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?

It depends on your definition of inflection point. You have given this as 
"Where curve changes its concavity".
In this case checking $f'''(x)$ is not necessary since for example,
$$f(x) = x^3 \text{ gives } f'''(x) = 6 \neq 0 $$ however $x  = 0$ is an inflection point since there is a change in concavity here.
