function analysis - classworksks23cu.net/mt/files/calculus/mmm/3m86functionanalysis.pdf · function...
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www.MasterMathMentor.com!!!!!"#! $%&'()%*+! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!R-!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Stu Schwartz
Function Analysis - Classwork
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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
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!!!!!!!!!!!!!!!!
@30(!)+!(1'2!08%'(!(32!+&%F2!%:!(32!(0*=2*(!&)*2+!Y)>2>!(32!721)D0()D2I!0(!0&&!%:!(32+2!F%)*(+_!!all positive
\29120+)*=!9'1D2+!9%<2!)*!.!D01)2()2+!0+!+3%6*>!!Z*!2093!90+25!&2(`+!7106!(0*=2*(!&)*2+!0(!F%)*(+!0&%*=!(32!9'1D2>
!!!!!!!!!!!!!!!
@30(!)+!(1'2!08%'(!(32!+&%F2!%:!(32!(0*=2*(!&)*2+!Y)>2>!(32!721)D0()D2I!0(!0&&!%:!(32+2!F%)*(+_!all negative
www.MasterMathMentor.com!!!!!"#! $%&'()%*+! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!R^!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Stu Schwartz
$%!62!90*!<0B2!(32!:%&&%6)*=!+(0(2(<2*(+!08%'(!)*9120+)*=!0*7!729120+)*=!:'*9()%*+A!!V2(!f!82!0!:'*9()%*!(30(!)+9%*()*'%'+!%*!(32!9&%+27!)*(21D0&!ma, bn!0*7!7)::212*()08&2!%*!(32!%F2*!)*(21D0&!Ya5!bI5
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f x x a b f a b
f x x a b f a b
f x x a b f a b
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!!!!!!@2!90&&!(3)+!9'1D2!concave up>!!N32!(21<!Z @2!90&&!(3)+!9'1D2!concave down>!N32!(21<!Z!!!!!!'+2!)+!;3%&7+!60(21>?!Z:!60(21!)+!)+!F%'127 '+2!)+!;+F)&&+!60(21>?!Z:!60(21!)+!F%'127!)*(%
!!!!!!!!!!!!!!!! !!)*(%!(3)+!+(1'9('125!)(!6)&&!9%&&29(!0*7!;3%&7>? (3)+!+(1'9('125!)(!6)&&!+F)&&!%::>
!!!!!c*!(32!9%*90D2!'F!9'1D25!7106!(0*=2*(!&)*2+ c*!(32!9%*90D2!7%6*!9'1D25!7106!(0*=2*(!&)*2+!!!!!0(!(32!F%)*(+!+3%6*>!!@30(!)+!(1'2!08%'(!(32 0(!(32!F%)*(+!+3%6*>!!@30(!)+!(1'2!08%'(!(32!!!!!+&%F2+!%:!(32+2!&)*2+!0+!4%'!=%!&2:(!(%!1)=3(> +&%F2+!%:!(32+2!&)*2+!0+!4%'!=%!&2:(!(%!1)=3(>
!!!!!!!!!!!!!!!!!slopes increasing !!!!!!!!!!!slopes decreasing
!!!!$2!62!90*!72:)*2!9%*90D)(4!0+!:%&&%6+A!Z:!f!)+!7)::212*()08&2!%*!0*!)*(21D0&!I>!N32!=10F3!%:!f!)+!9%*90D2!'F!!!!%*!I!):! -f !)+!)*9120+)*=!%*!(32!)*(21D0&!0*7!9%*90D2!7%6*!%*!I ):! -f !)+!729120+)*=!%*!(32!)*(21D0&>
!!!!@32*!4%'!(3)*B!;+&%F2!%:!(0*=2*(!&)*2?5!4%'!(3)*B!!derivative
!!!!@32*!4%'!(3)*B!)*9120+)*=5!4%'!(3)*B @32*!4%'!(3)*B!729120+)*=5!4%'!(3)*B!!!!!!!!!!!!!!!!!derivative derivative
$%!62!012!(0&B)*=!721)D0()D2+!%:!721)D0()D2+5!63)93!<20*+!!2nd derivative>!$%!9%*90D)(4!)+!12&0(27!(%!(32!+)=*!%:
(32!+29%*7!721)D0()D25! --# $f x >!Z:!
!! --# $ %f x 2 5!:%1!0&&!x!)*!0*!)*(21D0&5!(32*!(32!=10F3!%:!f!)+!9%*90D2!'F!%*!(32
)*(21D0&>!Z:!!! --# $ &f x 2 !:%1!0&&!x!)*!0*!)*(21D0&5!(32*!(32!=10F3!%:!f!)+!9%*90D2!7%6*!%*!(32!)*(21D0&>!!$(10)=3(!&)*2+
30D2!*%!9%*90D)(4!Y9%*90D)(4!d!PI>!N30(!<0B2+!+2*+2!0+!0!&)*2!30+!*%!9'1D0('12>!"&&!%:!(3)+!90*!82!+'<<01)T27A
@32*!4%'!012!=)D2*!)*:%1<0()%*!08%'(! -# $f x !0*7! --# $f x 5!4%'!90*!72(21<)*2!(32!+30F2!%:! f x# $ >
!! -# $ %f x 2
!! -# $ &f x 2
! -# $ !f x 2
--# $ %f x 2 f x# $ !)*9120+)*=5!!!!!!!!!9%*90D2!'F
f x# $ !729120+)*=5!!!!!!!!!9%*90D2!'F
12&0()D2!<)*)<'<59%*90D2!'F
!! --# $ &f x 2
f x# $ !)*9120+)*=5!!!!!9%*90D2!7%6*
f x# $ !729120+)*=5!!!!!!!!!9%*90D2!7%6*
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--# $ !f x 2 f x# $ !)*9120+)*=5!!!)*:&29()%*!F%)*(
f x# $ !729120+)*=5!!!)*:&29()%*!F%)*(
f x# $ !&2D2&+!%::!!!!F%++)8&2!)*:&29()%*!F%)*(
www.MasterMathMentor.com!!!!!"#! $%&'()%*+! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!RR!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Stu Schwartz
]212!012!+%<2!(21<+!63)93!4%'!<'+(!B*%6>!N324!012!80+)9!(%!0*4!90&9'&'+!9%'1+2>
Critical Values (points)A!x,D0&'2+!%*!(32!:'*9()%*!63212!(32!:'*9()%*!30+!0!+&%F2!2E'0&!(%!T21%!%1!63212!(32:'*9()%*!)+!*%(!7)::212*()08&2>!!X)+'0&&45!)(!)+!63212!(3212!)+!0!3%1)T%*(0&!(0*=2*(!&)*2!%1!0!D21()90&!(0*=2*(!&)*2>
Stationary PointA!"!F%)*(!%*!(32!:'*9()%*!63212!(3212!)+!0!3%1)T%*(0&!(0*=2*(!0(!(32!x,D0&'2>
Relative Minimum: Z*:%1<0&!72:)*)()%*A!(32!8%((%<!%:!0!3)&&>!S2&0()D2!<)*)<'<+!%99'1!63212!(32!9'1D2++6)(932+!:1%<!729120+)*=!(%!)*9120+)*=>!!N32 x,D0&'2!)+!where (32!12&0()D2!<)*)<'<!%99'1+>!N32!y,D0&'2!)+!what(32!12&0()D2!<)*)<'<!)+>
Relative MaximumA!Z*:%1<0&!72:)*)()%*A!(32!(%F!%:!0!3)&&>!S2&0()D2!<0G)<'<+!%99'1!63212!(32!9'1D2!+6)(932+:1%<!)*9120+)*=!(%!729120+)*=>!!N32!x,D0&'2!)+!where (32!12&0()D2!<0G)<'<!%99'1+>!N32!y,D0&'2!)+!what!(3212&0()D2!<0G)<'<!)+>
Relative Extrema:!/)(321!0!12&0()D2!<)*)<'<!%1!12&0()D2!<0G)<'<>
Absolute maximum or minimum>!N32!3)=32+(!Y&%62+(I!F%)*(!%*!(32!9'1D2>!!N32!x,D0&'2!)+!where (32!08+%&'(2<0G)<'<Y<)*)<'<I!%99'1+>!N32!y,D0&'2!)+!what!(32!08+%&'(2!<0G)<'<!Y<)*)<'<I!)+>!!N32!08+%&'(2!<)*)<'<!%1<0G)<'<!<'+(!%99'1!0(!12&0()D2!2G(12<0!%1!0(!(32!2*7F%)*(+!%:!0*!)*(21D0&>
Inflection point:"!N32!x,D0&'2!63212!(32!9'1D2!+6)(932+!9%*90D)(4>!!Z*:&29()%*!F%)*(+!90*!%99'1!63212!(32!+29%*7721)D0()D2!2E'0&+!T21%!%1!:0)&+!(%!2G)+(>
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www.MasterMathMentor.com!!!!!"#! $%&'()%*+! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!Ra!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Stu Schwartz
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--# $f x 222222222222
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--# $f x PPPPPPPPPPPP
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SSSSSSSS2SSSSSSSSS --# $f x PPPPPPSSSSSSSS
2
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--# $f x PPPPPPSSSSSSSSS
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www.MasterMathMentor.com!!!!!"#! $%&'()%*+! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!aO!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Stu Schwartz
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!!!!!!!!!!To determine relative extrema of a function , do the following steps.
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!!!!!!!!!!!!!!c
++++++++++----------------
c
--------------- +++++++++++
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!!!!!!!!!!!!!!c
++++++++++----------------
c
--------------- +++++++++++
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