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Chapter 14 – Partial Derivatives14.7 Maximum and Minimum Values
14.7 Maximum and Minimum Values
Objectives: Use directional derivatives to
locate maxima and minima of multivariable functions
Maximize the volume of a box without a lid if we have a fixed amount of cardboard to work with
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14.7 Maximum and Minimum Values 2
Absolute & Local MaximumThere are two points (a, b) where f has a local maximum
—that is, where f(a, b) is larger than nearby values of f(x, y).
◦ The larger of these two values is the absolute maximum.
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14.7 Maximum and Minimum Values 3
Absolute & Local MinimaLikewise, f has two local minima—where f(a, b) is
smaller than nearby values.
◦ The smaller of these two values is the absolute minimum.
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14.7 Maximum and Minimum Values 4
Definition – Local Max and Min
If the inequalities in Definition 1 hold for all points (x, y) in the domain of f, then f has an absolute maximum (or absolute minimum) at (a, b).
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14.7 Maximum and Minimum Values 5
Theorem
Notice that this theorem can be stated in the notation of gradient vectors as ∇f(a, b) = 0.
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14.7 Maximum and Minimum Values 6
Definition – Critical PointA point (a, b) is called a critical point (or stationary
point) of f if either:
◦ fx(a, b) = 0 and fy(a, b) = 0
◦One of these partial derivatives does not exist.
Theorem 2 says that, if f has a local maximum or minimum at (a, b), then (a, b) is a critical point of f.
At a critical point, a function could have a local maximum or a local minimum or neither.
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14.7 Maximum and Minimum Values 7
Saddle PointA function need not have a maximum or minimum value
at a critical point.The figure shows how this is possible.◦ The graph of f is the hyperbolic paraboloid
z = y2 – x2.◦ It has a
horizontal tangent plane (z = 0) at the
origin.
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14.7 Maximum and Minimum Values 8
Saddle PointYou can see that f(0, 0) = 0 is:◦A maximum in the direction of the x-axis.◦A minimum in the direction of the y-axis.
Near the origin, the graph has the shape of a saddle.◦ So, (0, 0) is
called a saddle
point of f.
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14.7 Maximum and Minimum Values 9
Extreme Value at a Critical PointWe need to be able to determine whether
or not a function has an extreme value at a critical point.
The following test is analogous to the Second Derivative Test for functions of one variable.
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14.7 Maximum and Minimum Values 10
Second Derivative Test
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14.7 Maximum and Minimum Values 11
Notes on Second Derivative TestNote 1: In case c, ◦ The point (a, b) is called a saddle point of f .◦ The graph of f crosses its tangent plane at
(a, b).
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Notes on Second Derivative TestNote 2: If D = 0, the test gives no information: ◦ f could have a local maximum or local
minimum at (a, b), or (a, b) could be a saddle point of f.
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14.7 Maximum and Minimum Values 13
Notes on Second Derivative TestNote 3: To remember the formula for D, it’s helpful to
write it as a determinant:
2( )xx xy
xx yy xyyx yy
f fD f f f
f f
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14.7 Maximum and Minimum Values 14
VisualizationCritical Points from Contour MapsFamilies of Surfaces
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14.7 Maximum and Minimum Values 15
Example 1Find the local maximum and minimum
values and saddle point(s) of the function.
yxyxyxf 812),( 23
2223 52),( yxxyxyxf
y
xyxyxf
,sinsin),(
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14.7 Maximum and Minimum Values 16
Definition – Closed Set Just as a closed interval contains its endpoints, a closed
set in ℝ2 is one that contains all its boundary points.
NOTE: A boundary point of D is a point (a, b) such that every disk with center (a, b) contains points in D and also points not in D.
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Closed SetIf even one point on the boundary curve
were omitted, the set would not be closed.
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Definition – Bounded SetA bounded set in ℝ2 is one that is contained within some
disk.
◦ In other words, it is finite in extent.
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Theorem
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Closed Interval Method
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Example 2 – pg. 954Find the absolute maximum and
minimum values of f on the set D.
#32.
#34.
50,40|),(
64),( 22
yxyxD
yxyxyxf
2
2 2
( , )
( , ) | 0, 0, 3
f x y xy
D x y x y x y
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14.7 Maximum and Minimum Values 22
Example 3 – pg.955 # 48Find the dimensions of the rectangular
box with the largest volume if the total surface area is given as 64 cm2.
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14.7 Maximum and Minimum Values 23
Example 4 – pg. 955 #51A cardboard box without a lid is to have a
volume of 32,000 cm3. Find the dimensions that minimize the amount of cardboard used.
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14.7 Maximum and Minimum Values 24
More Examples
The video examples below are from section 14.6 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦Example 3◦Example 5◦Example 6
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14.7 Maximum and Minimum Values 25
DemonstrationsFeel free to explore these demonstrations
below.◦Second Order Partial Derivatives
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