True or False. If f ''(a) = f '(a) = 0, then f(x) cannot have a maximum at x = a .?

Answer 1

False

pablo's answer is what you would usually expect, i.e., that a is an inflection point, where the second derivative (f'') changes from positive to negative. (Example: y = x^3, where a = 0.)

However, it is quite possible to find examples where f''(a) = 0 does not represent an inflection point, and f(x) does have a maximum at x = a. (Example: y = -x^4, where a = 0.)

Answer 2

True. It will be an inflection point.

Edit: actuator is correct. y=-x^4 is a good example that has f"(a)=f'(a)=0, but there is a maximum.

Answer 3

uh???