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Maximization of a function of two variables.
Jery Stedinger
October 2006
Consider the maximization of L[
a,b
].
The general solution recommended by multivariate
calculus is to solve for a stationary point defined by the two equations:
e
L
[
a
,
b
]
a
= 0
L
[
a
,
b
]
b
= 0
(1)
One needs to check the second derivatives to confirm a maximum has been found.
Alternatively,
one could decide that one wanted to first solve for the best value of b*(
a
) for every
a. This best value would be obtained by for each a solving the equation:
e
L
[
a
,
b
]
b
= 0
(2)
One could then attempt to maximize over
a
the function L[a, b*(a) ]; thus for every a, the second
parameter b takes the best possible value: now L is a function only of a.
Maximizing L as a function of a, using the chain rule, we seek the a value where:
dL
[
a
,
b
*(
a
)]
da
=
Λ
[
α
,
β
(
29 ]
+
[
,
(
29 ]
δβ
δα
= 0
(3)
Recall that the function solves b*(
a
) the equation
e
L
[
a
,
b
]
b
= 0
, so that the second term in the
equation above vanishes; thus we obtain
dL
[
a
,
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 Fall '08
 Stedinger

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