To determine the average rate of change of the function [tex]\( f(x) = 17 - x^2 \)[/tex] over the interval [tex]\([1, 5]\)[/tex], we first evaluate the function at the endpoints of the interval. Here are the steps:
1. Evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 17 - 1^2 = 17 - 1 = 16 \][/tex]
2. Evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = 17 - 5^2 = 17 - 25 = -8 \][/tex]
3. Calculate the average rate of change using the formula:
[tex]\[ \text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \][/tex]
where [tex]\( x_1 = 1 \)[/tex] and [tex]\( x_2 = 5 \)[/tex].
Substituting the values we found:
[tex]\[ \text{Average rate of change} = \frac{f(5) - f(1)}{5 - 1} = \frac{-8 - 16}{5 - 1} = \frac{-24}{4} = -6 \][/tex]
Thus, the average rate of change of [tex]\( f(x) \)[/tex] over the interval [tex]\([1, 5]\)[/tex] is [tex]\( -6 \)[/tex].
The correct answer is:
[tex]\[ -6 \][/tex]
