Answer
See graph and explanations.
Work Step by Step
a. Step 1. Examine the graph given by the Exercise. We can identify that the rate of change started with a small positive value (increasing), and the slope increases rapidly until around 25 days; the rate of change begins to decrease passed that point and slowly reaches a minimum towards the end (50 days).
Step 2. With the signs determined, for each horizontal value (days), we can estimate the amplitude of the growth rate by its slope value $f'(x)$ (x-days).
Step 3. The change of $f'(x)$ is represented by the change of the slope and can also be estimated as mentioned in step 1.
Step 4. Graph the derivative $f'(x)$ as a function of days as shown in the figure.
b. The fastest increase is represented by the maximum region in the derivative graph, which is around 25 days.
The slowest days of growth should be towards the end region, such as $x\gt45$ days (please note that the starting growth rate is also low when $x\lt5$ days).
