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Chapter 13: Problem 8

Examine the function for relative extrema and saddle points. $$ f(x, y)=-x^{2}-5 y^{2}+8 x-10 y-13 $$

Short Answer

Expert verified
The examination of the function uncovers a relative maximum at (4,1) and there are no saddle points.

Step by step solution

01

Find the partial derivatives

First, compute the partial derivatives of the function \(f(x, y)\) with respect to \(x\) and \(y\), resulting in \(f_x(x, y)\) and \(f_y(x, y)\). Set these equal to zero to find critical points.
02

Find the critical points

Solve the system of equations created in step 1 to find the values of \(x\) and \(y\) that make the gradients zero.
03

Compute the second partial derivatives

Calculate the second order partial derivatives (\(f_{xx}, f_{yy}, f_{xy} = f_{yx}\)) of the function.
04

Calculate the determinant of the Hessian Matrix

Form a Hessian matrix \(H\) using the second order partial derivatives and calculate its determinant, often referred to as \(D\). The Hessian Matrix is a square matrix of second-order partial derivatives. The determinant \(D\) is defined as \(D = f_{xx} f_{yy} - (f_{xy})^2\).
05

Determine the nature of critical points

The determinant identifies the nature of the critical points. If \(D > 0\) and \(f_{xx} < 0\), the point is a relative maximum. If \(D > 0\) and \(f_{xx} > 0\), the point is a relative minimum. If \(D < 0\), the point is a saddle point.

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