Suppose f '' is continuous on (−∞, ∞).

Question

Suppose f '' is continuous on (−∞, ∞).

(a)

 If f '(−5) = 0 and f ''(−5) = −9,

 what can you say about f ?

1. At x = −5, f has a local maximum.
2. At x = −5, f has a local minimum.
3.   At x = −5, f has neither a maximum nor a minimum.
4. More information is needed to determine if f has a maximum or minimum at x = −5.

If f '(−1) = 0 and f ''(−1) = 0,

 what can you say about f ?

1. At x = −1, f has a local maximum.
2. At x = −1, f has a local minimum.
3.   At x = −1, f has neither a maximum nor a minimum.
4. More information is needed to determine if f has a maximum or minimum at x = −1.

Answer 1

Question A

We can use the second derivative test to determine the nature of the critical point at x = -5.

Since f'(−5) = 0, x = -5 is a critical point of f

If f''(-5) < 0, then f has a local maximum at x = -5

If f''(-5) > 0, then f has a local minimum at x = -5

If f''(-5) = 0, then the test is inconclusive.

In this case, f''(-5) = -9, which is less than 0. Therefore, we can conclude that f has a local maximum at x = -5, or option A

Question B

As the first and second derivatives are both equal to 0, as mentioned in the previous question, it would mean the second derivative test would be inconclusive, and more information would be needed to determine if f has a maximum or minimum at x=-1, or option D.

Hope the answers and explanations helped!