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?

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.