Question

the graph of \\(f\\) is below. use it to determine the intervals where \\(f\\) is increasing.

Explanation:

Identify the condition for increasing intervals

To find where the original function \(f\) is increasing, we look for intervals where its derivative is positive:

$$f'(x) > 0$$

Using the Increasing Functions knowledge point, a function increases when its rate of change is positive.

Locate positive regions on the graph

We look at the given graph of \(f'(x)\) and identify where the curve lies strictly above the \(x\)-axis:

$$f'(x) > 0 \iff y > 0$$

The graph is above the \(x\)-axis between the \(x\)-intercepts \(x = -3\) and \(x = -1\), and also to the right of \(x = 2\).

Write the intervals in interval notation

The intervals where the graph of \(f'\) is above the \(x\)-axis are:

$$(-3, -1) \cup (2, \infty)$$

These are the open intervals where \(f'(x) > 0\), meaning \(f\) is increasing.

Answer:

\((-3, -1) \cup (2, \infty)\)