To calculate the average rate of change for the function [tex]\( f(x) = -3x^4 + 2x^3 - 5x^2 + x + 5 \)[/tex] on the interval from [tex]\( x = -1 \)[/tex] to [tex]\( x = 1 \)[/tex], follow these steps:
1. Evaluate [tex]\( f(x) \)[/tex] at the endpoints of the interval:
First, find [tex]\( f(-1) \)[/tex]:
[tex]\[
f(-1) = -3(-1)^4 + 2(-1)^3 - 5(-1)^2 + (-1) + 5
\][/tex]
[tex]\[
f(-1) = -3(1) + 2(-1) - 5(1) - 1 + 5
\][/tex]
[tex]\[
f(-1) = -3 - 2 - 5 - 1 + 5
\][/tex]
[tex]\[
f(-1) = -6
\][/tex]
Next, find [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = -3(1)^4 + 2(1)^3 - 5(1)^2 + 1 + 5
\][/tex]
[tex]\[
f(1) = -3(1) + 2(1) - 5(1) + 1 + 5
\][/tex]
[tex]\[
f(1) = -3 + 2 - 5 + 1 + 5
\][/tex]
[tex]\[
f(1) = 0
\][/tex]
2. Calculate the average rate of change:
The average rate of change of [tex]\( f(x) \)[/tex] on the interval [tex]\([x_1, x_2]\)[/tex] is given by:
[tex]\[
\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
For our interval, [tex]\( x_1 = -1 \)[/tex] and [tex]\( x_2 = 1 \)[/tex], so:
[tex]\[
\text{Average rate of change} = \frac{f(1) - f(-1)}{1 - (-1)}
\][/tex]
[tex]\[
\text{Average rate of change} = \frac{0 - (-6)}{1 - (-1)}
\][/tex]
[tex]\[
\text{Average rate of change} = \frac{0 + 6}{1 + 1}
\][/tex]
[tex]\[
\text{Average rate of change} = \frac{6}{2}
\][/tex]
[tex]\[
\text{Average rate of change} = 3
\][/tex]
Therefore, the correct answer is [tex]\(3\)[/tex].
