calculus help would be greatly appreciated

Question

If the graph of f " (x) is continuous and has a relative maximum at x = c, which of the following must be true?

The graph of f has an x-intercept at x = c.

The graph of f has an inflection point at x = c.

The graph of f has a relative minimum at x = c

None of the above is necessarily true

Answer 1

None of the above -- a relative max or min for f^'(x) indicates an inflection point for f, and a relative min or max of f indicates a change of sign for f^'. And the sign of f^" indicates the concavity of f. But a relative max or min for f " bears no discernible relation to the graph of f -- certainly not any of those listed.

Answer 2

I believe the answer is none of the above is necessarily true.

When given info about f '', the most you can really do usually is think about f '' = 0 to look for concavity or points of inflection. Extrema of f '' do not give any of the information listed in the options.

For f to have an x-intercept at c that means f(x) = 0.

For f to have a point of inflection at x = c means f ''(x) must change sign there (meaning f changes concavity).

For f to have a relative minimum at x = c, one of the following must be true

1st Derivative Test: f '(c) = 0 and changes from negative to positive

2nd Derivative Test for Extrema: f '(c) = 0 and f '' (c) > 0