Problem 4 - College Algebra and Calculus: An Applied Approach

Question

Find any critical points and relative extrema of the function. $$ f(x, y)=\sqrt{25-(x-2)^{2}-y^{2}} $$

Step-by-Step Solution

Answer

The solution will vary depending on the values of the derivatives and their evaluation at the critical points. Be sure to solve for the critical points using the partial derivatives, and then apply the second derivative test to determine whether each point is a relative extremum.

Step 1: Calculate Partial Derivatives

To find critical points, calculate the first order partial derivatives of the function, \( f(x, y) \), with respect to each variable x and y. You can denote these as \( f_x \) and \( f_y \).

Step 2: Solve for Critical Points

Set \( f_x = 0 \) and \( f_y = 0 \) and solve for x and y. The solutions to these equations give the critical points.

Step 3: Apply Second Derivative Test

To determine whether the critical points are relative maxima, minima or saddle points, apply the Second Derivative Test. This involves calculating the second order partial derivatives: \( f_{xx} \), \( f_{yy} \), and \( f_{xy} \), and evaluating them at the critical points. Then, compute the determinant of the Hessian matrix \( D = f_{xx}*f_{yy} - (f_{xy})^2 \). If \( D > 0 \) and \( f_{xx} > 0 \), then the point is a relative minimum. If \( D > 0 \) and \( f_{xx} < 0 \), then the point is a relative maximum. If \( D < 0 \), then the point is a saddle point. If \( D = 0 \), then the test is inconclusive.