https://file.aiquickdraw.com/mathbot/file/1733946256166ck5nar75.jpg

Let's analyze the graph and solve each part of the question.

The graph shown is of \( y = f'(x) \), the derivative of the function \( f(x) \). Here's the reasoning for each question:

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### (a) On what intervals is \( f(x) \) increasing?
- \( f(x) \) increases when \( f'(x) > 0 \) (above the x-axis).
- From the graph:
  - \( f'(x) > 0 \) on \( (0, 2) \cup (4, 6) \).
  
**Answer**: \( (0, 2) \cup (4, 6) \)

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### (b) On what intervals is \( f(x) \) decreasing?
- \( f(x) \) decreases when \( f'(x) < 0 \) (below the x-axis).
- From the graph:
  - \( f'(x) < 0 \) on \( (2, 4) \cup (6, \infty) \).

**Answer**: \( (2, 4) \cup (6, \infty) \)

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### (c) At what value(s) of \( x \) does \( f(x) \) have a local maximum?
- A local maximum occurs when \( f'(x) \) changes from positive to negative (crosses the x-axis from above to below).
- From the graph:
  - \( f'(x) \) changes from positive to negative at \( x = 2 \).

**Answer**: \( 2 \)

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### (d) On what interval(s) is \( f(x) \) concave upward?
- \( f(x) \) is concave upward when \( f''(x) > 0 \), which occurs when \( f'(x) \) is increasing.
- From the graph:
  - \( f'(x) \) increases on \( (3, 5) \).

**Answer**: \( (3, 5) \)

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Let me know if you want further clarification or detailed steps for any part!  

### Related Questions:
1. How can we identify concavity using the second derivative of \( f(x) \)?
2. Why does the sign of \( f'(x) \) determine whether \( f(x) \) is increasing or decreasing?
3. What happens to \( f(x) \) at points where \( f'(x) = 0 \)?
4. How do we interpret the graph of \( f''(x) \) if it were provided instead of \( f'(x) \)?
5. Can \( f(x) \) have a point of inflection if \( f'(x) \) doesn't cross the x-axis?

### Tip:
Always examine where the derivative \( f'(x) \) is zero or undefined to locate critical points for \( f(x) \).

The graph of the derivative f'(x) of a function f(x) is shown. Answer the following: (a) On what intervals is f increasing? (b) On what intervals is f decreasing? (c) At what value(s) of x does f have a local maximum? (d) On what interval(s) is f concave upward?

Analyze the graph of the derivative f'(x) to determine where f(x) is increasing, decreasing, has local maxima, and is concave upward. This problem applies the First and Second Derivative Tests and is suitable for advanced calculus students.

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