IF f’’ is continuous everywhere

Question

IF f’’ is continuous everywhere If f’(-3) = 0 and f’’(-3) = 5, what can you say about f(x)? b. If f’(5) = 0 and f’’(5) = -7, what can you say about f(x)?

Answer 1

If f'(x) = 0 then f(x) has either a local minimum or a local maximum at that point. And if f''(x) > 0, then f(x) is concave up at the point. Therefore, in this case, since f'(-3) = 0 and f""(-3) > 0, then the function f(x) has a local minimum at -3 (where the slope is zero) and is concave up around x = -3.

The same holds true in b) except since f'' < 0, f(x) is concave down so f(5) is a local maximum and f(x) is concave down around x = 5.