If f is a differentiable function, and f ''(c)= 0 , then f has an inflection point at x=c . true /false

Question

If f is a differentiable function, and f ''(c)= 0 , then f has an inflection point at x=c . true /false

 

is there a theorem that proves or disproves this?

is there a counter example? 

Answer 1

Yes, the statement is true.  The Inflection Point Theorem (usually discussed with the Second Derivative Test) states that an inflection point exists where f"(c) equals 0.  The function must be differentiable and the second derivative must changes signs at x=c.