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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