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, 0); relative minimum value: f(0, 0) = 4
B) (0, 0); saddle point: f(4, 0) = 2
C) (0, 0); relative maximum value: f(0, 0) = 2

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, 0); relative minimum value: f(0, 0) = 4 
B) (0, 0); saddle point: f(4, 0) = 2 
C) (0, 0); relative maximum value: f(0, 0) = 2

Quizplus · Accepted Answer

The critical point is (0, 0). To use the second derivative test, we need to find the Hessian matrix:

Hessian = |1 2|
                 |2 2|

The determinant of the Hessian is -2, which is negative, so (0,0) is a saddle point. Therefore, the only valid choice is A.

Find the Critical Point(s) of the Function