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Let $x' = f(x)$ be autonomous first–order equation differential with an equiliburiium point $x_0$.

Suppose $f'(x_0) = 0$ what can I say about the behavior of soluton near $x_0$?

If $f'(x_0) ≠ 0$ and $f''(x_0) = 0$ then what is the dynamical behavior near this point. And identically I have above question for this $f'(x_0) ≠ 0$ and $f''(x_0) ≠ 0$, but $f'''(x_0) ≠ 0$.

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What are your own thoughts... –  akkkk Dec 23 '12 at 15:18
    
All-caps is interpreted as shouting on the internet - please don't do it again. –  Zev Chonoles Dec 23 '12 at 15:19
    
Why not make up some examples that are simple enough to solve and see what kind of behavior you can get? –  Gerry Myerson Dec 23 '12 at 16:46

2 Answers 2

The question is stated in a weird way, I suspect the derivatives should be put differently. Having said this,

Suppose $f′(x_0)=0$ what can I say about the behavior of soluton near $x_0$?

Answer: Nothing. Any type of behavior possible.

If $f'(x_0)\neq 0$ and $f''(x_0)=0$ then what is the dynamical behavior near this point?

Answer: You don't need the second derivative. If $f'(x_0)>0$ the point is the source (it repels the orbits), if $f'(x_0)<0$ the point is a sink (it attracts the orbits). The same applies yo your third question.

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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.)

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