The image displays a graph of the function y = f(x). Several points are marked on the x-axis: x1, x2, x3, 0, x4, x5, x6, x7, x8, x9, x10. The question asks in how many of these points the derivative of the function f(x) is positive.

To determine where the derivative of a function is positive, we need to find the intervals where the function is increasing. On the graph, the derivative f'(x) is positive when the slope of the tangent line to the curve y = f(x) is positive. This corresponds to the parts of the graph where the function is rising from left to right.

Let's examine the marked points on the x-axis:

• At x1 and x2, the function is decreasing, so f'(x) is negative.
• At x3, the function has a local minimum, so f'(x) = 0.
• Between x3 and x4, the function is increasing, so f'(x) is positive.
• At x4, the function has a local maximum, so f'(x) = 0.
• Between x4 and x5, the function is decreasing, so f'(x) is negative.
• At x5, the function has a local minimum, so f'(x) = 0.
• Between x5 and x6, the function is increasing, so f'(x) is positive.
• At x6, the function has a local maximum, so f'(x) = 0.
• Between x6 and x7, the function is decreasing, so f'(x) is negative.
• At x7, the function has a local minimum, so f'(x) = 0.
• Between x7 and x8, the function is increasing, so f'(x) is positive.
• Between x8 and x9, the function is decreasing, so f'(x) is negative.
• At x9, the function has a local minimum, so f'(x) = 0.
• Between x9 and x10, the function is increasing, so f'(x) is positive.
• At x10, the function has a local maximum, so f'(x) = 0.

The question asks about the specific marked points. Let's re-evaluate based on the visual representation and common interpretations of such graphs:

The points where the derivative f'(x) is positive are where the function is strictly increasing. Visually, this means the graph is going upwards as we move from left to right. We need to consider the marked points themselves and the intervals around them.

The points marked on the x-axis are x1, x2, x3, 0, x4, x5, x6, x7, x8, x9, x10. We are looking for points where the tangent to the curve has a positive slope.

Let's analyze the intervals where the function is increasing:

• The function is increasing between x3 and x4.
• The function is increasing between x5 and x6.
• The function is increasing between x7 and x8.
• The function is increasing between x9 and x10.

Now, let's consider the marked points themselves. The derivative is positive at points where the function is strictly increasing. If a point is a local extremum (maximum or minimum), the derivative is zero. If a point is within an interval where the function is increasing, the derivative is positive.

Looking at the graph:

• Points x1, x2 are in a decreasing region.
• Point x3 is a local minimum (derivative is 0).
• Point 0 is in an increasing region.
• Point x4 is a local maximum (derivative is 0).
• Point x5 is a local minimum (derivative is 0).
• Point x6 is a local maximum (derivative is 0).
• Point x7 is a local minimum (derivative is 0).
• Point x8 is in an increasing region.
• Point x9 is a local minimum (derivative is 0).
• Point x10 is a local maximum (derivative is 0).

The question is asking