8/3/2019 C3L4 Maxima and Minima of Function of Two Variables

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Maxima and Minima ofFunction of Two Variables

C3L4

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Learning Outcomes

At the end of the lesson youshould be able to:

1. Define relative maximum andabsolute maximum.

2. Define relative minimum andabsolute minimum.

3. Find the critical points.

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Definition

A function f of two variables is said tohave a relative a relative minimum at apoint (xo,yo) if there is a disk centered at

(xo.yo) such that forall points (x,y) that lie in the disk, and f is said to have an absolute minimum at(xo,yo) if for all points

(x,y) in the domain of f.

),(),( 00 y x f  y x f 

),(),(00 y x f  y x f 

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z

Z= f(x,y)

Absolute Min; Relative Min;Absolute Max; Relative Max

A

BC

D

y

x

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Finding Relative Extrema

Theorem is f has a relative extremum at apoint (xo,yo), and if the first partialderivatives of f exist at this point, then

 0),(0),(

0000 y x f and  y x f   y x

Note:

1.These are also called critical points.2. From the Extreme Value Theorem, if f(x,y) iscontinuous on a closed bounded set region R, thenf has both an absolute maximum and absolute

minimum.

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Second Partials Test

Theorem

Let f be a function of two variables withcontinuous second order PD in some disk

centered at a critical point (xo,yo), andlet

.,0.4

).,(int,0.3

).,(max,0),(&0.2

).,(min,0),(&0.1

,),(),(),(

00

0000

0000

00

2

0000

drawnbecanconclusionno D

 y xat  posaddleahas f then D

 y xat imumrelativeahas f  y x f  D y xat imumrelativeahas f  y x f  D

 If  y x f  y x f  y x f  D

 xx

 xx

 xy yy xx

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Example 1

Locate all relative maxima, relativeminima, and saddle points, if any

Given:

a.

b. More problems!

 y x y x y x  f   33),( 33

22

44

33

cos),(

44),(

3),(

 y x y x f 

 y x xy y x f 

 y xy x y x f 

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Answer:

a)

.max.)1,1(,036,036

.min.)1,1(,06,036

int)1,1(&)1,1(,0361;1

rel f  D

rel f  D

 posaddle D y x

 xx

 xx

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Answer

b)

(x,y)=(0,0)

D=-2<0 , therefore (0,0) is a saddlepoint.

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Absolute Extrema on Closed andBounded Sets

Steps to follow:

1. Find the critical points of f that is in theinterior of R.

2. Find all boundary points at which theabsolute extrema can occur.

3. Evaluate f(x,y) at the points obtained inthe previous step. The largest of thesevalue is the absolute maximum and thesmallest the absolute minimum.

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Example 2

)0,5(,)4,0(,)0,0(

,3),( verticeswithstriangleais R y x xy y x f 

(0,0) (5,0)

(0, 4)

Find the extreme values given

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Answer (x,y)=(3,1)

When y=0, (x-axis), u(x)= -x no CP

When x=0 , (y-axis) v(y) =-3y no CP

Along y=-4/5 x +4 hence,

1280 / 231503)()4,0()10 / 13,8 / 27()0,5()0,0()1,3(),(

)10 / 13,8 / 27(:,125

27

5

4)(

2

 xw y x

CP x x xw

  Abs. Max  Abs Min.

)0,5(,)4,0(,)0,0(

,3),( verticeswithstriangleais R y x xy y x f 

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Practice More!!!

On page 1008 do the

following numbers:

11 , 13, 14, 17, 29, 31

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Simple Recall!!

Find the absolute extrema of the givenfunction

on the indicated closed and bounded regionR that satisfies the inequalities

2),( y x y x f 

1,0,022 y xand  y x

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Answer:

3

2,

3

1(x,y ) (0,0) (0, 1) (1,0) 

f (x,y) 0 0 0 

9

32

Abs. MinimumAbs. Maximum

A point in R

Points alongthe boundary

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Practice More!

On page 986-987 do the following:

Nos. 11, 13 , 33, 39