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Chapter 13: Problem 20

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

Short Answer

Expert verified
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 by step solution

01

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.
02

Find the Critical Points

Set \( f_x = 0 \) and \( f_y = 0 \) and solve the resulting system of equations to identify the critical points.
03

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.
04

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.
05

Find the Determinant of the Hessian Matrix

Compute the determinant of the Hessian matrix. This determinant is called the discriminant, denoted as \( D \).
06

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.

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