MATH 120 2020-2

Recitation Problems Week 08

1. Find and classify the critical points of the function

f(x, y) = 3y2 � 2y3 � 3x2 + 6xy.

2. Find the absolute maximum and minimum values of the function

f(x, y) = 2x2 � 4x+ y2 � 4y + 1

on the triangular region bounded by x = 0, y = 2, y = 2x.

3. Find the absolute maximum and minimum values of the function

f(x, y, z) = 2x+ 3y + z

subject to the unit sphere x2 + y2 + z2 = 1.

4. Using the Lagrange multiplier method, find the point Q on the plane P : x+2y+2z = 3 that is closestto the origin.

Alternatives/Exercies:

1. Find the absolute maximum and minimum values of the function

f(x, y) = x4 + y2

on the unit disk D = {(x, y) : x2 + y2 1}. (Use the Lagrange multiplier method on @D.)

2. Let f(x, y) = x3y5 and g(x, y) = x+ y.

(a) Find the absolute maximum of f subject to the constraint g(x, y) = 8 using the Lagrange multi-plier method.

(b) By restricting f onto the line x+y = 8, verify that the value obtained in the first part is indeed theabsolute maximum of f , and also that f does not have any absolute minimum value on x+y = 8.

yay!ndependent var!ables

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f x y 3ya 2g 3 46 3

ftp0fxlx.y

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6 63 0

by 6yztbx oksubs.by65 4 0

12g byle O

2 gY012

g 2 2

The cr!t!cal po!nts are 0,07 and 2,2

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DM so f so e f !s bulm!n

f µ o ftp.l!slocdmaBIR test fa!ls

fucky 6 Check far C P 10.07

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9tl22 DCZ.zt fff.su 6Efzfofxxl2.2 60

so we have local max

fckylr3yz 2ys 3xztbxyoo."s C P

alay flay 35 25 143 2

70

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dany

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0,0 cpfcx!yl.rs local m!n

fcoth.o.tk flaşf all

flxth.ytk.IE Cx.y hık small katk!steşt

flkyl.rs local Max mfqytxty70 frdfor all KI ab!

mm

2 F!nd the absolute max!mum and m!n values ofthe funct!on fcky 2 2 4 5 4ytl on the tr!angular

reg!on bounded by o yaz y 2

y 2Xb _closed and bounded reg!onçağay f !s a polynom!al s c

t.everywheresbyextremevalvefzv.setrmfk.y has abs max and

closedbdd!ntervel m!n!mum at one of thecant fmc! CP follow!ng!!l SP!E ! Cr!t!cal pa!nt Of ö

A S!ngular pts

f"snotdftbkfxcx.gl4 4Boundary po!nts

Part!als da

fylx.gl 2y 4CPIII z fd

Ç Ka y C 0.23 qbs m!n

fCaylıya 4ytlgly yı 4ytl 0,23efrad max m!n

absMAX

g ay 4 yaz cpno SPendpo!nt g 1 g 2

Ç g 2 C G 1 HK 2 2 4 3 C G 13

fIX 2 2 44 3 h 4 4 CP x 1 Ktl 5

3 y ZXXEG.lyEP h 3

f X 2 7 6 42 1k 6K 12 1 Calk 12 12 CP x 1 ktl 5

EP k 3 ktl75

If there were no reg!on

fcx.gl 2 x !IltCy zTt5

3 F!nd the absolute max and m!n valuesof the funct!on fçyız 3ytl subjecttathe un!t sphere 4y zz 1

Constra!nt !s x2 yat 1 0Inanhan

Glayız 7 f z 71 2 342 1

Glçyız 7 solut!on g!ves the poss!blepa!nts fu max m!n wrt

that cent!stro!nt! G 2 7 211

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!v Çy xztyzz.l" !!l 3 73 0 ay !!n 2

us!ng ! !! !ü !n !v 1 51 41 0

Ez x tarz

x yıztafazfı 25ft EFE Po

x y.zt ftzf"tzgf.tk R

f Pal Tl4 flpı Tud dm!n Max

4 Us!ng Lagrange mult!pl!er method f"nd thepo!nt Q on the plane P 2527 3 that!s closest to the or!g!n

Q x yız d ol d x yızm!n!m!letferf"nd"l w!th constra!nt

fcky"zlaxtykzzpo" X 2y 2z.ES