chapter 14 – partial derivatives 14.8 lagrange multipliers 1 objectives: use directional...
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Chapter 14 – Partial Derivatives14.8 Lagrange Multipliers
14.8 Lagrange Multipliers
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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14.8 Lagrange Multipliers 2
Lagrange MultiplierMany optimization problems have
restrictions or constraints on the values that can be used to produce the optimal solution.
These constraints tend to complicate optimization problems because the optimal solution can occur at a boundary point of the domain.
We use Lagrange Multipliers to simplify solutions.
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Lagrange Multiplier
λ is a Lagrange Multiplier
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Method of Lagrange Multipliers
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VisualizationLagrange Multipliers
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Example 1 – pg. 963 #6Use Lagrange multipliers to find the
maximum and minimum values of the function subject to the given constraint(s).
3 3( , ) ; 16xyf x y e x y
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Example 2 – pg. 963 #10Use Lagrange multipliers to find the
maximum and minimum values of the function subject to the given constraint(s).
2 2 2 2 2 2( , , ) ; 1f x y z x y z x y z
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Example 3Find the minimum value of the function
subject to the constraint2 2 2( , , ) 2 3f x y z x y z
2 3 4 49x y z
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Two ConstraintsThe numbers λ and µ are the Lagrange
multipliers such that
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Example 4 – pg. 963 #16Find the extreme values of f subject to
both constraints.
2 2
( , , ) 3 3 ;
0, 2 1
f x y z x y z
x y z x z
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Example 5 – pg. 964 #42Find the maximum and minimum
volumes of a rectangular box whose surface area is 1500 cm2 and whose total edge length is 200 cm.
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Example 6 – pg. 964 #44The plane intersects the
cone in an ellipse.a) Use Lagrange multipliers to find the highest
and lowest points on the ellipse.
4 3 8 5x y z 2 2 2z x y
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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 1◦Example 2◦Example 5
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DemonstrationsFeel free to explore these demonstrations
below.◦The Geometry of Lagrange Multipliers◦Constrained Optimization◦Visualizing the Gradient Vector
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