What does third derivative tell about inflection point? I was trying to find the nature (maxima, minima, inflection points) of the function $$\frac{x^5}{20}-\frac{x^4}{12}+5=0$$
But I faced a conceptual problem. It is given in the solution to the problem that $f''(0)=0$ and $f'''(0) \neq 0$ so $0$ is not an inflection point. But why should we check the third derivative?
Isn't checking first and second derivative sufficient for verifying an inflection point ? 
Why must the higher order odd derivatives be zero for an inflection point?

The given solution appears to be partially incorrect.
In fact
\begin{align}
f''(x) &= x^3 - x^2 = (x - 1)x^2 & f''(0) &= 0 & f''(1) &= 0 \\
f'''(x) &= 3x^2 - 2x & f'''(0) &= 0& f'''(1) &= 1, \\
\end{align}
and $f''$ has a double root (and local maximum) at $x=0$ and a single root at $x=1.$
At $x=0$, $f''$ is negative both on the left side and on the nearby right side of $x=0,$ but $f''$ is negative to the left of $x=1$ and positive to right, so $f$ has an inflection point at $x=1$ and not at $x=0.$
The value of the third derivative gives useful information at $x=1$ but not so much at $x=0.$ As noted in a comment, however, the fourth derivative at $x=0$ provides some useful information.
