This problem involves analyzing the graph of the derivative $y = f'(x)$ to determine where the original function $f(x)$ is increasing.

Key Concepts:

1. The function $f(x)$ is increasing where its derivative $f'(x) > 0$ (i.e., above the x-axis).
2. Conversely, $f(x)$ is decreasing where $f'(x) < 0$ (i.e., below the x-axis).

Step-by-Step Solution:

1. Identify the given points: $x_1, x_2, \dots, x_7$ are marked on the x-axis. We need to check if these points lie in regions where $f'(x) > 0$.

2. Analyze the graph:

   • $f'(x) > 0$ in the regions where the curve is above the x-axis.
   • $f'(x) < 0$ in the regions where the curve is below the x-axis.

3. Mark the increasing intervals:

   • The derivative $f'(x) > 0$ in the following intervals:
     • From $x_1$ to just before $x_2$,
     • From $x_3$ to $x_4$,
     • From $x_5$ to $x_6$.

4. Check the given points:

   • $x_1$ lies in an increasing interval.
   • $x_2$ lies in a decreasing interval.
   • $x_3$ lies in an increasing interval.
   • $x_4$ lies in an increasing interval.
   • $x_5$ lies in an increasing interval.
   • $x_6$ lies in an increasing interval.
   • $x_7$ lies in a decreasing interval.

5. Count the points in increasing intervals:

   • The points $x_1, x_3, x_4, x_5, x_6$ lie in intervals where $f'(x) > 0$.

Final Answer:

There are 5 points that lie on the intervals of increase of $f(x)$.

Would you like a detailed explanation of any step?

Related Questions:

1. What does the sign of $f'(x)$ tell us about the behavior of $f(x)$?
2. How do critical points (where $f'(x) = 0$) relate to the intervals of increase or decrease?
3. Could $x_7$ be a local maximum or minimum for $f(x)$? Why?
4. How can we determine concavity using $f''(x)$ from the graph of $f'(x)$?
5. Why is it important to carefully consider where $f'(x)$ crosses the x-axis?

Tip:

Always verify the derivative's sign in small intervals around key points to avoid misinterpreting the graph!