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

First, we find the critical point(s) by setting the partial derivatives of the function equal to zero:

fx = 2x = 0
fy = 2y - 2 = 0

Solving for x and y, we get (0, 1).

Next, we use the second derivative test by finding the determinant and the trace of the Hessian matrix:

Hessian matrix: H(f) = [fxx  fxy]
                                  [fxy  fyy]

For our function, the Hessian matrix is:

H(f) = [2   0]
            [0   2]

The determinant of H(f) is 4, which is positive, and the trace of H(f) is 4, which is also positive. Therefore, we have a relative minimum at (0, 1).

Since there are no other critical points, this is the only relative extremum of the function.

Find the Critical Point(s) of the Function