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Summary: Max-MinTofindminimumandmaximumvaluesofafunctiony(x)
dy Solve =0tofindpointsx whereslope=zerodx Testeachx
forapossibleminimumormaximum
dyExample y(x) =x312x = 3x212
dxdy The slope is = 0 atx =2andx =2dx
Atthosepointsy(2)=824=16andy(2)=8 + 24=16
d dyx
=2isaminimum Lookat =2nd derivative
dx dxd2y
=derivativeof3x212.2nd derivativeis6x.dx2d2y dy increases
slopegoesfromdowntoupatxdx2 >0 dx
Thebendingisupwardsandthisx isaminimumd2y dy decreases
slopegoesfromuptodownatxdx2
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d2y d2ydx2 >0thecurvebendsupand dx2 0soparabolabendsup.
dy23.Findthemaximumheightofy(x) = 2 + 6xx .Solve =0.
dxd2y
4.Finddx2 toshowthatthisparabolabendsdown.
5.Fory(x) =x42x2 showthat dy = 0atx=1,0,1.dx
Findy(1), y(0), y(1).dy d2y
6.Now = 4x34x. Whatisthesecondderivativedx dx2?
dy d2y7.Ataminimumpointexplainwhy =0and
dx2 >0.dxd2
y8.Bendingdown changestobendingupdx2 0Doesy=x2 havesuchapoint?
Doesy=sinxhavesuchapoint?9.Supposex+X=12.
WhatisthemaximumofxtimesX?Thisquestionasksforthemaximumofy=x(12x)=12xx
.
dyFindwheretheslope =122xiszero. WhatisxtimesX?
dx2
PLAY PAUSE Click PLAY to hear Professor Strang walk you through
this summar
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Resource: Highlights of Calculus
Gilbert Strang
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