15. 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 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}\neq0) (for (a\neq b) and (a,b\neq0)).

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\neq0) (for (a\neq - 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\neq b)).

Step5: Analyze option d

For (f(x)=x), (\frac{f(b)-f(a)}{b - a}=\frac{b - a}{b - a}=1\neq0) (for (a\neq b)).

Answer:

C. (f(x) = 7)