Question

determine all intervals on which the graph of f is increasing.
answer
additional solution no solution

Explanation:

Step1: Recall increasing function definition

A function \( f(x) \) is increasing on an interval if, as \( x \) increases, \( f(x) \) also increases (i.e., the graph rises from left to right on that interval).

Step2: Analyze the graph's slope

• First, look at the leftmost part: from \( x = -\infty \) (or the left end of the visible graph) up to \( x = -4 \), the graph is decreasing (going down). Then from \( x = -4 \) to \( x = -3 \), the graph rises (so increasing here).
• Next, from \( x = -3 \) to \( x = \) (let's see the minimum after that), the graph decreases until around \( x = 3 \) (or the local minimum point). Then from \( x = 3 \) to \( x = 6 \), the graph rises (increasing here). After \( x = 6 \), it decreases again.

Wait, let's re - examine the graph coordinates:

• The first increasing interval: The graph has a minimum around \( x=-4 \) (y - value around - 5) and then increases to \( x = - 3 \) (where it has a local maximum or a peak before decreasing again).
• The second increasing interval: After a local minimum around \( x = 3 \) (y - value around - 3), the graph increases from \( x = 3 \) to \( x = 6 \) (where it reaches a local maximum).

So the intervals where the function is increasing are when the derivative (slope of the tangent) is positive, which corresponds to the graph rising. So we identify the intervals:

• From \( x=-4 \) to \( x = - 3 \): as \( x \) goes from - 4 to - 3, \( y \) increases.
• From \( x = 3 \) to \( x=6 \): as \( x \) goes from 3 to 6, \( y \) increases.

Answer:

The function \( f(x) \) is increasing on the intervals \((-4, - 3)\) and \((3,6)\).