Problem 20 - College Algebra and Calculus: An Applied Approach

Question

Examine the function for relative extrema and saddle points. $$ f(x, y)=-\frac{3}{x^{2}+y^{2}+1} $$

Step-by-Step Solution

Answer

The short answer will be determined by the outcome of step 6 and will describe whether the critical points are relative minimums, maximums, or saddle points.

Step 1: Find the First Partial Derivatives

The first partial derivatives of the function \( f(x, y)=-\frac{3}{x^{2}+y^{2}+1} \) are needed. Find \( f_x \) and \( f_y \) by applying the quotient rule.

Step 2: Find the Critical Points

Set \( f_x = 0 \) and \( f_y = 0 \) and solve the resulting system of equations to identify the critical points.

Step 3: Find the Second Partial Derivatives

The second partial derivatives, \( f_{xx}, f_{yy}, f_{xy}, f_{yx} \), are computed by differentiating \( f_x \) and \( f_y \) partially with respect to \( x \) or \( y \).Note that \( f_{xy} = f_{yx} \) due to Schwarz's Theorem.

Step 4: Construct the Hessian matrix

The Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function. In this case, it becomes a 2x2 matrix, with \( f_{xx}, f_{xy}, f_{yx}, f_{yy} \) as its elements.

Step 5: Find the Determinant of the Hessian Matrix

Compute the determinant of the Hessian matrix. This determinant is called the discriminant, denoted as \( D \).

Step 6: Classify the Critical Points

We use the discriminant \( D \) and the value of \( f_{xx} \) at the critical point to categorize it. If \( D > 0 \) and \( f_{xx} > 0 \), the point is a relative minimum. If \( D > 0 \) and \( f_{xx} < 0 \), the point is a relative maximum. If \( D < 0 \), the point is a saddle point. If \( D = 0 \), the test is inconclusive.