maxima and minima. article -1 definite,semi-definite and indefinite function

MAXIMA AND MAXIMA AND MINIMAMINIMA

ARTICLE -1ARTICLE -1

Definite,Semi-Definite and Indefinite

Function

DEFINITE FUNCTION

A real valued function f with domain is said to be positive definite if f(x)>0 and negative definition if f(x)<0

fDx

fDx

EXAMPLE The function defination by is positive

definite. Here The function defined by is negative definite. Here

RRf 3:3222 ),,(9),,( Rzyxzyxzyxf

3),,(09),,( Rzyxzyxf RRf 2: 222 ),()9(),( Ryxyxyxf

2),(09),( Ryxyxf

NOTENOTE• A positive definite or a negative

definite function is said to be a definite function.

Semi-Definite Semi-Definite A real valued function A real valued function f f with domain is said to with domain is said to

be semi-definite if it vanishes at some points of and be semi-definite if it vanishes at some points of and when it is not zero, it is of the same sign throughoutwhen it is not zero, it is of the same sign throughout

nf RD

fD

EXAMPLE The function defined by

is semi definite as and when then where

RRf 3:

3222 ),,(),,( Rzyxzyxzyxf

0)0,0,0( f

)0,0,0(),,( zyx 0),,( zyxf3),,( Rzyx

INDEFINITE FUNCTIONINDEFINITE FUNCTION A real valued function A real valued function f f with domainwith domain is said to be indefinite if it can take values which is said to be indefinite if it can take values which

have different signs have different signs i.e.,i.e., it is nether definite nor it is nether definite nor semi definite function. semi definite function.

nf RD

EXAMPLE The function defined by is an indefinite

function.

Here can be positive, zero or negative.

RRf 2:2),(754),( Ryxyxyxf

),( yxf

ARTICLE -2ARTICLE -2

CONDITIONS FOR A DEFINITE FUNCTION

Quadratic Expression of Two Real Quadratic Expression of Two Real Variables Variables

LetLet

]2[10,2),( 22222 abyahxyxaa

abyhxyaxyxf

])()[(1 222 yhabhyaxa

casescases

CASES- 1CASES- 1 If and ,then

f is positive semi-definite.

02 hab0a

2),(0),( Ryxyxf

CASES- 2CASES- 2If and ,then

f is negative semi-definite.

02 hab0a

2),(0),( Ryxyxf

CASES- 3CASES- 3 If ,then f(x,y) can be of any sign.

f is indefinite.

02 hab

QUADRATIC EXPRESSION OF THREE QUADRATIC EXPRESSION OF THREE REAL VARIABLESREAL VARIABLES

LetLet

When then When then

If are all positiveIf are all positive

ff is positive semi-definite. is positive semi-definite.

When ,then if the above three When ,then if the above three expression are alternately negative and positive .expression are alternately negative and positive .

f f is negative semi-definite.is negative semi-definite.

hxygzxfyzczbyaxzyxf 222),,( 222

0a 3),,(0),,( Rzyxzyxf

0a3),,(0),,( Rzyxzyxf

cfgfbhgha

bhha

a ,,

ARTICLE -3ARTICLE -3

MAXIMUM VALUE

A function f(x,y) is said to have a maximum value at x=a, y=b

if f(a,b)>f(a+h,b+k) for small values of h and k, positive or negative.

MINIMUM VALUE

A function f(x,y) is said to have a maximum value at x=a, y=b

if f(a,b)<f(a+h,b+k) for small values of h and k, positive or negative.

EXTREME VALUE

A maximum or a minimum value of a function is called an extreme value.

NOTE and are necessary but not

sufficient conditions.

02

2

xf

0yf

ARTICLE -4ARTICLE -4

WORKING METHOD FOR MAXIMUM AND MINIMUM

Let f(x,y) be given functions

STEPS

STEP-1 Find and

xf

yf

STEP-2STEP-2 Solve the equations and simultaneously Solve the equations and simultaneously

for for xx and and y y ..

Let be the pointsLet be the points

xf

...),........,(),,( 221 yxyx

0

xf 0

yf

...),........,(),,( 221 yxyx

STEP-3 Find and calculate

values of A,B,C for each points.

2

22

2

2

,,y

fCyxfB

xfA

STEP-4 If for a point ,we have and then f(x,y) is a maxima for this pair and maximum value is

If for point ,we have

and then f(x,y) is a minimum for this pair and minimum value is

If for point then there is neither max. nor minimum of f(x,y) . In this case f(x,y) is said to have a saddle at

),( 11 yx 02 BAC

02 BAC

02 BAC

0A

0A

),( 11 yxf),( 11 yx

),( 11 yx

),( 11 yx),( 11 yxf

0),(),( kbhafbaf

If for some point If for some point (a,b)(a,b) the case is doubtful the case is doubtful

In this case,In this case, if for small values ofif for small values of h h and and kk, positive or , positive or

negative, then negative, then ff is max. at is max. at (a,b).(a,b).if for small values of if for small values of h h and k, positive or and k, positive or

negative,then negative,then f f is min . at is min . at (a,b).(a,b).If dose not keep the same sign for small If dose not keep the same sign for small

values of values of hh and and kk, then there is neither max.nor minimum value., then there is neither max.nor minimum value.

02 BAC

0),(),( kbhafbaf

),(),( kbhafbaf

NOTEThe point are called

stationary or critical points and values of f(x,y) at these points are called stationary values.

.......).........,(),,( 2211 yxyx

LAGRANGE’S METHOD OF LAGRANGE’S METHOD OF UNDETERMINRD MULTIPLERSUNDETERMINRD MULTIPLERS

Let f(x,y,z) be a function of x,y,z which is to be examied for maximum or minimum value and let the variable be connected by the relation

…….(1)

Since f(x,y,z) is to have a maximum or minimum value

0),,( zyx

0,0,0

zf

yf

xf

0,,

dz

zfdy

yfdx

xf

Multiplying(2) by (3) by and adding, we get,

For this equation to be satisfied identically, coeffs. of dx,dy,dz should be separately zero.

Equation (1),(4),(5) and(6) give us the value of x,y,z, for which f(x,y.z)is maximum and minimum.

0][][][

dz

zzfdy

yyfdx

xxf

0

xxf

0

yyf

0

zzf

)4.......(

)5.......(

)6.......(