Let's analyze the problem by interpreting the graph of the function $f(x)$ and then deducing the behavior of its derivative.

Steps:

1. Critical Points: Look for points where the slope of the graph is zero (flat tangent lines), indicating where the derivative is zero.

   • In the graph, the curve is flat (horizontal tangent) near $x \approx 2$ and $x \approx 5$. Therefore, the derivative $f'(x)$ should cross the x-axis at these points.

2. Increasing/Decreasing Behavior:

   • Where the original function is increasing, the derivative is positive. Where it's decreasing, the derivative is negative.
   • $f(x)$ is decreasing between $x = 1$ and $x \approx 2$, so $f'(x)$ should be negative in this interval.
   • Between $x = 2$ and $x = 3$, $f(x)$ is increasing, so $f'(x)$ should be positive.
   • Between $x = 3$ and $x \approx 5$, $f(x)$ is decreasing again, so $f'(x)$ should be negative.
   • After $x = 5$, $f(x)$ is increasing, so $f'(x)$ should be positive.

3. Concavity:

   • The derivative should reflect changes in the curvature of $f(x)$.

Matching with the Derivative Graphs:

Looking at the options, the correct derivative should:

• Cross the x-axis at $x \approx 2$ and $x \approx 5$.
• Be negative between $x = 1$ and $x = 2$, positive between $x = 2$ and $x = 3$, negative between $x = 3$ and $x = 5$, and positive after $x = 5$.

Answer: Based on these observations, Graph 3 seems to be the correct derivative of $f(x)$.

Would you like a more detailed explanation, or do you have any other questions?

Here are five related questions to deepen your understanding:

1. What is the significance of critical points in determining the derivative?
2. How can you determine the concavity of a function from its graph?
3. How does the behavior of the derivative graph reflect inflection points in $f(x)$?
4. How can you approximate the values of the derivative based on the steepness of $f(x)$?
5. How would the graph change if $f(x)$ had more extreme slopes?

Tip: The zeroes of the derivative correspond to local maxima or minima in the original function.