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Chapter 13: Problem 1

Find any critical points and relative extrema of the function. $$ f(x, y)=x^{2}-y^{2}+4 x-8 y-11 $$

Short Answer

Expert verified
The critical point of the function is (-2, -4), and it is a saddle point.

Step by step solution

01

Find the Partial Derivatives

The first step is to find the derivative of the function with respect to both variables. Given the function \(f(x, y) = x^{2} - y^{2} + 4 x - 8 y - 11\), the partial derivatives with respect to x, denoted \(f_x\), and with respect to y, denoted \(f_y\), are calulated as follows: \[f_x = 2x + 4\] \[f_y = -2y - 8\]
02

Set the Partial Derivatives Equal to Zero

Next, set the partial derivatives equal to zero and solve for the corresponding variables. \[2x + 4 = 0\] Solving for x gives \(x = -2\). \[-2y - 8 = 0\] Solving for y gives \(y = -4\)
03

Determine Critical Points

The critical points are then the intersection of these solutions, i.e., the point \((-2, -4)\). This gives us the critical points of the function.
04

Find the Second Derivatives

The second partial derivatives of the function are given by \(f_{xx}, f_{yy}\) and \(f_{xy}\) or \(f_{yx}\). \[f_{xx} = 2 \] \[f_{yy} = -2 \] \[f_{xy} = f_{yx} = 0 \]
05

Use the Second Derivatives Test

Using the second derivatives test, we will now classify the critical point found as a relative minimum, maximum, or saddle point. We will do this by computing the determinant of the Hessian matrix (D) given as: \[D = f_{xx}f_{yy} - (f_{xy})^2\] Substituting our values in, we get: \[D = 2*(-2) - (0)^2 = -4\] Since \(D < 0\), the critical point \((-2, -4)\) is a saddle point.

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