https://file.aiquickdraw.com/mathbot/file/17302664578777o69yfb1.jpg

To analyze this problem, let's interpret the graph provided and understand how we can sketch the first and second derivatives based on the behavior of the original function \( f(x) \):

### Analyzing the Original Function \( f(x) \)

1. **Critical Points (where the slope is zero):**
   - Around \( x = 2 \): The function reaches a local maximum.
   - Around \( x = 5 \): The function reaches a local minimum.
   - These points will be where the first derivative \( f'(x) = 0 \), as the slope is zero at maxima and minima.

2. **Intervals of Increase and Decrease:**
   - From \( x = 0 \) to \( x \approx 2 \): The function increases, so \( f'(x) > 0 \).
   - From \( x \approx 2 \) to \( x \approx 5 \): The function decreases, so \( f'(x) < 0 \).
   - From \( x \approx 5 \) to \( x \approx 7 \): The function increases again, so \( f'(x) > 0 \).

3. **Concavity (for the second derivative):**
   - Between \( x = 0 \) and \( x \approx 3 \): The graph is concave down, so \( f''(x) < 0 \).
   - Between \( x \approx 3 \) and \( x \approx 6 \): The graph is concave up, so \( f''(x) > 0 \).
   - Around \( x \approx 3 \) and \( x \approx 6 \): These points appear to be inflection points where concavity changes, so \( f''(x) = 0 \).

### Sketching the Derivatives

1. **First Derivative \( f'(x) \):**
   - Starts positive, reaches zero at \( x \approx 2 \) (maximum of \( f(x) \)), turns negative, and then zero again at \( x \approx 5 \) (minimum of \( f(x) \)).
   - The graph of \( f'(x) \) would cross the x-axis at \( x = 2 \) and \( x = 5 \).
   - Between these values, \( f'(x) \) is negative, and outside of them, it is positive.

2. **Second Derivative \( f''(x) \):**
   - The second derivative \( f''(x) \) indicates the concavity of \( f(x) \).
   - It is negative from \( x = 0 \) to \( x \approx 3 \) (concave down), zero at \( x \approx 3 \) (inflection point), positive from \( x \approx 3 \) to \( x \approx 6 \) (concave up), and zero again at \( x \approx 6 \) (another inflection point).

Would you like a more detailed explanation on how to construct these derivative graphs, or a visualization to aid in understanding?

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### Follow-Up Questions
1. What is the significance of critical points in determining the behavior of the first derivative?
2. How does concavity in \( f(x) \) relate to the behavior of the second derivative?
3. Can you explain why inflection points are where the second derivative equals zero?
4. How would the behavior of \( f'(x) \) change if there were additional extrema in \( f(x) \)?
5. Why is it important to consider both the first and second derivatives when analyzing the shape of a function?

#### Tip:
When sketching derivatives, always start by identifying maxima, minima, and inflection points in the original function. These points provide valuable anchors for the shapes of the first and second derivatives.

Copy the graph of each function given. On the same set of axes, sketch the graphs of the first and second derivatives of the function.

Learn how to sketch the first and second derivatives of a function based on its original graph. This exercise covers critical points, intervals of increase and decrease, concavity, and inflection points. Ideal for students in Grades 11-12 studying calculus.

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