Question

what does a negative average rate of change indicate about a function over an interval?
a. the function is decreasing.
b. the function is constant.
c. the function is increasing.
d. the function is undefined.

Explanation:

Step1: Recall the definition of average rate of change

The average rate of change of a function \(y = f(x)\) over an interval \([x_1,x_2]\) is given by \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\).

Step2: Analyze the sign of the average rate of change

If the average rate of change \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}<0\), then \(f(x_2)-f(x_1)<0\) (since \(x_2>x_1\) for a non - degenerate interval). This means \(f(x_2)<f(x_1)\). When \(x_2>x_1\) and \(f(x_2)<f(x_1)\), the function is decreasing over the interval \([x_1,x_2]\).

Answer:

a. The function is decreasing.