By Vinogradov V.
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Extra resources for A cookbook of mathematics
C. we can see that λ > 0, thus |J| < 0 at the optimal point. Therefore, in a neighborhood of the optimal solution, given by the system (5), the conditions of the implicit-function theorem are met, and in this neighborhood x and y can be expressed as functions of a. The first-order partial derivatives of x and y with respect to a are evaluated as: ∂x =− ∂a = ∂F 1 ∂a ∂F 2 ∂a ∂F 3 ∂a ∂F 1 ∂y ∂F 2 ∂y ∂F 3 ∂y ∂F 1 ∂λ ∂F 2 ∂λ ∂F 3 ∂λ |J| 1λ 0 −2x −2λy −2aλ −2ay y2 2ay 0 =− |J| = 4ay 2 (λx + a) 3xy 2 = > 0.
R (x∗ )| > 0 • x∗ is a local maximum point of f subject to the given constraint if (−1)r |H for all r = 2, . . , n. Economics Application 5 (Utility Maximization) In order to illustrate how the Lagrange-multiplier method can be applied, let us consider the following optimization problem: The preferences of a consumer over two goods x and y are given by the utility function U (x, y) = (x + 1)(y + 1) = xy + x + y + 1. The prices of goods x and y are 1 and 2, respectively, and the consumer’s income is 30.
Xi + ∆xi , . . , xn ) − f (x1 , . . , xi , . . , xn ) = lim = lim ∂xi ∆xi →0 ∆xi ∆xi →0 ∆xi for all i = 1, 2, . . , n. If these limits exist, our function is differentiable with respect to all ∂f arguments. g. ∂x = fxi (= fxi ). i 32 Example 54 If y = 4x31 + x1 x2 + ln x2 then ∂y ∂x1 = 12x21 + x2 , ∂y ∂x2 = x1 + 1/x2 . Definition 38 The total differential of a function f is defined as dy = ∂f ∂f ∂f dx1 + dx2 + . . + dxn . ∂x1 ∂x2 ∂xn The total differential can be also used as an approximation device.
A cookbook of mathematics by Vinogradov V.