Suppose $f′(x_0)=0$ what can I say about the behavior of soluton near $x_0$?
Since the derivative function $f'$ returns the rate of change of the tangent line at a point in f, then whenever $f'(x_0) = 0$, $f'(x_0)$ is (the slope of) a horizontal line tangent to point $(x_0, f(x_0))$. $f(x_0)$ is probably then going to be a local extremum (a local minimum or maximum), unless you have something such as the case of $y=x^3$ where you have none. Surrounding $f(x_0)$ will, if you zoom in far enough, look "level."
If $f′(x_0)\ne 0$ and $f′′(x_0)=0$ then what is the dynamical behavior near this point. And identically I have above question for this $f′(x_0)\ne 0$ and $f′′(x_0)\ne 0$, but $f′′′(x_0)\ne 0$.
If $f''(x_0) = 0$, then you have the same thing, but for the derivative function $f'$. This means that there is zero change in the rate of change of $f$ at that point. This is equivalent to linear behavior at $f(x_0)$. The graph of $f'$ near $x_0$ will appear to be a horizontal line (not on the $x$-axis, since $f'(x_0) \ne 0$) or will be a local minimum or maximum. Keep going for $f'''$, $f''''$, etc. (Sorry, I hope I answered your question; two days of self-taught Calculus doesn't exactly make you an expert.)