How can i solve this exercise? Find the maximum and minimum values of the function f(x, y)= etY subject to x³ + y3 = 16Maximum value:Minimum value:

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Description: How can i solve this exercise? Find the maximum and minimum values of the function f(x, y)= etY subject to x³ + y3 = 16Maximum value:Minimum value:

Answered: Find the maximum and minimum values of…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.1: Inner Product Spaces

Problem 14BEXP

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How can i solve this exercise?

Find the maximum and minimum values of the function f(x, y)
= etY subject to x³ + y3 = 16
Maximum value:
Minimum value:

Expert Solution

Step 1

Given data:

The expression for the first function is f(x, y)=e^x^y.

The expression for the second expression is x³+y³=16.

The second expression can be written as,

$g (x) = x^{3} + y^{3} - 16$

The expression in Lagrange multipliers term is,

$f_{x} (x, y) = λ g_{x} (x, y) f_{y} (x, y) = λ g_{y} (x, y)$

Substitute the given values in the above expression.

$\begin{array}{rcl} \frac{\partial}{\partial x} (e^{x y}) & = & λ \frac{\partial}{\partial x} (x^{3} + y^{3} - 16) \\ y e^{x y} & = & λ (3 x^{2}) \\ y e^{x y} & = & 3 λ x^{2} . . . . . . . . . . . . . . . . . . . . (I) \\ \frac{\partial}{\partial y} (e^{xy}) & = & λ \frac{\partial}{\partial y} (x^{3} + y^{3} - 16) \\ {xe}^{xy} & = & λ (3 y^{2}) \\ {xe}^{xy} & = & 3 {λy}^{2} . . . . . . . . . . . . . . . . . . . . (II) \end{array}$

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