Question

1. the graph of a function f is given. use the graph to find each of the following. reference hw 2.2 #s 7 and 10 a. the open intervals on which the function is increasing, if any. b. the open intervals on which the function is decreasing, if any. c. the open intervals on which the function is constant, if any. d. the numbers, if any, at which f has a relative maximum. what are these relative maxima? e. the numbers, if any, at which f has a relative minimum. what are these relative minima?

Explanation:

Step1: Identify increasing intervals

A function is increasing when the graph rises from left - to - right. By observing the graph, we can see the function is increasing on the intervals $(-1,0)$ and $(1,2)$.

Step2: Identify decreasing intervals

A function is decreasing when the graph falls from left - to - right. The function is decreasing on the intervals $(-5,-1)$ and $(2,5)$.

Step3: Identify constant intervals

There are no intervals where the function is a horizontal line, so there are no open intervals on which the function is constant.

Step4: Find relative maxima

Relative maxima occur at points where the function changes from increasing to decreasing. The function has a relative maximum at $x = 0$ and $x=2$, and the relative maxima are $y = 3$ and $y = 3$ respectively.

Step5: Find relative minima

Relative minima occur at points where the function changes from decreasing to increasing. The function has a relative minimum at $x=-1$ and $x = 1$, and the relative minima are $y=-1$ and $y=-1$ respectively.

Answer:

a. $(-1,0)$ and $(1,2)$
b. $(-5,-1)$ and $(2,5)$
c. None
d. $x = 0$ and $x = 2$; relative maxima are $y = 3$ and $y = 3$
e. $x=-1$ and $x = 1$; relative minima are $y=-1$ and $y=-1$