10.53 Gradients and Directional Derivatives2/1

Jtx ,y )

oftx,y)=µy¥g] Gradient is a vector

Directional Derivative

Dunflx . ,y . ) = oftx . ,y . ) .u^ Hunky^ "

EH nation a. ¥,¥%

22 flay )=e*Y ( 0,0 ) [ I, ]

point vector'¥

th ,y)=keIYy] oslo .FMhiktsttmtrz a=¥n=⇐fIffYYr]

oflaoju - ftp.fyhzgfi.fftl.fz-2#scalar

v. 6. Maxima and Minima

5- ( x,y ) ( x. ,y° )

offx . ,yD=O=8=[0o]=( 0,0 )

DHess -xx,y)=[ftp.f# =ytheinconclusive T sapdognlky

D= fxxfyy - H×y)2 yoyoT= fxxtfyy local localmin Max

10.6.

5. f( X. g) = -2×2 + y'

-6g

Htxykfzyh . ]=[ ;] ⇒�1� .

4x⇒⇒x=o@ zy -6=0 ⇒ y=3( 0,3 )

Hessfcx ,y)=ffIfHess f( 0,3 )=[ to{ ] D= C-4) (2) - (0×0)=-8-0

-saddlepoin-al.CO#

1-1

J ( x ,y)=xy - Zyz

osixytf,?4y]=[ :]�1� Y=0 @ × - 4y=o

( 0,0 )

g. 49=0x=o

{ 91,] D=o.t4 ) - a) ( D= -1<0 -→ saddlepoint.ae#

8.Hx.ykyxe-YIfgY9.txtdefTDHg5-tgtgsFx-YeTFy-lyjxeTtfxEYjy-1xeT-xeis.y-CxyYeT.Ce-yyxy-XEY-xye-ylxeiExje.d-fIxFIyoenooo@ye9-o20xeT.x

yet

yEo↳E#Xeo -

x. 0eY=o

tDfh)g(x)=oXil -0=0

-51×7=0 or g(x)=oX=0

Eg .( x - 1) ( x -4=0

E:*

:H::]

¥x=YeT ¥y=(yjxeT+(xEY)'y=1xe '- xeity

5×x=£x(¥x)=§×(yey)=o=×EY -

xye'T

fyy=Ey(¥y)=Fy( XEY - xyEY)= - XEY - ( xeihxye 's )

5yx=¥(¥y)=F×(xe.hn/yED=EY-yeTfxy=fyxf××( 0,4=0 kixfyy - #y)2=D

fyylo ,o)= - oeotoeao .oeo)=O 0.0 - (1) 2= -1<0

t×y( 0,0=50-0 .eo=1saddle point

a. th ,y)= xcosy

o*⇒=h%nIft :L

:BnT:�1� Cos y=O @ - × sin y =O

y=kntzDiI,nEk-

xfN=¥#( at Is ,

( QEI,

( 0,3¥ ), ... . all saddle points

Enginedoo

'iE¥'

0u - 1<0