Use Lagrange multipliers to identify the critical points of the function: f (x, y, z) = 3 − x2 − 2y2 − z2 subject to the constraint 2x + y + z = 2.You must show how you use Lagrange multipliers in your solution.

Answered: Use Lagrange multipliers to identify…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Use Lagrange multipliers to identify the critical points of the function:

f (x, y, z) = 3 − x² − 2y²− z²

subject to the constraint 2x + y + z = 2.

You must show how you use Lagrange multipliers in your solution.

Expert Solution

Step 1

Given function is $f (x, y, z) = 3 - x^{2} - 2 y^{2} - z^{2}$ subject to constraint $2 x + y + z = 2$ .

We have to determine the critical point of the given function by using Lagrange multipliers.

Let $g (x, y, z) = 2 x + y + z - 2 = 0$ .

Use Lagrange multipliers

$\nabla f = λ \nabla g$

$\begin{array}{l} \Rightarrow [\begin{matrix} f_{x} \\ f_{y} \\ f_{z} \end{matrix}] = λ [\begin{matrix} g_{x} \\ g_{y} \\ g_{z} \end{matrix}] \\ \Rightarrow [\begin{matrix} - 3 x \\ - 4 y \\ - 2 z \end{matrix}] = λ [\begin{matrix} 2 \\ 1 \\ 1 \end{matrix}] \\ \Rightarrow [\begin{matrix} - 3 x \\ - 4 y \\ - 2 z \end{matrix}] = [\begin{matrix} 2 λ \\ λ \\ λ \end{matrix}] \end{array}$