Question

which of the following functions has an average rate of change equal to zero over any interval?
a. $f(x)=x^{2}$
b. $f(x)=7$
c. $f(x)=x$
d. $f(x)=\frac{1}{x}$

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 b

Let \(f(x)=7\). Then for any interval \([a,b]\), \(f(a) = 7\) and \(f(b)=7\).
So \(\frac{f(b)-f(a)}{b - a}=\frac{7 - 7}{b - a}=0\).

Step3: Analyze option a

Let \(f(x)=x^{2}\). Then \(\frac{f(b)-f(a)}{b - a}=\frac{b^{2}-a^{2}}{b - a}=\frac{(b - a)(b + a)}{b - a}=b + a
eq0\) (unless \(a=-b\) but not for any interval).

Step4: Analyze option c

Let \(f(x)=x\). Then \(\frac{f(b)-f(a)}{b - a}=\frac{b - a}{b - a}=1
eq0\).

Step5: Analyze option d

Let \(f(x)=\frac{1}{x}\). Then \(\frac{f(b)-f(a)}{b - a}=\frac{\frac{1}{b}-\frac{1}{a}}{b - a}=\frac{\frac{a - b}{ab}}{b - a}=-\frac{1}{ab}
eq0\).

Answer:

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