To solve this problem, we need to determine the number of integer points where the derivative $f'(x)$ of the given function $f(x)$ is positive. In terms of the graph, $f'(x) > 0$ corresponds to intervals where the graph of $f(x)$ is increasing.

Steps:

1. Analyze the graph to identify the intervals where $f(x)$ is increasing.
2. Count the integer $x$-values within these intervals.

Key Observations:

• The graph increases when the slope of the curve is positive (moving upward as $x$ increases).
• The function has several critical points where the graph transitions between increasing and decreasing. These points are local maxima or local minima, where $f'(x) = 0$.

From the graph:

• The function increases in the following approximate intervals:
  • $(-1, 0)$
  • $(2, 5)$
  • $(8, 11)$

Count the Integer Points:

• In $(-1, 0)$, the integer point is $x = -1$.
• In $(2, 5)$, the integer points are $x = 3, 4$.
• In $(8, 11)$, the integer points are $x = 9, 10$.

Total Count:

The integer points where $f'(x) > 0$ are: $x = -1, 3, 4, 9, 10$ Thus, there are 5 integer points where the derivative of the function is positive.

Final Answer:

The number of integer points where $f'(x) > 0$ is 5.

Do you need more explanation or help on any step?

Related Questions:

1. How can you determine where a function is increasing or decreasing using its derivative?
2. What are critical points, and how do they help analyze the behavior of a function?
3. How do you find the intervals where $f'(x) > 0$ for a given function analytically?
4. What is the relationship between the derivative and the slope of a graph?
5. How can you approximate slopes visually from a function's graph?

Tip: To confirm your result, always check for $f'(x) = 0$ at local maxima or minima to identify transitions between increasing and decreasing behavior.