If the first derivative helps us to know whether the function is in increment or in decrement
The second one helps us to know wether the function is concave upward or downward
So what about the third derviative?! What does it help us to know
To this moment I don't have any guesses
same with the fourth derivative
The first non-zero derivative at a point tells you the approximate behaviour near to it. For example, $x^3$ has a stationary point of inflection at $0$, and near $0$ the second derivative had the same sign as $x$, whence the first derivative is minimal at $x=0$, making this point neither a maximum nor a minimum. By contrast, the least derivative of $x^4$ that doesn't vanish at $x=0$ is the fourth, so the same logic shows the second derivative is non-negative in the neighborhood, the first derivative has the same sign as $x$, and the function is minimised at $x=0$. It is the parity of the number of times we have to differentiate to get a non-zero result that matters, not the number itself.