Minima and Maxima - which of the following statements are true?

Question

A - If f'(a) = 0 then f(x) has either a local maximum or a local minimum at x = a

B - If f'(a) = 0 and f''(a) = 0, then f(x) has an inflection point at x = a

C - If f'(x) < 0 for all x in the interval (0, 1), then f(x) is decreasing on (0, 1)

Answer 1

A is false. It's almost true though. When the derivative is zero at a point, the point is either a local max, min or an inflection point.

B is true. (which also helps explain why A is false) If the second derivative were not zero, then it would be either a local max or min.

C is true. When the derivative is negative, the slope of the function is negative and the function is decreasing