



 
Ahson, I'm going to illustrate with f(x) = 3x^{4}  16x^{3} + 24 x^{2}  8. For monotonicity I need the first derivative.
Hence the critical points are at x = 0 and x = 2. These two points separate the line into three parts and I check the monotonicity in each of the three parts.
In my rough work I usually mark this on a diagram. For concavity I need the second derivative
Hence the curve may change concavity at x = 2/3 and x = 2. Now I check the concavity in the 3 intervals defined by these points.
I put this information on the same diagram. Now I'm ready to attempt a sketch but I need to plot some points to determine where on the plane the graph lies. I usually plot the critical points if this is feasible.
Here is my sketch. I hope this helps,  


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