Question

1. which of the following functions has an average rate of change equal to zero over any interval?

a. $f(x)=\frac{1}{x}$
b. $f(x)=x^{2}$
c. $f(x)=7$
d. $f(x)=x$

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is given by \(\frac{f(b)-f(a)}{b - a}\).

Step2: Analyze option a

For \(f(x)=\frac{1}{x}\), \(\frac{f(b)-f(a)}{b - a}=\frac{\frac{1}{b}-\frac{1}{a}}{b - a}=\frac{\frac{a - b}{ab}}{b - a}=-\frac{1}{ab}
eq0\) (for \(a
eq b\) and \(a,b
eq0\)).

Step3: Analyze option b

For \(f(x)=x^{2}\), \(\frac{f(b)-f(a)}{b - a}=\frac{b^{2}-a^{2}}{b - a}=\frac{(b - a)(b + a)}{b - a}=b + a
eq0\) (for \(a
eq - b\)).

Step4: Analyze option c

For \(f(x)=7\), \(\frac{f(b)-f(a)}{b - a}=\frac{7-7}{b - a}=0\) for any \(a\) and \(b\) (\(a
eq b\)).

Step5: Analyze option d

For \(f(x)=x\), \(\frac{f(b)-f(a)}{b - a}=\frac{b - a}{b - a}=1
eq0\) (for \(a
eq b\)).

Answer:

C. \(f(x) = 7\)