Recall that the derivative f'(x) measures the rate of change of that function with respect to the given variable. Thus, if for some x_0, f'(x_0)=k, that means that at x=x_0, the function is changing at a rate of k units for every one unit on the x-axis. If f'(x)>0, the function is increasing at that point; f'(x)<0 means it is decreasing, and f'(x)=0 means that it is neither increasing nor decreasing.
This in mind, we see that f'(x)>0 for 1 < x < 6 and x>8, and f'(x) < 0 for 0 < x<1 and 6 < x <8. Therefore, the answer to part 1 of (a) is (1,6) & (8,oo); the answer to the second part is (0,1) & (6,8)
For part c, the function is concave upwards when f'(x) is increasing, and downwards when decreasing. This would in many cases be most easily observed by graphing f''(x) the same way that f'(x) is already graphed, and declaring the original function concave upward for f''(x)>0 or concave downward for f''(x)<0. However, looking at the graph is only slightly more intensive.
Examining the graph, we see that f'(x) is increasing on (0,2), (3,5), & (7,oo), and decreasing on (2,3) & (5,7). The first set of intervals will answer part 1 of c, and the second will answer part 2 of c.
Inflection points occur where the function changes concavity; more generally, inflection points occur when f''(x) changes sign. Since a function is concave upward for f''(x)>0 and downward for f''(x)<0, all we need to do is find the points where concavity changes. From our answers to c, we can tell that these will be x= 2,3,5,7
