\emph on
and
\emph default
 at the endpoints.
 
\end_layout

\begin_layout Itemize
The smallest 
\begin_inset Formula $\ec y$
\end_inset

 value found is the minimum, and the largest 
\begin_inset Formula $\ec y$
\end_inset

 value found is the maximum.
 
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{method}
\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

First example
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout OverlayArea
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\backslash
textwidth}{6in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Recall the first example: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

 Find the maximum and minimum values of 
\begin_inset Formula $\ec xe^{x}$
\end_inset

 on the interval 
\begin_inset Formula $\ec[-10,10]$
\end_inset

.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset

 How would we find the extrema?
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
Let 
\begin_inset Formula $\ec f(x)=xe^{x}$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset VSpace bigskip
\end_inset


\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<2-3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
First we check where 
\begin_inset Formula $\ec f'(x)$
\end_inset

 is zero or undefined:
\begin_inset Formula 
\begin{align*}
\ec f'(x) & {\ec=xe^{x}+e^{x}=(x+1)e^{x}}.
\end{align*}

\end_inset


\end_layout

\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
This derivative is never undefined, and is zero when 
\begin_inset Formula $\ec x=-1$
\end_inset

:
\begin_inset Formula 
\[
{\ec(x+1)e^{x}=0}\mathrm{\ when\ }{\ec x+1=0}\mathrm{\ or\ }{\ec e^{x}=0}.
\]

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Remember that 
\begin_inset Formula $e^{x}$
\end_inset

 is never zero, so 
\begin_inset Formula $x+1=0$
\end_inset

 is the only equation that can give us a solution.
 Thus 
\begin_inset Formula $x=-1$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\end_deeper
\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Itemize
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dims
\end_layout

\end_inset

 There is one critical point at 
\begin_inset Formula $x=-1$
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
 
\end_layout

\begin_layout Standard
Now find the 
\begin_inset Formula $y$
\end_inset

 values at the critical point and at the endpoints:
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\align center
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ec
\end_layout

\end_inset


\begin_inset Tabular
<lyxtabular version="3" rows="2" columns="4">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $x$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-10 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-1 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
10 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $f(x)$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-10e^{-10}\approx0.00$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-e^{-1}\approx-0.37$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $10e^{10}\approx220264.66$
\end_inset

 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The minimum is 
\begin_inset Formula $\ec-e^{-1}\approx-0.37$
\end_inset

 at 
\begin_inset Formula $\ec x=-1$
\end_inset

.
 The maximum is 
\begin_inset Formula $\ec10e^{10}\approx220264.66$
\end_inset

 at 
\begin_inset Formula $\ec x=10$
\end_inset

.
\end_layout

\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Second Example
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Now we turn to the second example: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

 Where does the graph of a function change direction from increasing to
 decreasing or vice versa? 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset


\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
The graph of a function 
\emph on
increases
\emph default
 when, if the 
\begin_inset Formula $x$
\end_inset

 values are increasing, so are the 
\begin_inset Formula $y$
\end_inset

 values.
 The graph of a function 
\emph on
decreases
\emph default
 when, if the 
\begin_inset Formula $x$
\end_inset

 values are increasing, the 
\begin_inset Formula $y$
\end_inset

 values are decreasing.
 This is the same as finding the extrema.
 The graph of a function changes direction from increasing to decreasing
 at any point where the derivative is zero, or where it is undefined.
\end_layout

\begin_layout BeginFrame
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Third Example
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout OverlayArea
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\backslash
textwidth}{6in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
The third example was a word problem.
\begin_inset VSpace -1ex
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

 A manufacturer's profit depends on the volume of sales: if not enough units
 are sold, income is insufficient to cover the costs, but if too many units
 are sold, the costs of making too much of a product grow faster than income.
 The profit function for the manufacturer Widgets-R-Us is known to be 
\begin_inset Formula $\ec p(x)=-{x}^{3}+{2.66{x}^{2}}+{3.8649x}+{0.113526}$
\end_inset

, where 
\begin_inset Formula $\ec x$
\end_inset

 is the number of units sold, in thousands.
 What number of units sold should the manufacturer target in order to maximize
 profit? 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset


\begin_inset VSpace -1ex
\end_inset

We want to find the maximum value of 
\begin_inset Formula $\ec p(x)$
\end_inset

 on the interval 
\begin_inset Formula $\ec[0,\infty)$
\end_inset

.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<2>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
The first thing to do is to find the critical points.
 First, compute the derivative, and find where it is zero or undefined.
 
\size scriptsize

\begin_inset Formula 
\begin{align*}
\ec0 & \ec=p'(x)=-3x^{2}+5.32x+3.8649\ec x & {\ec=\frac{-5.32\pm\sqrt{5.32^{2}-4\cdot(-3)\cdot3.8649}}{2\cdot(-3)}\approx-0.55,2.33}.
\end{align*}

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Itemize
There is a critical point at 
\begin_inset Formula $x=2.33$
\end_inset

.
 
\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
The negative critical point 
\begin_inset Formula $\ec-0.55$
\end_inset

 is outside the domain 
\begin_inset Formula $\ec[0,\infty)$
\end_inset

.
 Since 
\begin_inset Formula $\ec p'(x)$
\end_inset

 is never undefined, there are no other critical points.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Standard
\begin_inset VSpace -1ex
\end_inset

Comparing the 
\begin_inset Formula $y$
\end_inset

 values, we see
\begin_inset VSpace -2ex
\end_inset


\begin_inset Formula 
\begin{align*}
\ec p(2.33) & \ec=10.91028\\
\ec p(0) & {\ec=0.113526}.
\end{align*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset VSpace -1ex
\end_inset

 The manufacturer should sell 2,300
\begin_inset space ~
\end_inset

units to maximize profit.
\end_layout

\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Local Extrema
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
One can distinguish in Calculus between 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 global
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 extrema and 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 local
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 extrema.
 An extremum is global if it the largest or smallest 
\begin_inset Formula $y$
\end_inset

 value on the entire interval specified.
 An extremum is local if it is the largest or smallest 
\begin_inset Formula $y$
\end_inset

 value on some interval, but not perhaps the one specified.
\end_layout

\begin_layout BeginFrame
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Local Extrema: Example
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
As an example, we compute the local extrema of 
\begin_inset Formula $\ec f(x)=x\sin x$
\end_inset

 on the interval 
\begin_inset Formula $\ec[0,2\pi]$
\end_inset

.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
The derivative is 
\begin_inset Formula $\ec f'(x)=\sin x+x\cos x$
\end_inset

.
 To find where this is zero, you will have to use some method of approximating
 the zero of a function.
 You can do this with the 
\begin_inset Quotes eld
\end_inset

Newton's Method
\begin_inset Quotes erd
\end_inset

 SAGElet.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
You should find that critical points occur at 
\begin_inset Formula $\ec x=0$
\end_inset

, 
\begin_inset Formula $\ec x\approx2.03$
\end_inset

, and 
\begin_inset Formula $\ec x\approx4.91$
\end_inset

.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
 Compute the 
\begin_inset Formula $y$
\end_inset

 values of the critical points and the endpoints: 
\end_layout

\begin_layout Standard
\align center
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ec
\end_layout

\end_inset


\begin_inset Tabular
<lyxtabular version="3" rows="2" columns="5">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $x$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2.03 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.91 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $2\pi$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $f(x)$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1.82 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-4.81 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
At 
\begin_inset Formula $\ec x=0$
\end_inset

, we have a 
\emph on
local
\emph default
 minimum; 
\end_layout

\begin_layout Itemize
at 
\begin_inset Formula $\ec x=2.03$
\end_inset

, we have a 
\emph on
global
\emph default
 maximum; 
\end_layout

\begin_layout Itemize
at 
\begin_inset Formula $\ec x=4.91$
\end_inset

, we have a 
\emph on
global
\emph default
 minimum; and 
\end_layout

\begin_layout Itemize
at 
\begin_inset Formula $\ec x=2\pi$
\end_inset

, we have a 
\emph on
global
\emph default
 minimum.
 
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Local Extrema: Caution
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Be careful when analyzing this, because some functions have critical points
 that are neither minima nor maxima! For a minimum, the function 
\emph on
must
\emph default
 decrease to the value, then increase.
 For a maximum, the function 
\emph on
must
\emph default
 increase the to value, then decrease.
\end_layout

\begin_layout Standard
\noindent
\begin_inset VSpace bigskip
\end_inset

For example, if 
\begin_inset Formula $\ec f(x)=x^{3}$
\end_inset

, there is a critical point at 
\begin_inset Formula $\ec x=0$
\end_inset

 (check this yourself!) but it cannot be a local minimum or maximum on the
 interval 
\begin_inset Formula $\ec[-1,1]$
\end_inset

: 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec f'(x)>0$
\end_inset

 for all 
\begin_inset Formula $\ec x\in[-1,0)$
\end_inset

; and 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec f'(x)>0$
\end_inset

 for all 
\begin_inset Formula $\ec x\in(0,1]$
\end_inset

.
 
\end_layout

\begin_layout Standard
\noindent
This does not meet the definition of either a minimum or a maximum, no matter
 how small you make the interval around 
\begin_inset Formula $\ec x=0$
\end_inset

.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Conclusion
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\noindent
In this module, we discussed: 
\end_layout

\begin_layout Itemize
How to find the extrema, or minima of a function 
\begin_inset Formula $\ec f$
\end_inset

:
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
setlength{
\backslash
shadeboxwidth}{4in}
\end_layout

\end_inset


\begin_inset VSpace -1ex
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{method}
\end_layout

\end_inset

 
\end_layout

\begin_deeper
\begin_layout Itemize
find all 
\begin_inset Formula $x$
\end_inset

 values where the derivative of 
\begin_inset Formula $\ec f$
\end_inset

 is zero or undefined; 
\end_layout

\begin_layout Itemize
compute the 
\begin_inset Formula $y$
\end_inset

 values of 
\begin_inset Formula $\ec f$
\end_inset

 at those points and at the endpoints; 
\end_layout

\begin_layout Itemize
select the largest 
\begin_inset Formula $y$
\end_inset

 value as the maximum, and the smallest 
\begin_inset Formula $y$
\end_inset

 value as the minimum.
 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{method}
\end_layout

\end_inset

 
\end_layout

\end_deeper
\begin_layout Itemize
How to find the extrema by guess and check, but that would be unreliable.
 
\end_layout

\begin_layout Itemize
How extrema are related to where the graph of a curve changes direction.
 
\end_layout

\begin_layout Itemize
How extrema can be used in applications of mathematics.
 
\end_layout

\begin_layout Standard
Later we will use the example of where a graph changes direction to help
 draw the sketches of curves.
 
\end_layout

\begin_layout EndFrame

\end_layout

\end_body
\end_document
