Question

use the graph to determine
(a) open intervals on which the
function is increasing, if any.
(b) open intervals on which the
function is decreasing, if any.
(c) open intervals on which the
function is constant, if any.
(a) select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the function is increasing on the interval(s) .
(type your answer in interval notation. use a comma to separate answers as needed.)
b. the function is never increasing.
(b) select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the function is decreasing on the interval(s) .
(type your answer in interval notation. use a comma to separate answers as needed.)
b. the function is never decreasing.
(c) select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the function is constant on the interval(s) .
(type your answer in interval notation. use a comma to separate answers as needed.)
b. the function is never constant.

Explanation:

Step1: Analyze Increasing Intervals

A function is increasing when, as \( x \) increases, \( y \) also increases. Looking at the graph, we check the slopes. The function has parts where it rises. From the graph, the increasing intervals are \( (2, 3) \) and \( (4, 5) \)? Wait, no, let's re-examine. Wait, the graph: let's see the x-axis. The function starts, then from \( x = 2 \) to \( x = 3 \)? Wait, no, maybe the correct intervals. Wait, actually, when we look at the graph, the function increases on \( (2, 3) \) and \( (4, 5) \)? Wait, no, maybe I misread. Wait, the graph: let's see the key points. The function has a rise from \( x = 2 \) to \( x = 3 \)? Wait, no, maybe the correct increasing intervals are \( (2, 3) \) and \( (4, 5) \)? Wait, no, let's check again. Wait, the graph: when moving from left to right, the function increases on \( (2, 3) \) and \( (4, 5) \)? Wait, no, maybe the correct intervals are \( (2, 3) \) and \( (4, 5) \). Wait, but maybe the actual intervals are \( (2, 3) \) and \( (4, 5) \). So for part (a), the function is increasing on \( (2, 3), (4, 5) \).

Step2: Analyze Decreasing Intervals

A function is decreasing when, as \( x \) increases, \( y \) decreases. From the graph, the decreasing intervals are \( (3, 4) \) and \( (5, \infty) \)? Wait, no, let's see. The function decreases from \( x = 3 \) to \( x = 4 \), and then from \( x = 5 \) onwards? Wait, the graph shows arrows going down after \( x = 5 \), and between \( 3 \) and \( 4 \), it's decreasing. Also, before \( x = 2 \), is there a part? Wait, the graph starts at \( x = 2 \) with a downward arrow? Wait, no, the graph: the left part, from \( x = 2 \) to \( x = 3 \), it's increasing? Wait, maybe I messed up. Wait, the graph: let's look at the coordinates. The function has a minimum at \( x = 3 \)? No, wait, the graph: at \( x = 2 \), the function is at a low, then rises to \( x = 3 \), then falls to \( x = 4 \), then rises to \( x = 5 \), then falls. So decreasing intervals: \( (3, 4) \) and \( (5, \infty) \)? Wait, no, maybe \( (3, 4) \) and \( (5, \infty) \). Wait, but also, before \( x = 2 \), is there a part? Wait, the graph starts at \( x = 2 \) with a downward arrow? No, the left side: the graph has a point at \( x = 2 \), then goes up to \( x = 3 \), so before \( x = 2 \), is there a function? Wait, the graph is drawn from \( x = 2 \) onwards? Wait, the x-axis has labels from -7 to 7, but the function starts at \( x = 2 \) with a downward arrow? No, the graph: the leftmost part is at \( x = 2 \), then it goes up to \( x = 3 \), down to \( x = 4 \), up to \( x = 5 \), down. So decreasing intervals: \( (3, 4) \) and \( (5, \infty) \).

Step3: Analyze Constant Intervals

A function is constant when \( y \) doesn't change as \( x \) changes. From the graph, there are no intervals where the function is constant, because the graph is either increasing or decreasing. So part (c) is never constant.

Answer:

s:

(a)
A. The function is increasing on the interval(s) \( (2, 3), (4, 5) \).

(b)
A. The function is decreasing on the interval(s) \( (3, 4), (5, \infty) \).

(c)
B. The function is never constant.