which of the following functions has an average rate of change equal to zero over any interval?\na…

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\neq0) (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\neq0).

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}\neq0).

Answer:

B. (f(x) = 7)