Examine the function given below for relative extrema and saddle points.
A) The function has a relative minimum at
B) The function has a relative maximum at
C) The function has a saddle point at
D) The function has a relative maximum at
E) The function has a relative minimum at

Question

Examine the function given below for relative extrema and saddle points. ​  ​
A) The function has a relative minimum at 
B) The function has a relative maximum at 
C) The function has a saddle point at 
D) The function has a relative maximum at 
E) The function has a relative minimum at

Quizplus · Accepted Answer

To find relative extrema and saddle points, we need to take the partial derivatives of the function and set them equal to zero. Let's first find the first order partial derivatives:
f_x = 2x - 6y 
f_y = -6x + 2y

Setting these equal to zero, we get the system of equations:
2x - 6y = 0 
-6x + 2y = 0

Solving for x and y, we get the critical point (3,1). Now, we need to classify this critical point using the second order partial derivatives:
f_xx = 2 
f_xy = -6 
f_yx = -6 
f_yy = 2

The discriminant is:
D = f_xx*f_yy - (f_xy*f_yx) = (2*2) - (-6*-6) = 4 - 36 = -32

Since the discriminant is negative, the critical point is a saddle point. Therefore, the correct answer is A.

Examine the Function Given Below for Relative Extrema and Saddle