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 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
f. (b,0) and (0,4)
c. choose the values at which the rate of change is 0.
a. x = 7 and x = 4
b. x = 0 and x = 7
c. x = 0 and x = -2
d. x = 4 and x = -2
e. x = 0 and x = 4

Explanation:

Step1: Recall rate - of - change concept

The rate of change of a function \(y = f(x)\) is positive when the function is increasing (graph going up as \(x\) increases), negative when the function is decreasing (graph going down as \(x\) increases), and zero when the function has a horizontal tangent.

Step2: Identify positive rate - of change interval

By observing the graph, the function \(y = f(x)\) is increasing on the interval \((b,0)\) and \((0,4)\).

Step3: Identify negative rate - of change interval

The function \(y = f(x)\) is decreasing on the interval \((4,c)\).

Step4: Identify zero rate - of change values

The function has horizontal tangents (rate of change is 0) at \(x = 0\) and \(x = 4\).

Answer:

a. F. \((b,0)\) and \((0,4)\)
b. A. \((b,0)\) and \((4,c)\)
c. E. \(x = 0\) and \(x = 4\)