Assuming we have a function
$$ F(x)=\frac{x^4}{12} $$ with second derivatie: $$ \frac{d^2}{dx^2}\left(\frac{x^4}{12}\right)=x^2 $$ At x=0 second derivative is 0, but since the sign does not change, we don't have an inflection point. Second derivative of 0 at x means, that at the function is neither concave up, nor concave down at x. Therefore, the function is concave up everywhere, except at x=0.
Concave up means that a line segment between any two points on the graph of the function lies above or on the graph. Since the function is not concave up at x=0, above segment property does not hold everywhere for such a function?
We have that since $f''(x)\ge 0$ and $f''(x)=0$ in a single point (not on an interval) then $f(x)=\frac{x^4}{12}$ is strictly convex.