Suppose that f'(c)=0, f''(c)=0 and f'''(c)>0 . What, if anything, happens to the graph of the function f at x = c? Explain clearly and completely.

Answer 1

the slope has a minimum at x=c.

f"(c)=0 means it's an inflection point.

Think of the graph f(x) = x^3

Answer 2

So is it just an inflection point?

Answer 3

yes. The f"' nonzero makes sure that it's not just a cup like x^4.