If f''(x)=0, how do you find the convexity of the function?

[LCKurtz]

Getting to your original question where $f''(x) = 0$ can the function be convex. Others have pointed out if you have this equation holding in an interval, the function is linear there. But if you just have $f''(c)=0$ for a single point, that just tells you that the function may be concave up, down, or neither depending on higher derivatives at that point. Think about $y = x^4$ at $x=0$. It is obviously concave up at $x=0$ but $f''(0) = f'''(0) = 0$ and $f^{4}(0)=24$. But $x^3$ has neither concavity at $0$. And for $x^n$ for larger values of $n$ it's the same story, only more so.

 

[archaic]

If $f''(c)=0$ but doesn't change sign before and after, then the concavity also doesn't change. If it changes its sign, then the concavity changes and $c$ is called an inflection point.
A linear function is both concave up and down, though.

 

[archaic]

or neither

How?
EDIT : Oh, you mean an inflection point.

 

[LCKurtz]

It could literally be neither without being an inflection point. Perhaps the graph has a straight line segment at $c$. Unless you want to consider a straight line to be both concave up and down, which I don't think is standard convention. Nor would you consider such a point an inflection point since there is not a change of concavity at $c$.