12Maximum and Minimum Values Lesson 12

8/6/2019 12Maximum and Minimum Values Lesson 12

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Maximum and Minimum ValuesLesson 12

Objectives:

At the end of the lesson you should be able to:

1. Define the maximum and minimum values of agiven function.

2. Find the maximum and minimum values usingSecond Derivative Test

3. Find the maximum and minimum values using theLagrange Multipliers

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Maximum and Minimum Values

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Definition:

A function of two variables has a local maximum at (a,b)if

If is near

is a local minimum value.

). ,( ) ,( ) ,( ) ,( banear is y x whenba f y x f e

) ,( ) ,( ) ,( y x eba f y x f u) ,( , ) ,( ba f thenba

Theorem

If f has a local maximum or minimum at (a,b) and the firstorder partial derivatives of f exists there, then

.0 ) ,( 0 ) ,( !! ba f a d ba f y x

N ote: (a,b) is called critical point or stationary point.


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S econd Derivative Test

If the second partial derivative of f is continuous on a disk

with center (a,b), and (a,b) is a critical point, let

and :

a. If

b. If

c. If

d. If D = 0, then no conclusion can be drawn.

? A2 ) ,( ) ,( ) ,( ) ,( ba f ba f ba f ba D D

y x y y x x !!

.min ) ,( 0 ) ,( 0 imumlocal aisba f thenba f and x x ""

.ax ) ,( 0 ) ,( ,0 i m mlocal i sba f t eba f a d D x x

.int),(,0 po saddleai sba f then D

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JBJ/UTP 4

To find D, use determinants y y x y

y x x x

f f

f f !

To find the absolute maximum and minimum values,

1. Find the value of f at the critical points of f in D.

2. Find the extreme values of f on the boundary of D.

3. The largest values from 1 and 2 is absolute maximum;the smallest value is the absolute minimum value.

Lets see the an example .


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JBJ/UTP 5

E xample1 : Maximum and Minimum Values

Find the local maximum ,minimum values and saddle

point(s) of the function

Solution:

.4429 ) ,( 22 y x y x y x f !

21

1

21010

84.22.

!!

!!

!!

y x

y x

y f b x f a y x

2.4)],([),(),(.

2

!!!

x x

x y yy xx

f d ba f ba f ba f Dc


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Example 2: Maximum and Minimum

Find the critical values given

.),(22

22 x ye y x y x f !

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Let s use the2nd DerivativeTest


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E xample 3: Maximum and Minimum

ValuesFind the absolute maximum and minimum values of f

on the set D given D is

a closed triangular region with vertices (0,0), (2,0)

and, (0,3).

y x y x f 541 ) ,( !

)0 ,0( )0 ,2(

)3 ,0(

y

x 1 L

2 L

3 L


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Values

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Determine and classify all the critical points for

.322),( 2423 y x y y x y x f !

S olve for algebraically:0;0 !! y x f f

Test the points using

2)},({),(),(),( ba f ba f ba f ba D x y yy xx!


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Your answers are the following:

.).(24)1,2(

.).(6)21

,1(

.).(0)0,0(

P M D

P S D

C N D

!

!

!

Le ts mov e on to the nex t topic!!!

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Lagrange Multipliers

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It is a method to find the maximum and minimum values of

a general function f(x,y,z) subject to a given condition of

the form g(x,y,z)=k.

Assume that the extreme values exist and that

k z y x g on g !{ ),,(,0

M ultiplier agrangetheiswhere

z y x g z y x f P

P

:

) , ,( ) , ,( 000000 !


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H ow to Use Lagrange Multipliers

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1. Find all values of

2. E valuate f at all the points (x,y,z) that result

fromS

tep 1.a. The largest value is the maximum value of f.

b. The smallest value is the minimum value of

f.

.),,(

),,(),,(

,

,,,

k z y x g and

z y x g z y x f

where

and z y x

!

P!

P

O ne Constraint


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Therefore,

There will be four unknowns,

when it is a function of three variables.For a function of two variables , i.e.,

subject to the constraint

There will be three unknowns. These areTherefore,

z z y y x x g f g f g f !!! PPP , ,

.,,, Pa nd z y x

) ,( y x f , ) ,( k y x g !

. , , ,P

and y x . , y y x x g f g f PP !!


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Use Lagrange Multipliers to find the maximum andminimum values of the function subject to a givenconstraint.

.6 2; ) ,( 222 !! y x y x y x f

y x x y x 4,2,2 2 P!

Answ er: 4)1,2(;4)1,2(

1,2),(,

!!

ss!!

f f

ba y x


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JBJ/UTP 15


Use Lagrange Multipliers to find the maximum andminimum values subject to a constraint.

.35;106 2 ) , ,( 222 !! z y x z y x z y x f

z y x ,,5,3,1 P!Equa tion :

5,3,1,, sss! z y x


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Use of Lagrange Multiplierswith Two Constraints

Find the maximum/minimum values of f(x,y,z) subject totwo constraints g(x,y,z)=k and h(x,y,z)= c. Therefore theformula becomes:

) , ,( ) , ,( ) , ,( 000000000 z y x z y x g z y x f ! QPThere will equations with FIVE (5) unknowns and these

are: . , , , , , z a d y x QP

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JBJ/UTP 17

The components are the following:

z z z

y y y

x x x

h g f

c z y x hh g f

k z y x g h g f

QP

QP QP

!

!!

!!

) , ,(

) , ,(

Lets solve anexample!!!


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JBJ/UTP 1 8


Find the maximum/minimum values of f( x, y , z

)given two constraints.

4 ) , ,( 1 ) , ,(

2 ) , ,(

22!

!

!

z y z y x h z y x z y x g

y x z y x f


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JBJ/UTP 19


Find the maximum values of

given two constraints

z y x z y x f 33 ) , ,( !

.12 , ,0 22 !! z x and z y x


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JBJ/UTP 2 0

Find the points on the surface that are

closest to the origin.

( Use the S econd Derivative Test )

122 ! z y x

2222 z y x D !


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JBJ/UTP 21

E xample 9: Use Lagrange Multipliers with TwoConstraints

Find the maximum and minimum volumes of arectangular box whose surface area is 1500 sq.cm.and whose total edge length is 200 cm.

z y x

z y x E EdgeTotal

yz xz xy

yz xz xy Aa SurfaceAre z y x V

!

!!!

!

!!!

!

50

200444

7 50

1500222

x

z


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x 6 27.2

y 22 11.4z 22 11.4

-484 -129.96

2904 3534.9

221!P 4.112 !P

Q

f

The values of x,y,and z incolumn 3

is maximum.

The valuesin the secondcolumn for x,yand z is

minimum.


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E xample 10. Find the extreme values of

y xe y x f !),(

on the region described by the inequality .14 22 ey x

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E xample 11: The plane intersects the

paraboloid in an ellipseFind the points on the ellipse that are nearest and farthestfrom the origin. (Lagrange Multipliers with two constraints).

22 !z y x

22 y x z !

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P ractice Task

G iven.1),,,(

;),,,(2222!

!

t z y xt z y x g

t z y xt z y x f

Find the maximum and minimum values.

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E xercisesS olve the following on page 998

Nos. 9, 30 , 38 and 41. ( S econd Derivative

Test)

O n page 9, 16, and 38 (Lagrange Multipliers)

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12Maximum and Minimum Values Lesson 12