Let's analyze the questions based on the provided graph $f(x)$.

Part (a): Average rate of change

To determine the average rate of change, we compute the slope between two points on the graph. The formula for the average rate of change between two points $(x_1, f(x_1))$ and $(x_2, f(x_2))$ is:

$\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

Compare between:

1. $x = 1$ and $x = 3$
2. $x = 2$ and $x = 5$

We'll observe the steepness of the curve from the graph to estimate which interval shows a larger change.

Part (b): Comparing $f(2)$ vs $f(5)$

Here, you need to determine which of the function values is larger, $f(2)$ or $f(5)$. You can estimate the values from the graph at $x = 2$ and $x = 5$.

Part (c): Comparing $f'(1)$ vs $f'(4)$

This question asks for the comparison of the instantaneous rates of change at $x = 1$ and $x = 4$. The derivative, $f'(x)$, represents the slope of the tangent to the curve at that point. Visually, compare how steep the curve is at $x = 1$ versus at $x = 4$.

Steps:

1. For part (a), check how steep the curve is between the pairs of points to compare the average rate of change.
2. For part (b), look at the function values at $x = 2$ and $x = 5$ on the graph.
3. For part (c), compare the slopes of the tangents to the curve at $x = 1$ and $x = 4$.

Would you like a detailed calculation or clarification on any part of this?

Additional Questions:

1. How is the average rate of change related to the slope?
2. Why is the derivative important for determining instantaneous rate of change?
3. What does it mean for the curve to be steeper between two points?
4. Can the average rate of change ever be zero? Why?
5. How does concavity affect the comparison of instantaneous rates of change?

Tip:

When comparing rates of change, steeper sections of the graph indicate higher values for both average and instantaneous rates.