Answered: Use the method of Lagrange multipliers…

Use the method of Lagrange multipliers to maximize the function subject to the given constraint. f(x, y, z) = xyz subject to 10x + 10 y+ z = 84 and a 2 0, y > 0, z > 0 maximum of ? at (x, y, z) = (

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter10: Inequalities

Section10.8: Systems Of Linear Inequalities

Problem 2E

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Question

Use the method of Lagrange multipliers to maximize the function subject to the given constraint.
f(x, y, z)
= xyz subject to 10x + 10 y + z = 84 and x > 0, y 2 0, z 2 0
maximum of
? at (x, y, z) = (
?)

Expert Solution

Step 1

Given:

The function is $f (x, y, z) = x y z$ .
Subject to constraint are $10 x + 10 y + z = 84, x \geq 0, y \geq 0, z \geq 0$ .

The lagrangian's function is written as:

$\begin{array}{rcl} f (x, y, z, λ) & = & f (x, y, z) + λ g (x, y, z) \\ = & x y z + λ (10 x + 10 y + z - 84) \end{array}$

Partially differentiate the function with respect to $x, y, z$ and $λ$ .

$f_{x} = y z + 10 λ f_{y} = x z + 10 λ f_{z} = x y + λ f_{λ} = 10 x + 10 y + z - 84$

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