How many inflection points are there in the graph? 
Answer:
Since $f'(x)$ has $5$ concave parts at $x=-2, \ x=-1, \ x=0, \ x=1, \ x=2$. 
So are there $5$ inflection points?
 A: I count 6 inflections points.  (But the graph is a little blurry.)  There is one at approximately $x=1/2$ where the graph changes from being concave down to concave up. There is another at $x=3/2$ where the graph changes from concave up to concave down.  Then (if I'm seeing the blurry parts right) there is another at $x=5/2$ where the concavity changes again.  
An inflection point is where a graph (smoothly) changes concavity.  So there are three more points on the negative $x$-axis.  An inflection point is also (and more properly) defined as a point where the tangent line crosses the graph.  
Edit:  I missed that the graph was of $f'(x)$ rather than $f(x)$.   If the slope is increasing, the graph of $f(x)$ is concave up.  If the slope is decreasing, the graph is concave down.   So at each max or min point of $f'(x)$ the slope changes from increasing to decreasing or the other way.  Since there are $5$ such place, that's how many inflection points there are.
A: According to Wikipedia

A differentiable function has an inflection point at (x, f(x)) if and only if its first derivative, f′, has an isolated extremum at x.

Since, $f'$ has local maxima/minima only at the points named by you(as per the graph), they are the required points of inflexion and you are correct.
Note, that the asymptotic portions are, well asymptotic (to the X-axis), so they can give no extra extrema (while the domain is real).
