Question

for the function shown in the graph, give the interval(s) where the rate of change of f(x) with respect to x is a. positive, b. negative, and c. zero.
a. choose the interval(s) on which the rate of change is positive.
a. (b,0) and (4,c)
b. (4,c) only
c. (b,0) and (0,4)
d. (0,4) only
e. (b,0) only
f. (0,4) and (4,c)
b. choose the interval(s) on which the rate of change is negative.
a. (b,0) and (4,c)
b. (0,4) and (4,c)
c. (4,c) only
d. (b,0) only
e. (0,4) only

Explanation:

Step1: Recall rate - of - change concept

The rate of change of a function \(y = f(x)\) is positive when the function is increasing. A function \(y=f(x)\) is increasing if as \(x\) increases, \(y\) also increases.

Step2: Analyze the graph for positive rate - of change

Looking at the graph, the function is increasing on the interval \((0,4)\). There is no other interval where the function is increasing.

Step3: Recall rate - of - change for negative case

The rate of change of a function \(y = f(x)\) is negative when the function is decreasing. A function \(y = f(x)\) is decreasing if as \(x\) increases, \(y\) decreases.

Step4: Analyze the graph for negative rate - of change

The function is decreasing on the intervals \((b,0)\) and \((4,c)\)

Answer:

a. D. \((0,4)\) only
b. A. \((b,0)\) and \((4,c)\)