Find the critical point(s) of the function.Then use the second derivative test to classify the nature of each point, if possible.Finally, determine the relative extrema of the function.
A)(0, - 1); relative minimum value: f(0, - 1) = - 4
B)(0, 0); saddle point: f(0, 0) = 4
C)(1, 0); relative maximum value: f(1, 0) = 1
D)there are no critical points

Question

Find the critical point(s) of the function.Then use the second derivative test to classify the nature of each point, if possible.Finally, determine the relative extrema of the function. 
A)(0, - 1); relative minimum value: f(0, - 1) = - 4 
B)(0, 0); saddle point: f(0, 0) = 4 
C)(1, 0); relative maximum value: f(1, 0) = 1 
D)there are no critical points

Quizplus · Accepted Answer

The function has critical points where the gradient is zero or undefined. Taking partial derivatives and setting them equal to zero, we get:

fx = 3x^2 - 3y^2 - 6x = 0
fy = -6xy + 6y = 0

From fy, we get either y = 0 or x = 1.

Case 1: y = 0
Substituting y=0 in fx, we get x=0. This gives us a critical point at (0,0).

Case 2: x = 1
Substituting x=1 in fx, we get y=±1. This gives us two critical points at (1,-1) and (1,1).

To determine the nature of these critical points, we need to compute the Hessian matrix and evaluate it at each point.
H = [fxx fxy; fyx fyy], where fxx, fxy, fyx, fyy are the second partial derivatives of f.

Evaluating the Hessian at each critical point:

At (0,0): H = [-6 0; 0 6], which has eigenvalues -6 and 6. Since the eigenvalues have opposite signs, this critical point is a saddle point.

At (1,-1): H = [6 6; -6 0], which has eigenvalues 3√2 and -3√2. Since the eigenvalues have opposite signs, this critical point is also a saddle point.

At (1,1): H = [6 -6; -6 0], which has eigenvalues 3√2 and -3√2. Since the eigenvalues have opposite signs, this critical point is also a saddle point.

Therefore, the function has no relative extrema, but has three critical points that are all saddle points. The correct answer is choice B.

Find the Critical Point(s) of the Function