https://file.aiquickdraw.com/mathbot/file/1729582229368fxto2h5q.jpg

Let's analyze the graph and solve the questions based on the provided graph of the derivative \( f'(x) \).

1. **Where is \( f(x) \) increasing?**  
   A function is increasing when its derivative is positive. From the graph, \( f'(x) > 0 \) between approximately \( (3, 5) \). Thus, the answer for this part is:
   \[
   (3, 5)
   \]

2. **Where is \( f(x) \) concave down?**  
   A function is concave down where its second derivative is negative, which corresponds to intervals where the slope of \( f'(x) \) is decreasing. In other words, \( f(x) \) is concave down where \( f'(x) \) has a negative slope. Based on the graph, the concave down interval seems to be approximately from \( (1, 4) \). So, the answer is:
   \[
   (1, 4)
   \]

3. **Local Minima:**  
   Local minima occur where \( f'(x) \) crosses the x-axis from negative to positive (i.e., where \( f'(x) = 0 \) and changes from negative to positive). From the graph, this happens at approximately \( x = 3 \), so the local minimum is:
   \[
   3
   \]

4. **Inflection Points:**  
   Inflection points occur where the concavity changes, which corresponds to where \( f'(x) \) has local maxima or minima (i.e., where the slope of \( f'(x) \) changes sign). From the graph, inflection points are at approximately \( x = 1 \), \( x = 4 \), \( x = 5 \), and \( x = 6 \). Thus, the inflection points are:
   \[
   1, 4, 5, 6
   \]

Would you like more details on any part of the solution or further explanation? Here are 5 related questions for deeper understanding:

1. What is the relationship between the derivative of a function and the concavity of the original function?
2. How can you determine if a critical point is a local minimum, maximum, or inflection point using the second derivative test?
3. What does it mean for a function to have a local maximum?
4. Why do we look at where the derivative changes sign to find inflection points?
5. How does the behavior of \( f'(x) \) near an inflection point help us understand the graph of \( f(x) \)?

**Tip:** Inflection points are not just where the second derivative equals zero, but where the sign of the concavity changes!

Refer to the graph of the derivative f'(x) to answer various questions regarding where f(x) is increasing, concave down, has local minima, and inflection points.

Learn how to analyze a graph of the derivative of a function to determine intervals where f(x) is increasing, concave down, and find local minima and inflection points. Suitable for calculus students in Grades 10-12.

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