To find the average rate of change of the function [tex]\( f(x) \)[/tex] over the interval [tex]\( 2 \leq x \leq 6 \)[/tex], follow these steps:
1. Identify the Values of [tex]\( x \)[/tex] and [tex]\( f(x) \)[/tex] at the Endpoints:
For [tex]\( x = 2 \)[/tex], [tex]\( f(2) = 54 \)[/tex].
For [tex]\( x = 6 \)[/tex], [tex]\( f(6) = 42 \)[/tex].
2. Formula for the Average Rate of Change:
The average rate of change of a function [tex]\( f(x) \)[/tex] over an interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}
\][/tex]
3. Apply the Formula Using the Given Values:
Substitute [tex]\( a = 2 \)[/tex], [tex]\( b = 6 \)[/tex], [tex]\( f(2) = 54 \)[/tex], and [tex]\( f(6) = 42 \)[/tex] into the formula:
[tex]\[
\text{Average rate of change} = \frac{f(6) - f(2)}{6 - 2} = \frac{42 - 54}{6 - 2}
\][/tex]
4. Simplify the Expression:
Calculate the difference in the function values and the difference in the [tex]\( x \)[/tex]-values:
[tex]\[
\text{Average rate of change} = \frac{42 - 54}{6 - 2} = \frac{-12}{4} = -3
\][/tex]
Thus, the average rate of change of the function [tex]\( f(x) \)[/tex] over the interval [tex]\( 2 \leq x \leq 6 \)[/tex] is [tex]\(-3\)[/tex].
Therefore, the average rate of change, in simplest form, is [tex]\(-3\)[/tex].
