extreme values and saddle points

8/10/2019 Extreme Values and Saddle Points

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FUNCTIONS OF TWO

VARIABLES


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Definition:

A functionf(x,y) has an absolute maximum (or global maximum)at(a,b) if f(a,b) f(x,y) for all (x,y) in D, where D is the domain of

f(x,y). The number f(a,b) is called maximum value offon D.

Similarly,f(x,y) has an absolute minimum (or relative maximum)at(a,b) iff(a,b) f(x,y) for all (x,y) in D and the number f(a,b) iscalled the minimum value offon D.

The maximum and minimum values offare called the extremevalues off.

Absolute Maximum and Minimum


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Local Maximum and Minimum

Definition:

A functionf(x,y) has a local maximum at (a,b) iff(a,b) f(x,y) when (x,y) is near(a,b).

Similarly,f(x,y) has a local minimum at(a,b) if

f(a,b) f(x,y) when (x,y) is near(a,b).


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Local Extrema

Point (a,b,f(a,b)) is a (local) maximum


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Point (a,b,f(a,b)) is a (local) minimum

Local Extrema


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Critical Points

An interior point of the domain of a functionf(x,y) where:

Both and are zeroor

Where one or both of and Dont exist

is called a critical point off(x,y).

xf

xfyf

yf

Local Extrema exist only at the critical points or at the

boundary points of the domain


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Point (a,b,f(a,b)) is a saddle point

A differentiable functionf(x,y) has a saddle point at a critical

point(a,b) if there are domain points(x,y) near(a,b) where

f(x,y) > f(a,b) and domain points(x,y) wheref(x,y) < f(a,b).

The corresponding point(a,b,f(a,b)) on the surfacez = f(x,y)

is called a saddle point of the surface.

Saddle Point


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LOCAL EXTREMA


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Second Derivative Test

2D xyyyxx

yyxy

xyxxfff

ff

ff

Iff(x,y) and its first and second derivatives are continuous near

a critical point(a,b) then

fhas local maximum at(a,b) if 0and02

xxxyyyxx ffff

fhas local minimum at(a,b) if 0and02

xxxyyyxx ffff

fhas saddle point at(a,b) if 02 xyyyxx fff

Saddle point at(a,b) is (a,b,f(a,b))

Test fails at (a,b) if 02 xyyyxx fff


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yxy

yxf 23

3),(

Example

Locate and identify the critical points of the following

function


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yxy

yxf 23

3),(

Solution

Example


function

Step I: Find the critical points

101

002

1,2

2

2y

yyf

xx-f

yfxf

y

x

x

yyfff

yfff

xyyyxx

yyxyxx

40)2(2D

202

2

Critical points are (0,1) and (0,-1)

Step II: Find the Second derivatives


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3

210

3

1)1,0(

04)1(4-4yD(0,1)At

f

-

Step III: Test the critical points

3

210

3

)1()1,0(

02)1,0(

04)1(4D(0,-1)At

3

f

f

-

xx

3

20,1, Is a saddle point

3

20,-1, Is a maximum point


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444),( yxxyyxf

Example


function


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444),( yxxyyxf

Solution

Example


function


1,1,00)1(

0044

044

44,44

8

93

33

3

y

3

xxx

xxyxf

xyxyf

yxfxyf

y

x

x

1614416)12(12D

12412

22222

22

yxyxfff

yffxf

xyyyxx

yyxyxx

Critical points are (0,0), (1,1) and (-1,-1)Step II: Find the Second derivatives


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0)0,0(

01616144D(0,0)At 22

f

-yx


00,0, Is a saddle point

21,1, Is a maximum point2)1,1(

01212012816144D(1,1)At

2

22

f

xf

yx

xx

21,-1,-Is a maximum point

2)1,1(

01212

012816144D(-1,-1)At

2

22

f

xf

yx

xx


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yxy

xyxf

11),(

Example


function


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yxy

xyxf

11),(

Solution

Example


function


1,00)1(

00/1

/10y/1

/1,y/1

3

42

22

2

y

2

xxx

xxyxf

xyxf

yxfxf

y

x

x

Critical point is (1,1)

Note: The point (0,0) is not in the domain of f(x,y), so it cant be critical point.


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21,1, Is a minimum point

2)1,1(

02/2

031/4D(1,1)At

3

33

f

xf

yx

xx

1/41)/2)(/2(D

/21/2

33332

33

yxyxfff

yffxf

xyyyxx

yyxyxx

Step II: Find the Second derivatives


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ABSOLUTE EXTREMA


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Example26),( 22 xyxyxyxf

Find the absolute extremum of the function

defined in the region R: 03,50:),( yxyxR

x

y

A B

CD


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Solution

Example26),( 22 xyxyxyxf


defined in the region R:

Interior points

06-3y062

202

2,62 y

yxf

yxxyf

xyfyxf

x

y

x

Critical point is (4,-2)

03,50:),( yxyxR

102)4(6)2()2(4)4()2,4( 22 f

x

y

A B

CD


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Boundary points

1: Along AB: y=0

3062 xx

dx

df

Critical point on AB is (3,0)

End points of AB

22)0(6)0()0,0( 2 f

26)0,( 2 xxxf

72)3(6)3()0,3( 2 f

32)5(6)5()0,5( 2 f


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2: Along BC; x=5

2/5052 yydy

df

Critical point on BC is (5,-5/2)

End points of BC

93)3(5)3()3,5( 2 f

33)0(5)0()0,5( 2 f

35),5( 2 yyyf

4/373)2/5(5)2/5()2/5,5( 2 f


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3: Along CD; y=-3

2/9092 xxdx

df

Critical point on AC is (9/2,-3)

End points of CD

1111)0(9)0()3,0( 2 f

911)5(9)5()3,5( 2 f

119)3,( 2 xxxf

4/3711)2/9(9)2/9()3,2/9( 2 f


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11)3,0( f

10)2,4( f

Absolute Maximum

Absolute Minimum

2: Along AD; x=0

002 yydy

df

Critical point on AD is (0,0)

End points of AD

112)3()3,0( 2 f

2),0(

2 yyf

220)0,0( 2 f

220)0,0( 2 f

x

y

A B

CD


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Example

1442),( 22 yyxxyxf


defined in the region R bounded by lines x=0,y=2 and y=2x.

y

)0,0(A

)2,0(B )2,1(C

x


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Solution

Example

1442),( 22 yyxxyxf


defined in the region R bounded by lines x=0,y=2 and y=2x.

Interior points

2042

1044

42,44 y

yyf

xxf

yfxf

y

x

x

Critical point is (1,2) BUT (1,2) IS NOT AN INTERIOR POINT

y

)0,0(A

)2,0(B )2,1(C

x


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Boundary points

1: Along AB; x=014),0(

2 yyyf

2042 yy

dy

df

Critical point on AB is (o,2)

31)2(4)2()2,0( 2 f

End points of AB

31)2(4)2()2,0( 2 f

11)0(4)0()0,0( 2 f


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2: Along BC; y=2

1044 xxdx

df

Critical point on BC is (1,2)

End points of BC

53)1(4)1(2)2,1( 2 f

33)0(4)0(2)2,0( 2 f

342)2,( 2 xxxf

51)2(4)2()1(4)1(2)2,1( 22 f


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3: Along AC; y=2x

101212 xxdx

df

Critical point on AC is (1,2)

End points of AC

53)1(4)1(2)2,1( 2 f

11)0(12)0(6)0,0( 2 f

51)2(4)2()1(4)1(2)2,1( 22 f

1126)2,( 2 xxxxf

1)0,0( f

5)2,1( f

Absolute Maximum

Absolute Minimum

y

)0,0(A

)2,0(B)2,1(C

x


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Example

2/2),( 2 yxxyxyxf


defined in the region R bounded by lines and .2xy 22 xy


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Example

2/2),( 2 yxxyxyxf


defined in the region R bounded by lines and .

y

)1,1(A )1,1(B

x

2xy 22 xy

2

xy

2

2 xy


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Solution

Example

2/2),( 2 yxxyxyxf


defined in the region R bounded by lines and .

Interior points

1022

2/102/1

2/1,22 y

yyxf

xxf

xfyxf

x

y

x

Critical point is (1/2,1)

y

)1,1(A )1,1(B

x

2xy 22 xy

2

xy

2

2 xy

4/32/1)2/1(21)2/1()2/1()1,2/1( 2 f


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2: Along

1,0033 2 xxxdx

df

Critical point on are (0,2) and (1,1)

22 xy

12/3)2,( 232 xxxxf

22 xy

112/)0(3)0()2,0( 23 f

2/112/)1(3)1()1,1( 23 f

2/3)1,1( f

1)2,0( f

Absolute Maximum

Absolute Minimum

y

)1,1(A )1,1(B

x

2xy

22 xy

extreme values and saddle points

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