Find the maximum value of the following function:ƒ (x, y, z) = z − x − y − x² - y² — z² – xz.-

Given information:A function .To find:The maximum value of the function .Concept used:To find the…

Answered: Find the maximum value of the following…

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

Find the maximum value of the following function:
ƒ (x, y, z) = z − x − y − x² - y² — z² – xz.
-

Expert Solution

Step 1: Introduction

Given information:

A function $f left parenthesis x comma y comma z right parenthesis equals z minus x minus y minus x squared minus y squared minus z squared minus x z$

To find:

The maximum value of the function $f left parenthesis x comma y comma z right parenthesis$ .

Concept used:

To find the maximum value of a function, take the partial derivatives with respect to each variable (in this case, $x comma space y comma$ and $z$ ), set them equal to zero to find critical points, and then use the second partial derivative test to determine if these critical points correspond to maxima.

The Hessian matrix is $A equals H open parentheses f close parentheses equals open vertical bar table row cell fraction numerator partial differential squared f over denominator partial differential x squared end fraction end cell cell fraction numerator partial differential squared f over denominator partial differential x partial differential y end fraction end cell cell fraction numerator partial differential squared f over denominator partial differential x partial differential z end fraction end cell row cell fraction numerator partial differential squared f over denominator partial differential x partial differential y end fraction end cell cell fraction numerator partial differential squared f over denominator partial differential y squared end fraction end cell cell fraction numerator partial differential squared f over denominator partial differential y partial differential z end fraction end cell row cell fraction numerator partial differential squared f over denominator partial differential x partial differential z end fraction end cell cell fraction numerator partial differential squared f over denominator partial differential y partial differential z end fraction end cell cell fraction numerator partial differential squared f over denominator partial differential z squared end fraction end cell end table close vertical bar$

Find the Hessian matrix at the critical point.

First, compute $text det end text space A space equals space text det end text left parenthesis H space f right parenthesis$ . If $text det end text space A space not equal to space 0$ , then proceed with the next steps.

Existence of relative minimum:

If all principal minors $open vertical bar fraction numerator partial differential squared f over denominator partial differential x squared end fraction close vertical bar comma space open vertical bar table row cell fraction numerator partial differential squared f over denominator partial differential x squared end fraction end cell cell fraction numerator partial differential squared f over denominator partial differential x partial differential y end fraction end cell row cell fraction numerator partial differential squared f over denominator partial differential x partial differential y end fraction end cell cell fraction numerator partial differential squared f over denominator partial differential y squared end fraction end cell end table close vertical bar$ and $A$ at the critical point are postive, then the critical point is said to be a minima. The function attains the relative minimum at the critical points.

Existence of relative maximum:

If all principal minors $open vertical bar fraction numerator partial differential squared f over denominator partial differential x squared end fraction close vertical bar comma space open vertical bar table row cell fraction numerator partial differential squared f over denominator partial differential x squared end fraction end cell cell fraction numerator partial differential squared f over denominator partial differential x partial differential y end fraction end cell row cell fraction numerator partial differential squared f over denominator partial differential x partial differential y end fraction end cell cell fraction numerator partial differential squared f over denominator partial differential y squared end fraction end cell end table close vertical bar$ and $A$ at the critical point are alternative in sign as - + -, then the critical point is said to be a maxima. The function attains the relative maximum at the critical points.

Formula used:

The derivative of $x to the power of n$ with respect to $x$ is $n x to the power of n minus 1 end exponent$ .