Question

1. let ( f(x)=(x - 7)^{6} ). find an interval of ( x ) on which the average rate of change of ( f ) is 0.

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 \(\frac{f(b)-f(a)}{b - a}\).

Step2: Simplify the function \(f(x)=(x - 7)^0\)

By the zero - exponent rule \(a^0=1\) for \(a
eq0\). So \(f(x)=(x - 7)^0 = 1\) for \(x
eq7\).

Step3: Calculate the average rate of change

Let \(a\) and \(b\) be two values in the domain of \(f(x)\) (i.e., \(a
eq7\) and \(b
eq7\)). Then \(f(a)=1\) and \(f(b)=1\). The average rate of change \(\frac{f(b)-f(a)}{b - a}=\frac{1 - 1}{b - a}=0\) for any interval \([a,b]\) where \(a
eq7\) and \(b
eq7\). For example, take the interval \([0,1]\).

Answer:

Any interval \([a,b]\) where \(a
eq7\) and \(b
eq7\). For instance, the interval \([0,1]\) (the answer is not unique).