To match the description given, we need to sketch a graph of the function $g(x)$ that shows the following behavior:

1. Positive Derivative on $(-\infty, -2)$: This indicates that the graph is increasing in this interval, so the slope of $g(x)$ is positive for values less than -2.

2. Negative Derivative on $(-2, \infty)$: This means the graph is decreasing in this interval, so the slope of $g(x)$ is negative for values greater than -2.

Interpretation of Graph Characteristics

Based on these points:

• There should be a maximum point at $x = -2$, as the function changes from increasing to decreasing at this point.
• The graph would look like an "uphill" segment on the left of $x = -2$ and a "downhill" segment on the right.

If given options, you should look for a graph that has a peak at $x = -2$ and shows this increasing-then-decreasing pattern.

Would you like further details on interpreting derivative behavior or assistance in identifying the matching graph from your options?

Related Questions

1. What is the significance of a positive derivative in a graph?
2. How does the point $x = -2$ act as a critical point in this context?
3. What is the effect of a sign change in a derivative on the shape of a graph?
4. How would the second derivative help identify concavity in this function?
5. How does this function’s graph compare to one with a minimum point instead?

Tip: When the derivative changes sign, look for a peak or trough at that point.