of 15
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x remumx remum oo
functionsfunctionsObjectives:Objectives:
Compute and classify critical pointsCompute and classify critical points
Hasfazilah Ahmat
Jan 2010
Compute the relativeCompute the relative extremaextrema for afor a
function of two variablesfunction of two variablesSolve appliedSolve applied extremaextrema problemsproblems
Graph of function of twoGraph of function of two
variablesvariables
Hasfazilah Ahmat
Jan 2010
8/13/2019 8 Extremum of Function
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Mat235 4/13/20
Critical pointsCritical points
Let f(x,y) be a function of two variables. Apoint (a, b) is a critical point of f(x,y) if either
fx (a,b) = 0 and
f a,b = 0 or
Hasfazilah Ahmat
Jan 2010
( ) 2
f x , y 4 xy 2 x y= +fx (a,b) = 0 orfy (a,b) = 0 does not exist
Critical pointsCritical points -- stepssteps
x y
fx = 0 or fy = 0
Solve fx = 0 or fy = 0
Hasfazilah Ahmat
Jan 2010
Determine the critical points
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Example 1Example 1Find the critical points for the functionFind the critical points for the function
Solution:Solution:
( ) 2f x , y 4 xy 2 x y= +
LetLet andand
x 4 y 4 x=
yf 4 x 1= +
xf 0= yf 0=
xf 4 y 4 x 0= =
4 x 1 0= + =
Hasfazilah Ahmat
Jan 2010
Solve and simultaneouslySolve and simultaneously
y
xf 0= y 0=
4 y 4 x 0 =
y x=
4 x 1 0+ =1
x4
=
Example 1Example 1Find the critical points for the functionFind the critical points for the function
Solution:Solution:
( ) 2f x , y 4 xy 2 x y= +
Determine the critical pointsDetermine the critical points
y x= 1x
4=
1y
4=
Hasfazilah Ahmat
Jan 2010
Therefore, the critical point isTherefore, the critical point is
1 1,
4 4
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Find the critical points for the functionFind the critical points for the function
Solution:Solution:
( ) 2 3x , y 3 4 xy 2 x y= + +
Example 2Example 2
LetLet andand
x 4 y 4 x= + 2
yf 4 x 3 y=
xf 0=
yf 0=
xf 4 y 4 x 0= + =
Hasfazilah Ahmat
Jan 2010
yf 4 x 3 y 0= =
Find the critical points for the functionFind the critical points for the function
Solution:Solution:
( ) 2 3x , y 3 4 xy 2 x y= + +
Example 2 (cont.)Example 2 (cont.)
Solve and simultaneouslySolve and simultaneouslyxf 0= yf 0=
4 y 4 x 0+ =
y x=
( )224 x 3 y 4 x 3 x 0 = =
( )x 4 3 x 0 =4
x 0 ,3
=
Hasfazilah Ahmat
Jan 2010
Determine the critical pointsDetermine the critical pointsy 0=x 0=
4y
3=
4x
3=
( )0 ,04 4
,3 3
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Find the critical points for the functionFind the critical points for the function
Solution:Solution:
( ) 3 3 2 2f x , y 2 x y 3 x 18 y 81 y 5= + + + +
Example 3Example 3
From (1)From (1)
2
xf 6 x 6 x 0 ( 1 )= + =
2
yf 3 y 36 y 81 0 ( 2 )= + =
( )6 x x 1 0+ =x 0 , 1=
Hasfazilah Ahmat
Jan 2010
From (2)From (2)2
y 12 y 27 0 + =( ) ( )y 9 y 3 0 =
y 9 ,3=
Therefore, thecritical pointsare: (0,3), (0,9),(-1,3) and (-1,9)
Warm up exerciseWarm up exercise
( ) 3 3x , y 3 y x y 3 x= +
Hasfazilah Ahmat
Jan 2010
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Mat235 4/13/20
ExtremumExtremum
Minimum point Maximum point
Hasfazilah Ahmat
Jan 2010
Saddle point
Second partial derivative testSecond partial derivative testSuppose the second partial derivativesSuppose the second partial derivatives of f areof f are
continuoucontinuous on critical point (s on critical point (a,ba,b) and suppose) and supposethatthat ffxx((a,ba,b) = 0 and) = 0 and ffyy((a,ba,b)=0. Let)=0. Let
2
xx yy xya , a , a , a ,= =
Hasfazilah Ahmat
Jan 2010
D > 0 And
fxx > 0
D > 0 And
fxx< 0
D < 0 D = 0 Inconc
lusive
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Mat235 4/13/20
Identify critical point, (a,b)
Second partial derivative testSecond partial derivative testDiscriminantDiscriminant :: ( ) ( ) ( ) ( )
2
xx yy xyD a ,b f a ,b f a ,b f a ,b = =
Compute fxx, fyy and fxy
At critical point, evaluate Fxx (a,b), Fyy (a,b) and Fxy(a,b)
2
D =
Hasfazilah Ahmat
Jan 2010
,
Check D and fxx (a,b) (if necessary)
Classify as minimum, maximum, saddle point or noconclusion
xx yy xy
The critical points for the functionThe critical points for the function
are (0, 0) and (4/3,are (0, 0) and (4/3, --4/3). Classify the critical points4/3). Classify the critical points
( ) 2 3x , y 3 4 xy 2 x y= + +
Example 4Example 4
Solution:Solution:
Identify theIdentify the critical pointscritical points
ComputeCompute ffxxxx,, ffyyyy andand ffxyxy
( )0 ,0 4 4,3 3
=
xf 4 y 4 x= + 2
y 4 x 3 y=
Hasfazilah Ahmat
Jan 2010
xx yy xy
Critical point
(0,0) 4 0 4
(4/3, -4/3) 4 8 4
xxf 4= yyf 6 y= xyf 4=
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Mat235 4/13/20
The critical points for the functionThe critical points for the function
are (0, 0) and (4/3,are (0, 0) and (4/3, --4/3). Classify the critical points4/3). Classify the critical points
( ) 2 3x , y 3 4 xy 2 x y= + +
Example 4 (cont.)Example 4 (cont.)
ComputeCompute discriminantdiscriminant
Critical pointD
(0,0) 4 0 4 -16
(4/3, -4/3) 4 8 4 16
xxf 4= yyf 6 y= xyf 4=
Hasfazilah Ahmat
Jan 2010
Check D andCheck D and xxxx (if necessary) and classify nature(if necessary) and classify nature
Critical point D Classification
(0,0) -16 < 0 Not necessary to check Saddle point
(4/3, -4/3) 16 4 > 0 Relative minimum
xxf 4=
Locate and classifyLocate and classify the critical point forthe critical point for
Solution:Solution:
( ) 2 2x , y 3 x 2 xy y 8 y= +
Example 5Example 5
xf 6 x 2 y 0= =
yf 2 x 2 y 8 0= + =
( )y 3 x ..... 1=
( )2 x 2 3 x 8 0 + =x 2= Into (1)
( )y 3 2 6= = Critical point
Hasfazilah Ahmat
Jan 2010
xxf 6=
yyf 2= xy 2=
Critical point
(2,6) 6 2 -2
xxf 6= yyf 2= xyf 2=
(2,6)
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Locate and classify the critical point forLocate and classify the critical point for
ComputeCompute discriminantdiscriminant
Example 5 (cont.)Example 5 (cont.)
( ) 2 2x , y 3 x 2 xy y 8 y= +
Check D andCheck D and ffxxxx (if necessary) and classify nature(if necessary) and classify nature
Critical pointD
(2,6) 6 2 -2 8
xxf 6= yyf 2= xyf 2=
Hasfazilah Ahmat
Jan 2010
Critical point D Classification
(2,6) 8 > 0 Fxx > 0 Minimum point
xxf 6=
Locate allLocate all relativerelative extremumextremum and saddle points ofand saddle points of
Solution:Solution:
ExampleExample 66
( ) 3 3x , y x y 3 xy= +
2
xf 3 x 3 y 0= + =
2
yf 3 y 3 x 0= + =
( )2y x ..... 1=
( )2
23 x 3 x 0 + =
4x x 0 + =
3x x 1 0 + =
Hasfazilah Ahmat
Jan 2010
Into (1)x 0 ,1=
Critical point (0, 0) and (1, -1)
y 0=x=0
y 1= X=1
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Locate all relativeLocate all relative extremumextremum and saddle points ofand saddle points of
Example 6 (cont.)Example 6 (cont.)
( ) 3 3x , y x y 3 xy= +
xx 6 x= 6 y=
xf 3=
Critical pointD
(0,0) 0 0 3 -9
(1,-1) 6 6 3 27
xxf 6 x= yyf 6 y= xyf 3=
2
xx yy xyD f f f=
Hasfazilah Ahmat
Jan 2010
ass ca on
(0,0) -9 Saddle point
(1,-1) 27 Fxx > 0 Minimum point
xx
( ) 3 2 3f x , y 2 x 6 xy 3 y 150 x= +
Find the stationary point of the function andFind the stationary point of the function anddeterminedetermine the naturethe nature
ExampleExample 77
Solution:Solution:
2 2
xf 6 x 6 y 150 0= + =
212 x 9 0= =
2 26 x 6 y 150+ =
( )2 2y 25 x .... 1=
Hasfazilah Ahmat
Jan 2010
y
Into (1)
( )3 y 4 x 3 y 0 =4 x
y 0 ,3
=
225 x 0 =y=0 x 5=
8/13/2019 8 Extremum of Function
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Mat235 4/13/20
( ) 3 2 3f x , y 2 x 6 xy 3 y 150 x= +
Find the stationary point of the function andFind the stationary point of the function and
determinedetermine the naturethe nature
ExampleExample 7 (cont.)7 (cont.)
Solution:Solution:
4 xy
3=
2216 x 25 x
9=
3 3=
225 x25 0
9 =
5 x 5 x5 5 0
+ =
Critical point :(5, 0), (-5, 0),(3 4) (-3 -4)
Hasfazilah Ahmat
Jan 2010
x 3,=
3 3
4 xy3=
( )4 3y 43= =
x 3,= ( )4 3
y 43
= =
( ) 3 2 3f x , y 2 x 6 xy 3 y 150 x= +
Find the stationary point of the function andFind the stationary point of the function anddeterminedetermine the naturethe nature
ExampleExample 7 (cont.)7 (cont.)
Solution:Solution:
point :(5, 0), (-5, 0),(3, 4), (-3, -4)
Criticalpoint D Conclusion
(5, 0) 60 60 0 3600 Minimum
xxf 12 x= yyf 12 x 18 y= xyf 12 y=
xxf 12 x=
yyf 12 x 18 y=
xyf 12 y=
2
xx yy xyD f f f=
Hasfazilah Ahmat
Jan 2010
(-5, 0) -60 -60 0 3600 Maximum
(3, 4) 36 -36 48 -3600 Saddle
(-3, -4) -36 36 -48 -3600 Saddle
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Find all the criticalFind all the critical points of the functionpoints of the function
Determine which give relative minima, maxima orDetermine which give relative minima, maxima orsaddle pointssaddle points
( ) 4 4x , y 4 xy x y=
ExampleExample 88
Solution:Solution:
3
xf 4 y 4 x 0= =
3f 4 x 4 y 0= =
( )3y x .... 1=
Hasfazilah Ahmat
Jan 2010x 0 ,1 , 1=
( )3
34 x 4 x 0 =
9x x 0 =
( )8x 1 x 0 =x 0 , y 0
x 1, y 1
x 1, y 1
= =
= =
= =
Find all the critical points of the functionFind all the critical points of the function
Determine which give relative minima, maxima orDetermine which give relative minima, maxima orsaddle pointssaddle points
ExampleExample 88 (cont.)(cont.)
( ) 4 4x , y 4 xy x y=
Solution:Solution:
point :(0, 0), (1, 1),(-1, -1)
Critical2=
2= 4=
2
xx 12 x= 2
yy 12 y=
xyf 4=
2
xx yy xyD f f f=
Hasfazilah Ahmat
Jan 2010
po n
(0, 0) 0 0 4 -16 saddle
(1, 1) -12 -12 4 128 Maximum
(-1, -1) -12 -12 4 128 maximum
xx yy xy
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Mat235 4/13/20
Find all the criticalFind all the critical points of the functionpoints of the function
Determine which give relative minima, maxima orDetermine which give relative minima, maxima orsaddle pointssaddle points
( ) 4 4x , y 4 xy x y=
ExampleExample 99
Solution:Solution:
3
xf 4 y 4 x 0= =
3f 4 x 4 y 0= =
( )3y x .... 1=
Hasfazilah Ahmat
Jan 2010x 0 ,1 , 1=
( )3
34 x 4 x 0 =
9x x 0 =
( )8x 1 x 0 =x 0 , y 0
x 1, y 1
x 1, y 1
= =
= =
= =
A rectangular boxA rectangular box opeopen at the top is to have an at the top is to have acolumecolume of 32 cmof 32 cm33. What must be the dimensions so. What must be the dimensions sothat the total surface is minimum?that the total surface is minimum?
ExampleExample 99
Solution:Solution:
V lwh 32= =
( )S l ,w ,h lw 2lh 2wh= + + lw
h32h
lw=
( ) 32 32
S l , w lw 2l 2wlw lw
= + +
w 264
S l 0= =
Hasfazilah Ahmat
Jan 2010
( ) 64 64
S l , w lww l
= + +
l 2
64S w 0
l= = ( )2
64w ..... 1
l=
w
( )264
l ..... 2w
=
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Mat235 4/13/20
A rectangular boxA rectangular box opeopen at the top is to have an at the top is to have a
columecolume of 32 cmof 32 cm33. What must be the dimensions so. What must be the dimensions sothat the total surface is minimum?that the total surface is minimum?
ExampleExample 9 (cont.)9 (cont.)
Solution:Solution:
V lwh 32= = 32
hlw
=
( )264
w ..... 1l
=64
l=
4ll 0
64 =
3
(1) Into (2)
Hasfazilah Ahmat
Jan 2010
2l ..... 2
w=
So, l = 4
( )264 l4l
l64
=
l 1 064
=
l 0 ,4=
2
64w 4
4= =
( ) ( )32
h 24 4
= =
A rectangular boxA rectangular box opeopen at the top is to have an at the top is to have acolumecolume of 32 cmof 32 cm33. What must be the dimensions so. What must be the dimensions sothat the total surface is minimum?that the total surface is minimum?
ExampleExample 9 (cont.)9 (cont.)
Solution:Solution:
ll 3
128S
l= ww 2
128S
w= lwS 1=
2
ll ww lwD S S S=
Hasfazilah Ahmat
Jan 2010
( )23 3D 1l w= ( )2
3 3D 1 34 4= =
llD 3 0 , f 0= > > The computed value produce
minimum total surface
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FindFind the dimensions of a sixthe dimensions of a six--faced box that has thefaced box that has the
shape of a rectangular prism with the largestshape of a rectangular prism with the largestpossible volume that you can make with 12 squarepossible volume that you can make with 12 square
ExampleExample 1010
. .
Solution:Solution:A 2 xy 2 yz 2zx 12= + + =
V xyz=
6 xyz
x y
=
+
( ) 6 xy
V x , y xyx y
=
+
( )2 2x 2
y x 2 xy 6V 0
x y
+ = =
+
Hasfazilah Ahmat
Jan 2010
( )
( )
2 2
y 2
x y 2 xy 6V 0
x y
+ = =
+2
x 2 xy 6 0+ =
2y 2 xy 6 0+ =
2 2x y 0 =
FindFind the dimensions of a sixthe dimensions of a six--faced box that has thefaced box that has theshape of a rectangular prism with the largestshape of a rectangular prism with the largestpossible volume that you can make with 12 squarepossible volume that you can make with 12 square
ExampleExample 10 (cont.)10 (cont.)
. .
Solution:Solution:2x 2 xy 6 0+ =
2y 2 xy 6 0+ =
2 2x y 0 =
x y , y=
Since it is a dimension, x -y
Hasfazilah Ahmat
Jan 2010
x y= 2 2x 2 x 6 0+ =
x 2= y 2= 6 2
z 22 2
= =