Let's analyze the problem step by step.

You are provided with a graph of the derivative $f'(x)$, and the task is to determine what the function $f(x)$ looks like near specific points based on this derivative.

Key Points of the Derivative Graph $f'(x)$:

• The derivative $f'(x)$ represents the slope of the original function $f(x)$.
• When $f'(x) > 0$, the graph of $f(x)$ is increasing.
• When $f'(x) < 0$, the graph of $f(x)$ is decreasing.
• Where $f'(x) = 0$, $f(x)$ has a critical point (potential maximum, minimum, or a horizontal inflection point).

Now let's interpret the derivative graph at each point and match it with the options provided:

1. Near $x = 1$:

   • At $x = 1$, $f'(x)$ is positive and decreasing, which means that $f(x)$ is increasing, but at a decreasing rate. This resembles a curve like B, where the slope starts positive but decreases.

2. Near $x = 3$:

   • At $x = 3$, $f'(x)$ is equal to 0, indicating a critical point. Just before and after $x = 3$, $f'(x)$ is negative, which means $f(x)$ is decreasing before and after the critical point. This suggests a local minimum, like F.

3. Near $x = 5$:

   • At $x = 5$, $f'(x)$ is 0 again, indicating another critical point. The derivative changes from positive before $x = 5$ to negative after, indicating that $f(x)$ reaches a local maximum here, similar to D.

4. Near $x = 7$:

   • At $x = 7$, $f'(x)$ is negative but increasing (getting less negative), meaning $f(x)$ is decreasing but starting to level out, which matches a shape like G, a curve that is concave up while decreasing.

Answers:

• Near $x = 1$, the graph of $f$ looks like B.
• Near $x = 3$, the graph of $f$ looks like F.
• Near $x = 5$, the graph of $f$ looks like D.
• Near $x = 7$, the graph of $f$ looks like G.

Would you like further clarification on any of these points?

Here are some follow-up questions that expand on this information:

1. What is the significance of the points where $f'(x) = 0$?
2. How does the concavity of $f(x)$ relate to the sign of $f'(x)$?
3. Can $f'(x)$ provide information about inflection points of $f(x)$?
4. What does it mean for $f(x)$ if $f'(x)$ is constant?
5. How would you sketch $f(x)$ if given a more complex $f'(x)$ graph?

Tip: Always check the behavior of the derivative just before and after critical points to accurately describe the local behavior of the function.