Intuition behind higher derivatives

Question

While I can easily imagine the second derivative conveying the concavity, and the first derivative conveying the slope of any function in a graph. How do I visually understand the meaning of higher derivatives apart from the fact that they represent the rate of $(n-1)^{th}$ derivatives.

Answer 1

I don't understand the relation, if you take $x \mapsto x^3$, you have :

$$ \left. \frac{d^2}{dx^2}x^3\right|_{x=0} = 0 $$

But :

$$ \frac{d^3}{dx^3}x^3 = 6 >0 $$

and $x \mapsto x^3$ have no maxima.