To sketch a possible graph of $f$ based on the given information, let's analyze the details provided in the problem.

Given Information

1. Function Values:

   • $f(-1) = 3$
   • $f(2) = -4$

   These are two points on the graph of $f$: $(-1, 3)$ and $(2, -4)$.

2. Sign of the Derivative:

   • The derivative $f'(x)$ changes sign at $x = -1$ and $x = 2$.
   • The sign of $f'(x)$ in different intervals is given:
     • For $x < -1$: $f'(x) > 0$ (positive slope, $f$ is increasing).
     • For $-1 < x < 2$: $f'(x) < 0$ (negative slope, $f$ is decreasing).
     • For $x > 2$: $f'(x) > 0$ (positive slope, $f$ is increasing).

Interpretation for Sketching $f$

Using the information about $f(-1)$, $f(2)$, and the sign changes in $f'(x)$:

• At $x = -1$: The derivative $f'(x)$ changes from positive to negative, indicating a local maximum at $x = -1$. Therefore, the point $(-1, 3)$ is a local maximum.
• At $x = 2$: The derivative $f'(x)$ changes from negative to positive, indicating a local minimum at $x = 2$. Therefore, the point $(2, -4)$ is a local minimum.

Steps to Sketch $f$

1. Draw Points: Plot the points $(-1, 3)$ and $(2, -4)$ on a coordinate system.

2. Behavior of $f(x)$ Between Points:

   • For $x < -1$: The function is increasing, approaching the local maximum at $(-1, 3)$.
   • For $-1 < x < 2$: The function is decreasing from the maximum at $(-1, 3)$ to the minimum at $(2, -4)$.
   • For $x > 2$: The function is increasing from the minimum at $(2, -4)$.

3. Sketch: Based on the increasing and decreasing behavior in each interval, sketch a smooth curve that:

   • Starts by increasing for $x < -1$,
   • Peaks at $(-1, 3)$,
   • Decreases to $(2, -4)$,
   • Then starts increasing again for $x > 2$.

This sketch will show a graph that reflects the specified behavior of $f(x)$ and $f'(x)$ over the intervals.

Would you like a more detailed explanation, or do you have any questions?

Here are 5 related questions to deepen your understanding:

1. What does it mean for a function to have a local maximum or minimum?
2. How can we determine if a point is a maximum or minimum based on the derivative?
3. How would the graph change if $f'(x)$ did not change signs at $x = -1$ or $x = 2$?
4. What additional information could we use to further refine the graph of $f(x)$?
5. How can the second derivative test help in identifying concavity near critical points?

Tip: When sketching the graph of a function based on derivative information, focus on identifying intervals of increase and decrease, as well as the points where these intervals change.