16. Find the minimum value of f(x, y, z) = x² + 2y² + 3z?subject to the constraint x + 2y + 3z = 10. Show that fhas no maximum value with this constraint.There is only one critical point, so the minimum value of the function occurs atEnter your answer in the form:(,7/5,2/9,1/3) with the parentheses, but no spaces.Note: The fact that we don't have both a maximum and a minimum does not violate the Extreme Value Theorem because the constraint isn't bounded in R^3.

Consider the given: fx,y,z=x2+2y2+3z2Subject to constraints:x+2y+3z=10gx,y,z=x+2y+3z−10 Use…

Answered: Find the minimum value of f(x, y, z) =…

Find the minimum value of f(x, y, z) = x² + 2y² + 3z? subject to the constraint x + 2y + 3z = 10. Show that f %3D has no maximum value with this constraint.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 17EQ

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Question

16. Find the minimum value of f(x, y, z) = x² + 2y² + 3z?
subject to the constraint x + 2y + 3z = 10. Show that f
has no maximum value with this constraint.
There is only one critical point, so the minimum value of the function occurs at
Enter your answer in the form:(
,7/5,2/9,1/3) with the parentheses, but no spaces.
Note: The fact that we don't have both a maximum and a minimum does not violate the Extreme Value Theorem because the constraint isn't bounded in R^3.

Expert Solution

Step 1

Consider the given:

$\begin{array}{l} f (x, y, z) = x^{2} + 2 y^{2} + 3 z^{2} \\ Subject to constraints: \\ x + 2 y + 3 z = 10 \\ g (x, y, z) = x + 2 y + 3 z - 10 \end{array}$

Use Lagrange’s Multipliers:

$\begin{array}{l} \nabla f = λ \nabla g \\ This can be written as: \\ f_{x} = λ g_{x} \\ f_{y} = λ g_{y} \\ f_{z} = λ g_{z} \\ Find the partial derivatives: \\ f_{x} = 2 x \\ f_{y} = 4 y \\ f_{z} = 6 z \\ g_{x} = 1 \\ g_{y} = 2 \\ g_{z} = 3 \end{array}$

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