The exercise consists of the following instructions:

1. Define the function $f$: The function $f$ is defined and differentiable on the interval $\mathbb{R} \setminus \{-1, 3\}$.

2. Analyze the graph: Use the provided graph of $f$ to:

   • Determine the signs of $f$ and $f'$ (the derivative of $f$) over the interval $\mathbb{R} \setminus \{-1, 3\}$.

3. Construct a sign table: Set up a table of signs for $f'$ based on its behavior.

Solution Steps

1. Observation of the Graph:

   • The graph shows $f(x)$ with asymptotes or undefined points at $x = -1$ and $x = 3$.
   • Identify intervals where $f(x)$ is increasing or decreasing based on the slopes of $f(x)$.
   • Check for the behavior of $f(x)$ as $x$ approaches these undefined points, as well as around the other areas of the graph.

2. Determine the Sign of $f'$:

   • $f'$ will be positive where $f(x)$ is increasing and negative where $f(x)$ is decreasing.
   • Identify turning points or inflection points to understand where $f'$ changes signs.

3. Create a Table of Signs for $f'$:

   • Divide the interval $\mathbb{R} \setminus \{-1, 3\}$ into smaller sub-intervals.
   • Indicate where $f' > 0$ (increasing regions) and $f' < 0$ (decreasing regions) in these intervals.

Would you like me to go into further detail on the process, such as interpreting the graph further or setting up the table of signs based on observations?

Related Questions

1. How do we determine intervals of increase and decrease using the derivative?
2. What is the significance of undefined points in the function’s domain?
3. How does the behavior of a function near asymptotes affect the derivative?
4. What additional information could we infer if we had $f'(x)$ explicitly?
5. How would inflection points impact the interpretation of $f'$?

Tip: When analyzing graphs, always look for intervals of monotonicity (increasing or decreasing behavior) and any discontinuities, as these can provide insights into the derivative's behavior.