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217 286 "There appear to be two local minima, a local maximum close to
 x = 1, and possibly another local maximum near x = -3. At  points whe
re there is a local maximum or minimum the derivative is 0. We next co
mpute the derivative and draw the graphs of f (x) and f '(x) on the sa
me set of axes:" }}{PARA 0 "" 0 "" {TEXT 256 0 "" }}}{EXCHG {PARA 0 ">
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ve 1" "Curve 2" }}{TEXT 257 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 218 
0 "" }{TEXT 204 97 "    To find the exact locations of the local maxim
a and minima we solve the equation  f '(x) = 0:" }}{PARA 0 "" 0 "" 
{TEXT 256 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(D(f)
(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&#\"\"\"\"\"#F%-I'RootOf
G6$%*protectedGI(_syslibG6\"6#,&I#_ZGF)F%*&\"\"'F%-I$cosGF)6#F/F%F%F%F
%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 219 0 "" }{TEXT 204 113 "    Appare
ntly, MAPLE does not know a general solution, so we will find the solu
tions using the \"fsolve\" command." }}{PARA 0 "" 0 "" {TEXT 256 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "X[1]:=fsolve(D(f)(x)=0,x=
-1..0);X[2]:=fsolve(D(f)(x)=0,x=0..1);X[3]:=fsolve(D(f)(x)=0,x=1..2);"
 }}{PARA 11 "" 1 "" {XPPMATH 20 "$!+F_vBn!#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "$\"+#[*fd%*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+.J\"H
*>!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 220 0 "" }{TEXT 204 83 "    To
 see if there is another solution near -3, we zoom in on the graph of \+
f '(x):" }}{PARA 0 "" 0 "" {TEXT 256 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "plot(D(f),-Pi..-2.8,-1..1,color=blue);" }}{PARA 13 ""
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"Curve 1" }}{TEXT 257 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 221 0 "" }
{TEXT 204 92 "    Since the graph of f '(x) does not touch the X-axis \+
there is not an additional solution." }}{PARA 0 "" 0 "" {TEXT 256 5 " \+
    " }{TEXT 222 74 "Note that, from the graph of f '(x), f '(x) is ne
gative in the intervals (" }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetti
ng:-mi(\"\1700\", italic = \"false\", mathvariant = \"normal\"));" "-I
%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q%&pi;F
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223 3 " , " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-msub(Typese
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{TEXT 256 0 "" }{TEXT 224 8 ") and ( " }{XPPEDIT 18 0 "Typesetting:-mr
ow(Typesetting:-msub(Typesetting:-mi(\"X\", italic = \"true\", mathvar
iant = \"italic\"), Typesetting:-mrow(Typesetting:-mn(\"2\", mathvaria
nt = \"normal\")), subscriptshift = \"0\"));" "-I%mrowG6#/I+modulename
G6\"I,TypesettingGI(_syslibGF'6#-I%msubGF$6%-I#miGF$6%Q\"XF'/%'italicG
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 0 "Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(\"X\", italic \+
= \"true\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-
mn(\"3\", mathvariant = \"normal\")), subscriptshift = \"0\"));" "-I%m
rowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I%msubGF$6%-I#miGF
$6%Q\"XF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-F#6#-I#mnGF$6$Q\"
3F'/F6Q'normalF'/%/subscriptshiftGQ\"0F'" }{TEXT 256 0 "" }{TEXT 226 
75 "), and thus f (x) is decreasing on these intervals. f '(x) is posi
tive in (" }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-msub(Typeset
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F'6#-I%msubGF$6%-I#miGF$6%Q\"XF'/%'italicGQ%trueF'/%,mathvariantGQ'ita
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{TEXT 256 0 "" }{TEXT 227 3 " , " }{XPPEDIT 18 0 "Typesetting:-mrow(Ty
pesetting:-msub(Typesetting:-mi(\"X\", italic = \"true\", mathvariant \+
= \"italic\"), Typesetting:-mrow(Typesetting:-mn(\"2\", mathvariant = \+
\"normal\")), subscriptshift = \"0\"));" "-I%mrowG6#/I+modulenameG6\"I
,TypesettingGI(_syslibGF'6#-I%msubGF$6%-I#miGF$6%Q\"XF'/%'italicGQ%tru
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riptshiftGQ\"0F'" }{TEXT 256 0 "" }{TEXT 228 7 ") and (" }{XPPEDIT 18 
0 "Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(\"X\", italic =
 \"true\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-m
n(\"3\", mathvariant = \"normal\")), subscriptshift = \"0\"));" "-I%mr
owG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I%msubGF$6%-I#miGF$
6%Q\"XF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-F#6#-I#mnGF$6$Q\"3
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" , " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\1700\", ita
lic = \"false\", mathvariant = \"normal\"));" "-I%mrowG6#/I+modulename
G6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q%&pi;F'/%'italicGQ&falseF'/
%,mathvariantGQ'normalF'" }{TEXT 256 0 "" }{TEXT 230 44 " ), and there
fore f (x) is increasing there." }}{PARA 0 "" 0 "" {TEXT 256 0 "" }
{TEXT 231 85 "    We now apply the First Derivative Test. Since f '(x)
 is negative to the left of  " }{XPPEDIT 18 0 "Typesetting:-mrow(Types
etting:-msub(Typesetting:-mi(\"X\", italic = \"true\", mathvariant = \+
\"italic\"), Typesetting:-mrow(Typesetting:-mn(\"1\", mathvariant = \"
normal\")), subscriptshift = \"0\"));" "-I%mrowG6#/I+modulenameG6\"I,T
ypesettingGI(_syslibGF'6#-I%msubGF$6%-I#miGF$6%Q\"XF'/%'italicGQ%trueF
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ptshiftGQ\"0F'" }{TEXT 256 1 " " }{TEXT 232 56 "and positive to the ri
ght, there is a local minimum at  " }{XPPEDIT 18 0 "Typesetting:-mrow(
Typesetting:-msub(Typesetting:-mi(\"X\", italic = \"true\", mathvarian
t = \"italic\"), Typesetting:-mrow(Typesetting:-mn(\"1\", mathvariant \+
= \"normal\")), subscriptshift = \"0\"));" "-I%mrowG6#/I+modulenameG6
\"I,TypesettingGI(_syslibGF'6#-I%msubGF$6%-I#miGF$6%Q\"XF'/%'italicGQ%
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bscriptshiftGQ\"0F'" }{TEXT 256 0 "" }{TEXT 233 41 "; similarly, there
 is a local minimum at " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting
:-msub(Typesetting:-mi(\"X\", italic = \"true\", mathvariant = \"itali
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tingGI(_syslibGF'6#-I%msubGF$6%-I#miGF$6%Q\"XF'/%'italicGQ%trueF'/%,ma
thvariantGQ'italicF'-F#6#-I#mnGF$6$Q\"3F'/F6Q'normalF'/%/subscriptshif
tGQ\"0F'" }{TEXT 256 2 " ." }{TEXT 234 37 " Since f '(x) is positive t
o the left" }{TEXT 256 2 "  " }{TEXT 235 4 "of  " }{XPPEDIT 18 0 "Type
setting:-mrow(Typesetting:-msub(Typesetting:-mi(\"X\", italic = \"true
\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mn(\"2\"
, mathvariant = \"normal\")), subscriptshift = \"0\"));" "-I%mrowG6#/I
+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I%msubGF$6%-I#miGF$6%Q\"XF
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normalF'/%/subscriptshiftGQ\"0F'" }{TEXT 256 1 " " }{TEXT 236 56 "and \+
negative to the right, there is a local maximum  at " }{XPPEDIT 18 0 "
Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(\"X\", italic = \"
true\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mn(
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." }}{PARA 0 "" 0 "" {TEXT 256 0 "" }}{PARA 0 "" 0 "" {TEXT 256 4 "   \+
 " }{TEXT 238 157 "The second derivative is used to find intervals of \+
concavity and points of inflection. We will compute f ''(x) and plot i
t and f (x) on the same set of axes." }{TEXT 256 0 "" }{TEXT 239 0 "" 
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{TEXT 204 300 "     Points of inflection occur at points where f ''(x)
 = 0 and the second derivative changes sign. The graph is concave down
 on intervals where f ''(x) < 0 and concave upward when f ''(x) > 0; t
hus the points of inflection are the points where the concavity change
s. Here there are four such points:" }}{PARA 0 "" 0 "" {TEXT 256 0 "" 
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0,x=-Pi..-3);Z[2]:=fsolve(D(D(f))(x)=0,x=-2..-1);Z[3]:=fsolve(D(D(f))(
x)=0,x=0..1);Z[4]:=fsolve(D(D(f))(x)=0,x=1..2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "$!+9'oy0$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "$!+m._a;!
\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+hRSs$)!#6" }}{PARA 11 "" 1 ""
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" }{TEXT 241 46 "    The graph of f(x) is concave up for x in (" }
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"" {TEXT 256 4 "    " }{TEXT 247 450 "In each problem graph the functi
on and its first and second derivatives as in the example, and determi
ne the locations of  all (a) local maxima and minima, (b) points of in
flection, (c) intervals where the function is increasing or decreasing
, and (d) intervals where the graph is concave up or down. In problem \+
2, also find all vertical and horizontal asymptotes. In problem 3, als
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}{TEXT 248 3 "not" }{TEXT 249 33 " have a point of inflection at c." }
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