The equation defines the function near the point .
The point is:
A) a local maximum of .
B) a local minimum of .
C) a saddle point of .
D) a global maximum of .
E) not a critical point of .

Question

The equation  defines the function  near the point  .
The point  is:
A) a local maximum of . 
B) a local minimum of . 
C) a saddle point of . 
D) a global maximum of . 
E) not a critical point of .

Quizplus · Accepted Answer

Since the equation defines a function, we can use the second derivative test to determine whether a critical point is a maximum or minimum. 
Evaluating the second partial derivative at (2,1), we get:
$$
\frac{\partial^2}{\partial x^2} f(2,1) \cdot \frac{\partial^2}{\partial y^2} f(2,1) - \left(\frac{\partial^2}{\partial x \partial y} f(2,1)\right)^2 = 12 \cdot (-2) - (-4)^2 = -8 < 0 
$$
Since the second partial derivative test is negative, (2,1) is a local minimum of the function. Therefore, the answer is B.

The Equation Defines the Function near the Point