maxima and minima. article -1 definite,semi-definite and indefinite function
DESCRIPTION
Definite,Semi-Definite and Indefinite FunctionTRANSCRIPT
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MAXIMA AND MAXIMA AND MINIMAMINIMA
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ARTICLE -1ARTICLE -1
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Definite,Semi-Definite and Indefinite
Function
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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
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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
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NOTENOTE• A positive definite or a negative
definite function is said to be a definite function.
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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
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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
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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
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EXAMPLE The function defined by is an indefinite
function.
Here can be positive, zero or negative.
RRf 2:2),(754),( Ryxyxyxf
),( yxf
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ARTICLE -2ARTICLE -2
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CONDITIONS FOR A DEFINITE FUNCTION
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Quadratic Expression of Two Real Quadratic Expression of Two Real Variables Variables
LetLet
]2[10,2),( 22222 abyahxyxaa
abyhxyaxyxf
])()[(1 222 yhabhyaxa
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casescases
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CASES- 1CASES- 1 If and ,then
f is positive semi-definite.
02 hab0a
2),(0),( Ryxyxf
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CASES- 2CASES- 2If and ,then
f is negative semi-definite.
02 hab0a
2),(0),( Ryxyxf
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CASES- 3CASES- 3 If ,then f(x,y) can be of any sign.
f is indefinite.
02 hab
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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 ,,
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ARTICLE -3ARTICLE -3
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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.
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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.
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EXTREME VALUE
A maximum or a minimum value of a function is called an extreme value.
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NOTE and are necessary but not
sufficient conditions.
02
2
xf
0yf
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ARTICLE -4ARTICLE -4
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WORKING METHOD FOR MAXIMUM AND MINIMUM
Let f(x,y) be given functions
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STEPS
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STEP-1 Find and
xf
yf
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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
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STEP-3 Find and calculate
values of A,B,C for each points.
2
22
2
2
,,y
fCyxfB
xfA
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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
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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
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NOTEThe point are called
stationary or critical points and values of f(x,y) at these points are called stationary values.
.......).........,(),,( 2211 yxyx
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LAGRANGE’S METHOD OF LAGRANGE’S METHOD OF UNDETERMINRD MULTIPLERSUNDETERMINRD MULTIPLERS
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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
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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.......(