8. let ( f(x)=(x - 7)^0 ). find an interval of ( x ) on which the average rate of change of ( f ) is 0.

Explanation:

Step1: Recall the average rate of change formula

The average rate of change of a function (y = f(x)) over the interval ([a,b]) is (\frac{f(b)-f(a)}{b - a}). Given (f(x)=(x - 7)^{0}), and by the zero - exponent rule (a^{0}=1) for (a\neq0). So (f(x)=1) for (x\neq7).

Step2: Calculate (f(b)-f(a))

Let (a) and (b) be two values in the domain of (f(x)) (where (a\neq7) and (b\neq7)). Then (f(a) = 1) and (f(b)=1). So (f(b)-f(a)=1 - 1=0).

Step3: Determine the interval

Since (\frac{f(b)-f(a)}{b - a}=\frac{0}{b - a}=0) (for (b\neq a) and (a\neq7), (b\neq7)). An example of an interval is ([1,2]) (any interval ([a,b]) where (a\neq7), (b\neq7) will work).

Answer:

An interval such as ([1,2]) (any interval ([a,b]) with (a\neq7) and (b\neq7) is valid).