Suppose (1,1)is a critical point of a function f with continuous second derivatives. In the case of , , what can you say about f ?
A) f has a local maximum at (1,1)
B) f has a saddle point at (1,1)
C) f has a local minimum at (1,1)

Question

Suppose (1,1)is a critical point of a function f with continuous second derivatives. In the case of  ,  ,  what can you say about f ?
A) f has a local maximum at (1,1) 
B) f has a saddle point at (1,1) 
C) f has a local minimum at (1,1)

Quizplus · Accepted Answer

Since (1,1) is a critical point, the first derivative at that point is 0. We can use the second derivative test to determine whether it is a local minimum or maximum or a saddle point.

For option 1, the second derivative test tells us that if the second derivative is negative, then the critical point is a local maximum. But in this case, the second derivative at (1,1) for all three functions is positive. Hence option 1 is not correct.

For option 2, the second derivative test tells us that if the second derivative is of opposite signs along the two axes, then the critical point is a saddle point. Here, the second derivative is negative along the x-axis and positive along the y-axis for all three functions. This means that the critical point is a saddle point. Hence, option 2 is not correct.

For option 3, the second derivative test tells us that if the second derivative is positive, then the critical point is a local minimum. Here, the second derivative is positive for all three functions. So, the critical point is a local minimum, and hence option 3 is correct.

Suppose (1,1)is a Critical Point of a Function F with Continuous