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regionaround(^ , ).
at , if , , forallpoints , insome
Afunctionoftwovariableshasa
a b
a b f x y ≤ f ab x y
local maximum
regionaround(^ , ).
at , if , , forallpoints , insome
Afunctionoftwovariableshasa
a b
a b f x y ≥ f ab x y
local minimum
- outside the region it is possible that the function could be larger
- outside the region it is possible that the function could be smaller
A point ( a , b ) iscalleda criticalpoint of f isoneofthefollowingistrue:
i ) ∇ f ( a , b ) = 0 ,thatisBOTH fx ( a , b ) = 0 and fy ( a , b ) = 0
ii ) fx ( a , b ) and/or fy ( a , b ) doesn'texist
( )
derivativesof exist ther e
andthefirstpartial
localminimumat ,
hasalocalmaximumor
f
ab
f
f x ( a , b ) = 0 and fy ( a , b ) = 0
( a , b ) isacriticalpoint
( a b )
f
localminimumat ,
hasalocalmaximumor
( a , b ) isacriticalpoint
notallcriticalpointsleadtoalocalmaximumorlocalminimum
2 2 2
1 z = y − x
maximum in the direction of the x-axis
minimum in the direction of the y-axis
f ( 0 , 0 ) = 0
the graph is in the shape of a saddle
the point (0,0,0) is called a saddlepoint
( )
( ) ( )
( ab )
f ab f ab
ab
x y
contiuousinsomeregionaround ,
thesecondpartialderivativesare
and
, 0 and , 0
Findallcriticalpoints , such that
( )( ) ( )
Evaluate at thesecritical points
Let
2
D
f f f f f
f f D (^) xx yy xy xy yy
xx xy = = −
D ( a , b ) > 0
f ( a , b ) > 0 xx
f ( a , b ) < 0 xx
f ( a , b ) isa localminimum
f ( a , b ) isa localmaximum
D ( a , b ) < 0 f (^ a , b )^ isa saddlepoint
D ( a , b ) = 0 thetestgivesnoinformation
at( a , b ) if f ( x , y ) f ( a , b ) forallpointsin thedomainof f
Afunctionoftwovariableshasan
absolute maximum
at ( a , b ) if f ( x , y ) f ( a , b ) forallpointsin thedomainof f
Afunctionoftwovariableshasan
absolute minimum
Usually the domain is restricted to some region
( )
1 1 and 1 1
RestrictedDomain:
, 4
2 2 2
− ≤ ≤ − ≤ ≤
= + + +
x y
f x y x y x y
absolute maximum
absolute minimum
iscalled ifitincludesits boundary.
A regioninR forusthiswillbethe plane
2
closed
xy
( inother wordsaregionisboundedifitisfinite)
iscalled ifitiscontainedwithinsomedisk
A regioninR forusthiswillbethe plane
2
bounded
xy
( )
2 closedboundedregion in
, iscontinuousinsome
ExtremeValueTheorem
S R
f xy ⇒
( , ) isanabsoluteminimum
sothat , isanabsolutemaximumand
therearepoints , and , in theregion
fc d
fab
ab cd S
This tells us that the points exist but it doesn’t tell us how to find them.
valuesof acontinuousfunction onaclosedregion :
Tofindtheabsolutemaximumandabsoluteminimum
f S
)
Evaluatethefunctionateachofthese points
1 Findallthecriticalpointsof f thatliein theregion S
)
theboundary. (This turnsintoaCalcIproblem)
2 Findallextremevaluesof f thatlieon
)
minimumvalueofthefunction f
steps 1 and 2 aretheabsolutemaximumvalueandabsolute
3 Thelargestandsmallestofthevaluesfoundin
( )
2 f x y , = x + xy
f (^) x = 2 x + y 0 f (^) y = x
set
( ) (^) {( ) }
2
Find the absolute maximum and absolute minimum values of
f x y , = x + xy on the region S : x y , x ≤ 2, y ≤ 1
1 ) Find possible critical pts. inside the region
set
Both have to be true at the same time plugging in x = 0 into f (^) x ⇒ y = 0
( )
Critical pt.
2 ) Find all extreme values of f that lie on the boundary.
L 1
L 2
L 3
L 4
a ) L 1 : x = − 2 ( ) ( ) ( )
2 f − 2, y = − 2 + − 2 y ⇒ f on L 1 = − 2 y + 4
f ′^ on L 1 = − 2 ≠ 0
no extreme points on L 1
c ) L 3 : x = 2 (^ )^
2 f 2, y = 2 + 2 y ⇒ f on L 3 = 2 y + 4 f ′ on L 3 = 2 ≠ 0
no extreme points on L 3
b ) L 2 : y = 1 (^ )^
2 f x ,1 = x + x^2 ⇒ f on L 2 = x + x f ′^ on L 2 = 2 x + 1 0
set = ⇒ x = − 21
1 2 ,
−
d ) L 4 : y = − 1 ( )
2 f x , − 1 = x − x ⇒ f on L 4 = x^2 − x
f ′^ on L 4 = 2 x − 1 1 ⇒ x = (^2)
1 2 ,^ −^1
f ( 0, 0 )= 0 ( )
1 1 f (^) 2 ,1 4 − −
1 1 f (^) 2 ,1 4 − − =
absolute maximum absolute minimum absolute minimum
