To find the average rate of change of the function [tex]\( f(x) = -2x^3 + 4x^2 \)[/tex] from [tex]\( x = -1 \)[/tex] to [tex]\( x = 3 \)[/tex], we will follow these steps:
1. Evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = -2(-1)^3 + 4(-1)^2
\][/tex]
[tex]\[
f(-1) = -2(-1) + 4(1)
\][/tex]
[tex]\[
f(-1) = 2 + 4 = 6
\][/tex]
2. Evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = -2(3)^3 + 4(3)^2
\][/tex]
[tex]\[
f(3) = -2(27) + 4(9)
\][/tex]
[tex]\[
f(3) = -54 + 36 = -18
\][/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]
Here, [tex]\( x_1 = -1 \)[/tex] and [tex]\( x_2 = 3 \)[/tex], so:
[tex]\[
\text{Average rate of change} = \frac{f(3) - f(-1)}{3 - (-1)}
\][/tex]
Substituting the values we computed:
[tex]\[
\text{Average rate of change} = \frac{-18 - 6}{3 - (-1)}
\][/tex]
[tex]\[
\text{Average rate of change} = \frac{-18 - 6}{3 + 1}
\][/tex]
[tex]\[
\text{Average rate of change} = \frac{-24}{4}
\][/tex]
[tex]\[
\text{Average rate of change} = -6
\][/tex]
So, the average rate of change of [tex]\( f(x) = -2x^3 + 4x^2 \)[/tex] from [tex]\( x = -1 \)[/tex] to [tex]\( x = 3 \)[/tex] is [tex]\(-6\)[/tex].
