To solve the problem:

The graph given is of the derivative $y = f'(x)$. The task is to determine how many of the marked points $x_1, x_2, \dots, x_7$ lie in intervals where the original function $f(x)$ is increasing.

Key Concept:

• The function $f(x)$ is increasing when its derivative $f'(x) > 0$, i.e., the graph of $y = f'(x)$ lies above the $x$-axis.

Step-by-step Solution:

1. Identify the intervals where $f'(x) > 0$: Look at the graph of $y = f'(x)$. The intervals where $f'(x) > 0$ are:

   • From $x_1$ to $x_2$ (positive region above the $x$-axis).
   • From $x_3$ to $x_4$.
   • From $x_5$ to $x_6$.

2. Check which points $x_1, x_2, \dots, x_7$ are in these intervals:

   • $x_1$: The starting point of the first interval where $f'(x) > 0$.
   • $x_2$: At the boundary of the interval where $f'(x) > 0$. (Technically, $f'(x_2) = 0$, so it’s not strictly increasing here.)
   • $x_3, x_4$: Both are within the second interval where $f'(x) > 0$.
   • $x_5, x_6$: Both are within the third interval where $f'(x) > 0$.
   • $x_7$: Lies outside any interval where $f'(x) > 0$.

3. Count the points: The points on intervals of increase are: $x_1, x_3, x_4, x_5, x_6$. Thus, 5 points lie on the intervals where $f(x)$ is increasing.

Final Answer:

There are 5 points where the function $f(x)$ is increasing.

Would you like a deeper explanation of the intervals or clarification on any step?

Here are 5 related questions to expand your understanding:

1. How does the behavior of $f'(x)$ determine where $f(x)$ is decreasing?
2. What happens to $f(x)$ when $f'(x) = 0$?
3. Can you describe the behavior of $f(x)$ when $f'(x) < 0$?
4. How can the second derivative $f''(x)$ help us understand $f'(x)$ further?
5. What is the significance of the turning points in $f'(x)$ for $f(x)$?

Tip:

Always visualize $f(x)$ by considering $f'(x)$'s sign changes—positive means increasing, negative means decreasing.