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{SECT 0 {EXCHG {PARA 216 "" 0 "" {TEXT 216 26 "Maximum and Minimum Val
ues" }}{PARA 217 "" 0 "" }}{SECT 0 {PARA 218 "" 0 "" {TEXT 215 85 "Der
ivatives can be used to find local maximum and local minimum values of
 functions. " }}{PARA 217 "" 0 "" {TEXT 244 70 "Example 1:  Find the a
bsolute and local maximum and minimum values of " }{XPPEDIT 241 0 "f(x
) = x^(2/5)*(2*x^2-3*x)^2+3;" "6#/-%\"fG6#%\"xG,&*&)F'*&\"\"#\"\"\"\"
\"&!\"\"F-*$,&*&F,F-*$F'F,F-F-*&\"\"$F-F'F-F/F,F-F-F5F-" }{TEXT 245 1 
" " }{TEXT 245 1 " " }{TEXT 245 1 " " }{TEXT 245 1 " " }{TEXT 245 1 " \+
" }{TEXT 246 23 "on the interval [0, 3]." }}{PARA 217 "" 0 "" }{EXCHG 
{PARA 217 "> " 0 "" {MPLTEXT 1 247 8 "restart;" }}}{EXCHG {PARA 217 ">
 " 0 "" {MPLTEXT 1 247 30 "f:=x->x^(2/5)*(2*x^2-3*x)^2+3;" }}}{EXCHG 
{PARA 217 "> " 0 "" {MPLTEXT 1 247 38 "plot(f(x),x=0..3,y=0..10,thickn
ess=2);" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 247 9 "fx:=D(f);" }}
}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 247 16 "s:=solve(fx(x));" }}}
{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 247 13 "ss:=evalf(s);" }}}
{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 247 75 "p1:=[0,f(0)];p2:=[ss[3],
f(ss[3])];p3:=[ss[2],f(ss[2])];p4:=evalf([3,f(3)]);" }}}{PARA 217 "" 0
 "" }{PARA 217 "" 0 "" {TEXT 248 7 "       " }{TEXT 244 123 "p1 and p2
 are both local and absolute minimum points, p3 is a relative minimum \+
point, and p4 is the absolute maximum point." }}{PARA 217 "" 0 "" 
{TEXT 244 77 "\nExample 2:  Find the local maximum and local minimum v
alues of the function " }{XPPEDIT 242 0 "f(x) = x*exp(x)-x^3*exp(x);" 
"6#/-%\"fG6#%\"xG,&*&F'\"\"\"-%$expGF&F*F**&F'\"\"$F+F*!\"\"" }{TEXT 
245 1 " " }{TEXT 245 1 " " }{TEXT 245 1 " " }{TEXT 245 2 " ." }}{PARA 
217 "" 0 "" }{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 247 26 "f:=x->x*exp
(x)-x^3*exp(x);" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 247 54 "plot
(f(x),x=-5..5,y=-2..2,thickness=2,numpoints=5000);" }}}{EXCHG {PARA 
217 "> " 0 "" {MPLTEXT 1 247 9 "fx:=D(f);" }}}{EXCHG {PARA 217 "> " 0 
"" {MPLTEXT 1 247 27 "s1:=fsolve(fx(x),x=-4..-2);" }}}{EXCHG {PARA 
217 "> " 0 "" {MPLTEXT 1 247 26 "s2:=fsolve(fx(x),x=-1..0);" }}}
{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 247 25 "s3:=fsolve(fx(x),x=0..1)
;" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 247 68 "local_max1:=[s1,f(
s1)];local_min:=[s2,f(s2)];local_max2:=[s3,f(s3)];" }}}{PARA 217 "" 0 
"" {TEXT 244 13 "\nExample 3:  " }{TEXT 248 30 "Find a polynomial of t
he form " }{XPPMATH 205 "6#/-%\"PG6#%\"xG,(*&%\"AG\"\"\"*$F'\"\"#F+F+*
&%\"BGF+F'F+F+%\"CGF+" }{TEXT 248 1 " " }{TEXT 248 1 " " }{TEXT 248 
11 " such that " }{XPPMATH 205 "6#/-%\"PG6#\"\"#\"\"&" }{TEXT 248 1 " \+
" }{TEXT 248 1 " " }{TEXT 248 1 " " }{TEXT 248 28 " , P'(2) = 3, and P
\"(2) = 2." }}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 249 8 "restart;" }
}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 249 18 "f:=x->A*x^2+B*x+C;" }}
}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 249 9 "fx:=D(f);" }}}{EXCHG 
{PARA 217 "> " 0 "" {MPLTEXT 1 249 11 "fxx:=D(fx);" }}}{EXCHG {PARA 
217 "> " 0 "" {MPLTEXT 1 249 39 "eq1:=f(2)=5;eq2:=fx(2)=3;eq3:=fxx(2)=
2;" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 249 21 "solve(\{eq1,eq2,e
q3\});" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 248 27 "So here is th
at polynomial;" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 249 14 "P(x):
=x^2-x+3;" }}}{PARA 217 "" 0 "" {TEXT 244 13 "\nExample 4:  " }{TEXT 
250 17 "For the function " }{XPPEDIT -1 0 "f(x) = 3*x^4/20-x^3+3*x^2/5
+24*x/5-2;" "6#/-%\"fG6#%\"xG,,*(\"\"$\"\"\"*$F'\"\"%F+\"#?!\"\"F+*$F'
F*F/*(F*F+*$F'\"\"#F+\"\"&F/F+*(\"#CF+F'F+F4F/F+F3F/" }{TEXT 248 1 " "
 }{TEXT 248 1 " " }{TEXT 248 1 " " }{TEXT 245 3 " , " }{TEXT 250 47 "p
lot it and its first and second derivatives.  " }}{PARA 217 "" 0 "" 
{TEXT 250 82 "a)  Find the coordinates of the points where the functio
n has horizontal tangents." }}{PARA 217 "" 0 "" {TEXT 250 180 "b)  Fin
d the coordinates of the points in the interval -2 < x < 4 where the f
unction is increasing and decreasing most rapidly and the rate of incr
ease or decrease at those points." }}{EXCHG {PARA 217 "> " 0 "" 
{MPLTEXT 1 249 8 "restart;" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 
249 36 "f:=x->3/20*x^4-x^3+3/5*x^2+24/5*x-2;" }}}{EXCHG {PARA 217 "> "
 0 "" {MPLTEXT 1 249 9 "fx:=D(f);" }}}{EXCHG {PARA 217 "> " 0 "" 
{MPLTEXT 1 249 11 "fxx:=D(fx);" }}}{EXCHG {PARA 217 "> " 0 "" 
{MPLTEXT 1 249 54 "plot(\{f(x),fx(x),fxx(x)\},x=-5..6,y=-8..8,thicknes
s=2);" }}}{EXCHG {PARA 217 "" 0 "" {TEXT 250 61 "This command will set
 the derivative equal to zero and solve." }}{PARA 217 "" 0 "" }{PARA 
217 "> " 0 "" {MPLTEXT 1 249 16 "s:=solve(fx(x));" }}}{EXCHG {PARA 
219 "" 0 "" {TEXT 251 72 "The following are the points where the funct
ion has a horizontal tangent" }}{PARA 219 "" 0 "" }{PARA 217 "> " 0 ""
 {MPLTEXT 1 249 81 "ht1:=[s[1],evalf(f(s[1]))];ht2:=[s[2],evalf(f(s[2]
))];ht3:=[s[3],evalf(f(s[3]))];" }}}{EXCHG {PARA 217 "" 0 "" }{PARA 
217 "" 0 "" {TEXT 250 68 "The next command sets the second derivative \+
equal to zero and solves" }}{PARA 217 "" 0 "" }{PARA 217 "> " 0 "" 
{MPLTEXT 1 249 18 "h:=fsolve(fxx(x));" }}}{EXCHG {PARA 217 "" 0 "" }
{PARA 217 "" 0 "" {TEXT 250 53 "These are the points of maximum increa
se and decrease" }}{PARA 217 "> " 0 "" {MPLTEXT 1 249 52 "r1:=[h[1],ev
alf(f(h[1]))];r2:=[h[2],evalf(f(h[2]))];" }}}{EXCHG {PARA 217 "> " 0 "
" {MPLTEXT 1 249 62 "max_rate_of_increase:=fx(h[1]);max_rate_of_decrea
se:=fx(h[2]);" }}}}{SECT 0 {PARA 218 "" 0 "" {TEXT 215 9 "Exercises" }
}{PARA 217 "" 0 "" {TEXT 244 83 "Exercise 1:  Find the absolute maximu
m and absolute minimum values of the function " }{XPPEDIT 240 0 "f(x) \+
= 1.2*sin(x-1)^2+cos(.9*x);" "6#/-%\"fG6#%\"xG,&*&$\"#7!\"\"\"\"\"*$-%
$sinG6#,&F'F-F-F,\"\"#F-F--%$cosG6#*&$\"\"*F,F-F'F-F-" }{TEXT 245 1 " \+
" }{TEXT 245 1 " " }{TEXT 245 1 " " }{TEXT 245 1 " " }{TEXT 245 1 " " 
}{TEXT 246 4 " on " }}{PARA 217 "" 0 "" {TEXT 244 31 "                \+
  the interval " }{XPPEDIT 238 0 "[0, 2*Pi];" "6#7$\"\"!*&\"\"#\"\"\"%
#PiGF'" }{TEXT 245 1 " " }{TEXT 245 1 " " }{TEXT 245 1 " " }{TEXT 245 
1 " " }{TEXT 245 1 "." }}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 247 8 "
restart;" }}}{PARA 217 "" 0 "" }{PARA 217 "" 0 "" {TEXT 244 210 "Exerc
ise 2:  Between 0 dergrees C and 30 degrees C, the volume  in cubic ce
ntimeters of 1 kg of water at a temperature T is given                \+
                                    approximately by the formula " }
{XPPEDIT 234 0 "V(T) = 999.87-.6426e-1*T+.85043e-2*T^2-.679e-4*T^3;" "
6#/-%\"VG6#%\"TG,*$\"&()***!\"#\"\"\"*&$\"%Ek!\"&F,F'F,!\"\"*&$\"&V])!
\"(F,*$F'\"\"#F,F,*&$\"$z'F5F,*$F'\"\"$F,F1" }{TEXT 245 1 " " }{TEXT 
245 1 " " }{TEXT 245 1 " " }{TEXT 245 1 " " }{TEXT 245 3 ".  " }{TEXT 
246 30 "Find the temperature at which " }}{PARA 217 "" 0 "" {TEXT 244 
118 "                  water has its maximum density and the density a
t that temperature.  Recall that density=mass/volume." }}{EXCHG {PARA 
220 "> " 0 "" {MPLTEXT 1 252 8 "restart;" }}}{PARA 217 "" 0 "" }{PARA 
217 "" 0 "" {TEXT 250 54 "Exercise 3:  Find a third degree polynomial \+
such that " }{TEXT 253 45 "P(1) = 1, P'(1) = 3, P\"(1) = 6, P'''(1) = \+
12." }}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 249 8 "restart;" }}}}
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