To solve this problem, let's carefully analyze the graph and the question.

Problem Details:

The graph represents the derivative of the function $f(x)$, denoted as $y = f'(x)$. We are tasked to find how many of the marked points $x_1, x_2, x_3, x_4, x_5, x_6, x_7$ lie on intervals where $f(x)$ is increasing.

Key Observations:

1. $f(x)$ is increasing where $f'(x) > 0$:

   • $f(x)$ is increasing on intervals where the derivative $f'(x)$ (the graph) is above the $x$-axis ($y > 0$).

2. Marked Points and Behavior of $f'(x)$:

   • Analyze the graph for each marked point $x_1, x_2, \ldots, x_7$ and determine whether $f'(x) > 0$ at those points.

Analysis of Points:

1. $x_1$: $f'(x) < 0$ (below the $x$-axis), so $f(x)$ is not increasing.
2. $x_2$: $f'(x) > 0$ (above the $x$-axis), so $f(x)$ is increasing.
3. $x_3$: $f'(x) > 0$ (above the $x$-axis), so $f(x)$ is increasing.
4. $x_4$: $f'(x) = 0$ (on the $x$-axis), so $f(x)$ is not increasing.
5. $x_5$: $f'(x) > 0$ (above the $x$-axis), so $f(x)$ is increasing.
6. $x_6$: $f'(x) = 0$ (on the $x$-axis), so $f(x)$ is not increasing.
7. $x_7$: $f'(x) < 0$ (below the $x$-axis), so $f(x)$ is not increasing.

Conclusion:

The points where $f(x)$ is increasing are $x_2, x_3, x_5$.
Number of points: $\boxed{3}$.

Would you like a detailed explanation of why $f'(x) > 0$ corresponds to $f(x)$ increasing? Here are related questions:

1. Why does the derivative being positive imply an increase in the function $f(x)$?
2. What happens to $f(x)$ when the derivative equals zero or becomes negative?
3. How do turning points relate to the behavior of $f'(x)$?
4. Can we infer concavity from this graph of $f'(x)$? Why or why not?
5. How would the graph of $f(x)$ look based on this $f'(x)$?

Tip: Always correlate the sign of the derivative $f'(x)$ with the slope of the original function $f(x)$.