Question

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

Short answer

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

Step-by-step solution

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: \[f_x = 2x + 4\] \[f_y = -2y - 8\]

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 \(y = -4\)

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.

Find the Second Derivatives

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

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.