Chapter 8 Multivariable Calculus

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Chapter 8Multivariable Calculus

• Section 4
• Maxima and Minima Using Lagrange Multipliers

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Learning Objectives for Section 8.4 Max/Min with
Lagrange Multipliers

• The student will be able to solve problems
  involving
• Lagrange multipliers for functions of two
  independent variables
• Lagrange multipliers for functions of three
  independent variables.

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Theorem 1 - Lagrange Multipliers
Step 1. Formulate the problem Maximize (or
minimize) z f (x, y) subject to g (x, y)
0. Step 2. Form the function F Step 3. Find
the critical points for F. That is,
solve the system Step 4. Conclusion If (x0,
y0, ?0) is the only critical point of F, we
assume that (x0, y0) is the solution. If F has
more than one critical point, we evaluate z f
(x, y) at each of them, to determine the maximum
or minimum.
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Example
Maximize f (x, y) 25 x2 y2, subject to 2x
y 10. Step 1. Formulate Maximize z f (x,
y) 25 x2 y2 subject to g(x,
y) 2x y 10 0 Step 2. Form the function F
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Example
Maximize f (x, y) 25 x2 y2, subject to 2x
y 10. Step 1. Formulate Maximize z f (x,
y) 25 x2 y2 subject to g(x,
y) 2x y 10 0 Step 2. Form the function
F Step 3. Find the critical points for F
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Example
Maximize f (x, y) 25 x2 y2, subject to 2x
y 10. Step 1. Formulate Maximize z f (x,
y) 25 x2 y2 subject to g(x,
y) 2x y 10 0 Step 2. Form the function
F Step 3. Find the critical points for
F Solving simultaneously yields one critical
point at (4, 2, 4). Step 4. Conclusion
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Example
Maximize f (x, y) 25 x2 y2, subject to 2x
y 10. Step 1. Formulate Maximize z f (x,
y) 25 x2 y2 subject to g(x,
y) 2x y 10 0 Step 2. Form the function
F Step 3. Find the critical points for
F Solving simultaneously yields one critical
point at (4, 2, 4). Step 4. Conclusion Since
(4, 2, 4) is the only critical point for F
Max of f (x, y) with constraints f (4, 2)
25 42 22 5.
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Example 2
The Cobb-Douglas production function for a
product is given by N(x, y) 10 x0.6 y0.4.
Maximize N under the constraint that 30x 60y
300,000. Step 1. Formulate Maximize N(x, y)
10 x0.6 y0.4 subject to g(x, y) 30x 60y
300,000 0 Step 2. Form the function F
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Example 2
The Cobb-Douglas production function for a
product is given by N(x, y) 10 x0.6 y0.4.
Maximize N under the constraint that 30x 60y
300,000. Step 1. Formulate Maximize N(x, y)
10 x0.6 y0.4 subject to g(x, y) 30x 60y
300,000 0 Step 2. Form the function F
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Example 2(continued)
Step 3. Find the critical points for F

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Example 2(continued)
Step 3. Find the critical points for
F Solving yields one critical point (6000,
2000, 0.1289) Step 4. Conclusion

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Example 2(continued)
Step 3. Find the critical points for
F Solving yields one critical point (6000,
2000, 0.1289) Step 4. Conclusion Since
(6000, 2000, 0.1289) is the only critical point
for F, we conclude that Max of N(x, y)
under the given constraints
N(6000, 2000) 10 6,0000.6 2,0000.4 38,664.

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Functions of Three Independent Variables
Any local extremum of w f (x, y, z) subject
to the constraint g(x, y, z) 0 will be among
the set of points (x0, y0, z0, ?0) which are a
solution to the system Whereprovided that
all the partial derivatives exist.This is an
extension of the two-variable case.

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Example
Find the extrema of f (x, y, z) 2x 4y
4z,subject to x2 y2 z2 9. Step 1.
Formulate Maximize w f (x, y) 25 x2 y2
subject to g(x, y) 2x y 10
0 Step 2. Form the function F
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Example
Find the extrema of f (x, y, z) 2x 4y
4z,subject to x2 y2 z2 9. Step 1.
Formulate Maximize w f (x, y) 2x 4y 4z
subject to g(x, y) x2 y2 z2
9 Step 2. Form the function F Step 3. Find
the critical points for F
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Example
Find the extrema of f (x, y, z) 2x 4y
4z,subject to x2 y2 z2 9. Step 1.
Formulate Maximize w f (x, y) 2x 4y 4z
subject to g(x, y) x2 y2 z2
9 Step 2. Form the function F Step 3. Find
the critical points for F
Solving yields two critical points ( 1, 2,
2, 1) and ( 1, 2, 2, 1)
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Example(continued)
Step 4. Conclusion. Since there are two
candidates, we need to evaluate the function
values f (1, 2, 2) 2 8 8 18 f (1, 2,
2) 2 8 8 18 We conclude that the
maximum of f occurs at (1, 2, 2), and the
minimum of occurs at (1, 2, 2).
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Summary

• We learned how to use Lagrange multipliers in the
  equationto find local extrema for two
  independent variable problems.
• We learned how to use Lagrange multipliers in the
  equationto find local extrema for three
  independent variable problems.