Lagrange Multipliers

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Section 15.8.

• Lagrange Multipliers

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FREE EXTREMUM VERSUS CONSTRAINED EXTREMUM PROBLEMS
To find the minimum value of f (x, y, z) x2
2y2 z4 4 is a free extremum problem. To find
the minimum of f (x, y, z) x2 2y2 z4
4 subject to the condition that x 3y - z 7 is
a constrained extremum problem.
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AN EXAMPLE
Maximize the area of the rectangle inscribed
inside the ellipse
We want to maximize the function f (x, y)
4xy. NOTE The function we are trying to
optimize is called the objective function.
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NOTES ON THE EXAMPLE
1. The maximum happens where a level curve of the
objective function is tangent to the constraint
curve. 2. If two curves are tangent, then their
gradients are parallel.
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THE EXAMPLE REPHRASED
Maximize f (x, y) 4xy subject to the
constraint g(x, y) 1, where
The objective function is maximized where the
level curve and constraint curve are tangent.
This happens when the gradients are parallel
that is, when
where ? is a real number. The number ? is called
a Lagrange multiplier.
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METHOD OF LAGRANGE MULTIPLIERS
To find the maximum and minimum values of f (x,
y, z) subject to the constraint that g(x, y, z)
k assuming that these extreme values exist
and on the surface g(x, y, z)
k (a) Find all the values of x, y, z, and ?
such that (b) Evaluate f at all the points
(x, y, z) that result from step (a). The
largest of these values is the maximum value of
f the smallest is the minimum value of f.
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THE METHOD OF LAGRANGE MULTIPLIERS WITH PARTIAL
DERIVATIVES
Step (a) in the Method of Lagrange Multipliers is
equivalent to simultaneously solving the
following system of equations
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EXAMPLES
1. Maximize f (x, y) 4xy subject to the
constraint x2/4 y2/9 1. 2. Minimize f (x,
y, z) x2 y2 z2 subject to the constraint
x y z 1.
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THE METHOD OF LAGRANGE MULTIPLIERS WITH TWO
CONSTRAINTS
To find the maximum and minimum values of f (x,
y, z) subject to the constraints that g(x, y,
z) k and h(x, y, z) c (a) Find all the
values of x, y, z, ?, and µ such
that (b) Evaluate f at all the points (x, y,
z) that result from step (a). The largest of
these values is the maximum value of f the
smallest is the minimum value of f.
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THE PARTIAL DERIVATIVE FORM FOR TWO CONSTRAINTS
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EXAMPLE
Let T(x, y, z) 20 2x 2y z2 represent
the temperature at each point on the sphere
x2  y2  z2 11. Find the extreme temperatures
on the curve formed by the intersection of the
plane x y z 3 and the sphere.