If f '(5) = 0 and f ''(5) = 0, what can you say about f ?

a. at x = 5, f has a local maximum.
b. at x = 5, f has a local minimum.
c. at x = 5, f has neither a maximum nor a minimum.
d. more information is needed to determine if f has a maximum or minimum at x = 5.

D)

Step-by-step explanation:

Remember, if a is critical point of f then f'(a)=0. And criterion of the second derivative says that if a is a critical point of f and

1. if then f has a relative maximum in (a,f(a)),

2. if then f has a relative minimum in (a,f(a)),

3. if Then the criterion does not decide. That is,  f may have a relative maximum at a, a relative minimum at (a, f (a)) or neither.

Since then 5 is a critical point of f. Now we apply the second criterium:

since then the criterium doesn't decide, that means, more information is needed to determine if f has a maximum or minimum at x = 5.

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step-by-step explanation: