C3L4 Maxima and Minima of Function of Two Variables
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Transcript of C3L4 Maxima and Minima of Function of Two Variables
8/3/2019 C3L4 Maxima and Minima of Function of Two Variables
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Maxima and Minima ofFunction of Two Variables
C3L4
8/3/2019 C3L4 Maxima and Minima of Function of Two Variables
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Learning Outcomes
At the end of the lesson youshould be able to:
1. Define relative maximum andabsolute maximum.
2. Define relative minimum andabsolute minimum.
3. Find the critical points.
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Definition
A function f of two variables is said tohave a relative a relative minimum at apoint (xo,yo) if there is a disk centered at
(xo.yo) such that forall points (x,y) that lie in the disk, and f is said to have an absolute minimum at(xo,yo) if for all points
(x,y) in the domain of f.
),(),( 00 y x f y x f
),(),(00 y x f y x f
8/3/2019 C3L4 Maxima and Minima of Function of Two Variables
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z
Z= f(x,y)
Absolute Min; Relative Min;Absolute Max; Relative Max
A
BC
D
y
x
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Finding Relative Extrema
Theorem is f has a relative extremum at apoint (xo,yo), and if the first partialderivatives of f exist at this point, then
0),(0),(
0000 y x f and y x f y x
Note:
1.These are also called critical points.2. From the Extreme Value Theorem, if f(x,y) iscontinuous on a closed bounded set region R, thenf has both an absolute maximum and absolute
minimum.
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Second Partials Test
Theorem
Let f be a function of two variables withcontinuous second order PD in some disk
centered at a critical point (xo,yo), andlet
.,0.4
).,(int,0.3
).,(max,0),(&0.2
).,(min,0),(&0.1
,),(),(),(
00
0000
0000
00
2
0000
drawnbecanconclusionno D
y xat posaddleahas f then D
y xat imumrelativeahas f y x f D y xat imumrelativeahas f y x f D
If y x f y x f y x f D
xx
xx
xy yy xx
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Example 1
Locate all relative maxima, relativeminima, and saddle points, if any
Given:
a.
b. More problems!
y x y x y x f 33),( 33
22
44
33
cos),(
44),(
3),(
y x y x f
y x xy y x f
y xy x y x f
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Answer:
a)
.max.)1,1(,036,036
.min.)1,1(,06,036
int)1,1(&)1,1(,0361;1
rel f D
rel f D
posaddle D y x
xx
xx
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Answer
b)
(x,y)=(0,0)
D=-2<0 , therefore (0,0) is a saddlepoint.
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Absolute Extrema on Closed andBounded Sets
Steps to follow:
1. Find the critical points of f that is in theinterior of R.
2. Find all boundary points at which theabsolute extrema can occur.
3. Evaluate f(x,y) at the points obtained inthe previous step. The largest of thesevalue is the absolute maximum and thesmallest the absolute minimum.
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Example 2
)0,5(,)4,0(,)0,0(
,3),( verticeswithstriangleais R y x xy y x f
(0,0) (5,0)
(0, 4)
Find the extreme values given
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Answer (x,y)=(3,1)
When y=0, (x-axis), u(x)= -x no CP
When x=0 , (y-axis) v(y) =-3y no CP
Along y=-4/5 x +4 hence,
1280 / 231503)()4,0()10 / 13,8 / 27()0,5()0,0()1,3(),(
)10 / 13,8 / 27(:,125
27
5
4)(
2
xw y x
CP x x xw
Abs. Max Abs Min.
)0,5(,)4,0(,)0,0(
,3),( verticeswithstriangleais R y x xy y x f
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Practice More!!!
On page 1008 do the
following numbers:
11 , 13, 14, 17, 29, 31
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Simple Recall!!
Find the absolute extrema of the givenfunction
on the indicated closed and bounded regionR that satisfies the inequalities
2),( y x y x f
1,0,022 y xand y x
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Answer:
3
2,
3
1(x,y ) (0,0) (0, 1) (1,0)
f (x,y) 0 0 0
9
32
Abs. MinimumAbs. Maximum
A point in R
Points alongthe boundary
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Practice More!
On page 986-987 do the following:
Nos. 11, 13 , 33, 39