(4) 5. given the graph of y = f(x), sketch the graph of its derivative y = f(x).

Explanation:

Step1: Identify increasing - decreasing intervals

Where (y = f(x)) is increasing, (f'(x)>0); where it is decreasing, (f'(x)<0). The function (y = f(x)) is decreasing on ((-\infty,- 1)) and ((1,2)), so (f'(x)<0) on these intervals. It is increasing on ((-1,1)) and ((2,\infty)), so (f'(x)>0) on these intervals.

Step2: Locate critical points

Critical points of (y = f(x)) occur where the slope is (0) or undefined. The function (y = f(x)) has critical points at (x=-1,1,2). At these points, (f'(x) = 0).

Step3: Analyze slope changes

Near (x=-1), the slope changes from negative to positive, so (f'(x)) crosses the (x) - axis from negative to positive at (x = - 1). Near (x = 1), the slope changes from positive to negative, so (f'(x)) crosses the (x) - axis from positive to negative at (x = 1). Near (x = 2), the slope changes from negative to positive, so (f'(x)) crosses the (x) - axis from negative to positive at (x = 2).

To sketch (y = f'(x)), we draw a curve that is below the (x) - axis on ((-\infty,-1)) and ((1,2)), above the (x) - axis on ((-1,1)) and ((2,\infty)), and crosses the (x) - axis at (x=-1,1,2).

Answer:

Sketch a curve for (y = f'(x)) that is negative on ((-\infty,-1)\cup(1,2)), positive on ((-1,1)\cup(2,\infty)) and crosses the (x) - axis at (x=-1,1,2).