Warmup i i Is there a function f x y s t Of Sy x

2 Can the vector field below be the gradient vector field ofa function

As youmove in directionof Tf thevalue i Pf fx fy

off Hint use 2nd partial deriincreases By Clairaut's TkmAs move f falongpink y yxoath fromP If f y E fy Xf increases

Pthen fxy fyx I

fCPI Ffp But It lBut that is go no such function fimpossible existso this isnot a gradientvector field

GeometricMethodsfortindingIClassifyingExtremafII Gradient Vector Fields

Local maximum occurs at pts towardswhich all

nearby gradient vectors point sink

Local minimum occurs at pts awayfromwhich all

nearby gradient vectors pointsource

Saddle pts occur at pts towards which some nearbyvectors point away from which others point

n tt r l T f Ie T i

I j d b TMax Min Saddle

teep

gottacrpifi.cat

t saddlePt

Ex a I all vectors same lengthpointing in same

directionno critical pts i.e

no maxima minima orsaddle pts

Contour map for this function

ex I Iitfit

Itv

ye ng

Often the length of a vector is shown by shadingNotedark blue Very very short

EXIT

Pinot aP

acritical w Pipt ie.IEiii i B i

rkI p Di I

f has a local Max at A CED are saddleptsf has a local min at B

Ex 2

No local Maxor min

iiFATT wso c T

gut FAt

saddle2 pairsofvectors onopposite sideswhichpoint inoppositedirection IAisasaddleptAbsoluteRecall Extreme Value Theorem States that if fix is

continuous on a closed interval ai b then f attains

an absolute Max and an absolute min on CaibTMethod a found critical pts on Laib

121 Tested endpts X a x b compared value off at pts in 4 to fca fcb

A closed set in 1122 is one that contains all of its

boundary points A boundary point of a set DE 1122 is

a point Cacb such that every diskcentered at Ca b

contains points in D points not in D bnoofhidnarfz.pt

of3

al t.jyn.inijmsPointb

closeo not dos HInot closed

seDH

Iclosed

a if

theonlyboundary ofD not closed

A boundedset in 1122 is one that is contained in

are bounded but g Isome disk lattenot bounded

b.yinfg.at

ExtremeValue Theorem for a function of 2 variables Itfis continuous on a closed bounded set D in 1122

then f attains an absolute Max an absolute min

on D

ttowtofindabs.extrema.ciFind critical pts on D

2 Find critical points on the boundary of D using singlevariable methods

3 Compare value of f at all points from Cil E Ca

the largest value is the abs max E the smallestvalue is the abs min of f on D

EI Find the absolute extrema of fcx.gl x 2xyt2y onD xy C1122 0 EXE

3 OE y E 2

Find critical pts of f Cin Dfx 2x 2 y O fy 2 2 0

X Ix y

Critical pt Chl in D

Draw Dmy boundary ofD L X o 0EYE22

2 13,2 y L 0 EX E3

4

g

x 3 0 Eye 2Too Ly Ly y O 0 EX E3

Find critical pts of f on boundary of D

L x 0 OE y E 2plug into f flog Ly gCy 0,2

Tiamat the endptsneed to check0,0 E 012

ftp.ffin3 gcxs Hxin YIIYzt4nvcritical pts 2,2 critical pet

endpoints 10,2 already at X 2

3,2 have 0121

Lz X 3 0EYE 2crit pts one

endpts 3,0312

24 y 0 OE XE 3crit pts co o

endpts cool030

Compare values of f at pts found

if i