03 relative and absolute extrema - handout
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Math 55 chapter 3TRANSCRIPT
Relative and Absolute Extrema
Math 55 - Elementray Analysis III
Institute of Mathematics
University of the Philippines
Diliman
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Recall
For a function f(x)
a is a critical number of f if either f 0(a) = 0 or f 0(a) doesnot exist.
The relative extrema of f occurs at the critical number, i.e.if a has a relative extremum at a, then a is a criticalnumber of f .
If f is continuous on a closed interval [a, b], then f attainsits abolute extrema on [a, b] at the critical points in [a, b] orat the endpoints a and b.
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Relative and Absolute Extrema
Definition
A function f(x, y) has a
relative minimum at (a, b) if f(a, b) f(x, y);
relative maximum at (a, b) if f(a, b) � f(x, y)
for all (x, y) on some disk centered at (a, b). If the aboveinequality holds for all (x, y) in the domain of f , then f has anabsolute minimum or absolute maximum at (a, b).
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Critical Point
Definition
The point (a, b) is a critical point or stationary point off(x, y) if either
rf(a, b) = ~0 (i.e., fx
(a, b) = 0 and f
y
(a, b) = 0),
one of fx
(a, b) or fy
(a, b) does not exist.
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Relative Extrema
Theorem
If f(x, y) has a relative extremum at (a, b), then (a, b) is a
critical point of f(x, y) and in fact, rf(a, b) = ~0.
Proof. Define g(x) = f(x, b) and suppose f has a relativeextremum at (a, b). Then g(x) has relative extremum (of thesame kind) at x = a, i.e., g0(a) = 0. Thus, g0(a) = f
x
(a, b) = 0.
By defining h(y) = f(a, y) and using similar aruments, we havef
y
(a, b) = 0.
This means rf(a, b) = ~0 and hence (a, b) is a critical point of f⌅
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Saddle Point
Consider f(x, y) = x
2 � y
2. Note that
f
x
(x, y) = 2x f
y
(x, y) = �2y
so the only critical point of f is (0, 0).
Graph of f(x, y) = x
2 � y
2
In this case we call (0, 0) a saddle point of f .
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Relative Extrema and Saddle point
Second Derivatives Test
Suppose (a, b) is a critical point of f(x, y), with second orderpartial derivatives of f continuous on some region containing(a, b). Define D = f
xx
(a, b)fyy
(a, b)� [fxy
(a, b)]2. Then f has
a relative minimum at (a, b) if D > 0 and f
xx
> 0;
a relative maximum at (a, b) if D > 0 and f
xx
< 0;
a saddle point at (a, b) if D < 0.
Remarks.
No conclusion can be made if D = 0.
The formula for D can be written as the determinant of the
matrix
f
xx
f
xy
f
yx
f
yy
�.
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Relative Extrema and Saddle point
Example
Find and classify all critical points of f(x, y) = 4+x
3+ y
3� 3xy
Solution. Solving the system
(f
x
(x, y) = 0
f
y
(x, y) = 0)
(3x2 � 3y = 0
3y2 � 3x = 0
gives the solutions (0, 0) and (1, 1).
Note that fxx
(x, y) = 6x; f
yy
(x, y) = 6y; f
xy
(x, y) = �3
f
xx
f
yy
f
xy
D
(0, 0) 0 0 �3 0� (�3)2 = �9(1, 1) 6 6 �3 6(6)� (�3)2 = 27
Hence f(x, y) has a saddle point at (0, 0) (sinceD < 0) and a relative minimum at (1, 1) (sinceD > 0 and f
xx
> 0)Graph of
f(x, y) = 4+x
3+y
3�3xy
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Example
Consider the function f(x, y) = x
2
ye
�x
2�y
2
.To find the critical points, we solve the system
(f
x
(x, y) = �2xy(x2 � 1)e�x
2�y
2
= 0
f
y
(x, y) = x
2(1� 2y2)e�x
2�y
2
= 0
Observe that 8y, (0, y) is a solution to the system above.
f
xx
= 2y(2x4 � 5x2 + 1)e�x
2�y
2
;
f
yy
= 2x2y(2y2 � 3)e�x
2�y
2
;
f
xy
= 2x(x2 � 1)(2y2 � 1)e�x
2�y
2
Note that when x = 0, f
yy
(0, y) = 0and f
xy
(0, y) = 0.Graph of f(x, y) = x
2ye
�x
2�y
2
Thus, D = f
xx
(0, y)fyy
(0, y)� f
xy
(0, y) = 0.
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Relative Extrema
Recall. If a continuous function f(x) has only one criticalnumber, then a relative maximum(minimum) has to be theabsolute maximum(minimum). However, this is not true for
f(x, y).
Consider f(x, y) = 3xey � x
3 � e
3y. The only critical point of fis (1, 0).
Observe from the graph that
f has a relative maximum at (1, 0) but it is not the absolutemaximum.
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Absolute Extrema
Definition
Let R ⇢ R2. Then R is said to be
closed if it contains all its boundary points; open if itdoes not contain any of its boundary points
bounded if it can be contained in a disk (of some radius).
Theorem
If f(x, y) is continuous on some closed and bounded region
R ⇢ R2
, then there exist points (x1
, y
1
), (x2
, y
2
) 2 R such that
f(x1
, y
1
) � f(x, y) and f(x2
, y
2
) f(x, y) for all (x, y) 2 R, i.e.
f attains its absolute maximum and absolute minimum in R at
(x1
, y
1
) and (x2
, y
2
), resp.
Remark. If f has an absolute extremum at (x1
, y
1
), then(x
1
, y
1
) is either a critical point or a boundary point of f .Math 55 Relative and Absolute Extrema 11/ 15
Absolute Extrema
Example
Find the absolute minimum and maximum values off(x, y) = x
2 � 2xy + 2y on the triangular region D with vertices(0, 0), (4, 0) and (0, 2).
Solution. We first find the critical points of f . The system(f
x
= 2x� 2y = 0
f
y
= �2x+ 2 = 0has a unique solution (1, 1) 2 D.
Next, consider the boundary of D composed of 3 line segments:
1 2 3 4
1
2
(1, 1)
l2
l1
l3
On l
1
, y = 0. Define g
1
(x) = f(x, 0) = x
2.Note that g(x) attains its extremum wheng
0(x) = 2x = 0, i.e. at the point (0, 0) andat the other endpoint (4, 0).
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Absolute Extrema
On l
2
, x = 0. Define g
2
(y) = f(0, y) = 2y. Note that g2
(y)attains its extremum at the endpoints (0, 0) and (0, 2).
On l
3
, y = 4�x
2
. Define
g
3
(x) = x
2 � 2x
✓4� x
2
◆+ 2
✓4� x
2
◆
= 2x2 � 5x+ 4
It attains its extremum wheng
0(x) = 4x� 5 = 0, i.e. at�5
4
,
11
8
�.
So the candidates for the absolute extrema are: (1, 1), (0, 0),(4, 0), (0, 2) and
�5
4
,
11
8
�
f(1, 1) = 1f(0, 0) = 0
f(4, 0) = 16f(0, 2) = 4
f
�5
4
,
11
8
�= 7
8
Hence f has a maximum value of 16 at (4, 0) and minimumvalue of 0 at (0, 0).
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Exercises
1 Find and classify all critical points ofa. f(x, y) = x
3y + 12x2 � 8y
b. f(x, y) = e
x cos y
c. f(x, y) = xy + x
�1 + y
�1
d. f(x, y) = (x2 + y
2)ey2�x
2
2 Show that⇣±1,±
p2
2
⌘are critical points of
f(x, y) = x
2
ye
�x
2�y
2
and classify each of them.
3 Find the points on the surface y
2 = 9 + xz that are closestto the origin. What is the distance of the surface from theorigin?
4 Find the absolute extrema of f(x, y) = 2x3 + y
4 over theunit disk centered at the origin.
5 Find the dimension of the rectangular box with largestvolume if the total surface area is 64 cm2.
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References
1 Stewart, J., Calculus, Early Transcendentals, 6 ed., ThomsonBrooks/Cole, 2008
2 Leithold, L., The Calculus 7
3 Dawkins, P., Calculus 3, online notes available athttp://tutorial.math.lamar.edu/
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