Problem 8 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: Compute the second partial derivatives

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

Step 4: 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 \(D = f_{xx} f_{yy} - (f_{xy})^2\).

Step 5: 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.