The problem asks to determine the intervals where the derivative of the given function is positive and where it is negative, based on the graph provided.

Steps:

1. Identify critical points (where the slope of the tangent line changes, usually turning points like peaks or valleys) and regions where the graph is increasing or decreasing.
2. Determine where the function is increasing (i.e., where $f'(x) > 0$) and decreasing (i.e., where $f'(x) < 0$).

From the graph, it appears that:

• The function increases for $-2 < x < 0$, $0 < x < 1$, $x > 5.3$.
• The function decreases for $x < -2$, $1 < x < 2.6$, $2.6 < x < 5.3$.

Corresponding answer option:

Based on this, the correct answer should be:

• $f'(x) > 0$ for $-2 < x < 0$, $0 < x < 1$, $x > 5.3$.
• $f'(x) < 0$ for $x < -2$, $1 < x < 2.6$, $2.6 < x < 5.3$.

Thus, the correct choice matches the second option:

$f'(x) > 0$ for $x < -2$, $0 < x < 1$, $1 < x < 2.6$, and $x > 5.3$.
$f'(x) < 0$ for $-2 < x < 0$ and $2.6 < x < 5.3$.

Would you like further clarification on any part of the solution?

Here are five related questions you might want to consider:

1. How do we find critical points for a given function?
2. What is the relationship between increasing/decreasing functions and their first derivative?
3. How does the second derivative help in determining concavity?
4. How can you sketch the graph of the derivative based on a given function graph?
5. What are the key characteristics of a function at a local maximum or minimum?

Tip: When solving problems involving derivatives, always check both increasing/decreasing behavior and points where the derivative equals