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consists of deriving the following first-order conditions to find the critical point(s) of the Lagrange function which are

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How can we be sure that for some arbitrarily given problem there is always a unique solution point at a tangency of the Constraint curve and contour of /, and that the Lagrange procedure always works in that it delivers this point? Exactly what role is played by the assumptions on the shapes of the level curves of the functions f and gl These questions will be considered later, in section 13.3. Here we will simply show the usefulness of the procedure by applying it to a number of examples.

Example 13.1 Solve the constrained maximization problem max y = .vf 25x? 75 S.t. 100 - lv, - 4xz = "


The Lagrange function is

and the first-order conditions are

Solving equations ( 13.1 ) and ( 13.2) to eliminate X gives

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