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 maximum value: f(0, 0) = 2
B)(2, 4); relative maximum value: f(0, 0) = 3
C)(3, 0); relative maximum value: f(0, 0) = 4

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 maximum value: f(0, 0) = 2 
B)(2, 4); relative maximum value: f(0, 0) = 3 
C)(3, 0); relative maximum value: f(0, 0) = 4

Quizplus · Accepted Answer

To find the critical points, we need to find where the partial derivatives of the function are equal to zero or undefined. 
Taking the partial derivative with respect to x, we get:
fx = 2x - 6y = 0
Taking the partial derivative with respect to y, we get:
fy = -6x + 2y = 0
Solving the system of equations, we get the critical point (0,0).

To use the second derivative test, we need to find the second partial derivatives:
fxx = 2, fyy = 2, fxy = -6
The discriminant is:
D = fxx*fyy - fxy^2 = 2*2 - (-6)^2 = -20
Since the discriminant is negative, the critical point is a saddle point.

Therefore, the relative extrema of the function is a saddle point at (0,0) and the maximum value is f(0,0) = 2.

Find the Critical Point(s) of the Function