3. maximum mark: 3 the graph of y = f(x) is shown here. sketch the graph of 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,- 5)), so (f'(x)<0) on this interval. It is increasing on ((-5,0)), so (f'(x)>0) on ((-5,0)), and decreasing on ((0,\infty)), so (f'(x)<0) on ((0,\infty)).

Step2: Locate critical points

The critical points of (y = f(x)) are at (x=-5) and (x = 0) (where the slope of (y = f(x)) is 0). So (f'(-5)=0) and (f'(0)=0).

Step3: Analyze slope changes

As (x) approaches (-\infty), the slope of (y = f(x)) is negative and getting less negative as (x) approaches (-5). As (x) moves from (-5) to (0), the slope is positive and then starts to decrease as (x) approaches (0). As (x) moves from (0) to (\infty), the slope is negative and getting more negative.

Answer:

Sketch a curve that is negative on ((-\infty,-5)), crosses the (x) - axis at (x = - 5), is positive on ((-5,0)), crosses the (x) - axis at (x = 0), and is negative on ((0,\infty)). The curve should show the appropriate changes in steepness corresponding to the concavity - like behavior of (y = f(x)) (for example, if (y = f(x)) is concave up on an interval, (f'(x)) is increasing on that interval and if (y = f(x)) is concave down, (f'(x)) is decreasing).