To answer this question, let's analyze the given functions [tex]\( f(x) = x^3 + 5x^2 - x \)[/tex] and the tabulated values for [tex]\( g(x) \)[/tex].
Given [tex]\( g(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & -2 & -1 & 0 & 1 & 2 & 3 \\
\hline
g(x) & -4 & 8 & 6 & 2 & -16 & -84 \\
\hline
\end{array}
\][/tex]
First, we need to calculate the values of [tex]\( f(x) \)[/tex] for the given [tex]\( x \)[/tex] values:
- For [tex]\( x = -2 \)[/tex]:
[tex]\( f(-2) = (-2)^3 + 5(-2)^2 - (-2) = -8 + 20 + 2 = 14 \)[/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\( f(-1) = (-1)^3 + 5(-1)^2 - (-1) = -1 + 5 + 1 = 5 \)[/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\( f(0) = 0^3 + 5(0)^2 - 0 = 0 \)[/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\( f(1) = 1^3 + 5(1)^2 - 1 = 1 + 5 - 1 = 5 \)[/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\( f(2) = 2^3 + 5(2)^2 - 2 = 8 + 20 - 2 = 26 \)[/tex]
Next, calculate the changes in [tex]\( f(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & -2 \rightarrow -1 & -1 \rightarrow 0 & 0 \rightarrow 1 & 1 \rightarrow 2 \\
\hline
\Delta f(x) & 14 - 5 = 9 & 5 - 0 = 5 & 0 - 5 = -5 & 5 - 26 = 21 \\
\hline
\end{array}
\][/tex]
Now calculate the changes in [tex]\( g(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & -2 \rightarrow -1 & -1 \rightarrow 0 & 0 \rightarrow 1 & 1 \rightarrow 2 \\
\hline
\Delta g(x) & -4 - 8 = -12 & 8 - 6 = 2 & 6 - 2 = 4 & 2 - (-16) = -18 \\
\hline
\end{array}
\][/tex]
Finally, determine the total changes for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] over the interval [tex]\([-2, 2]\)[/tex]:
- Total change in [tex]\( f(x) \)[/tex]:
[tex]\[
\Delta f \text{ (total)} = 9 + 5 - 5 + 21 = 30
\][/tex]
- Total change in [tex]\( g(x) \)[/tex]:
[tex]\[
\Delta g \text{ (total)} = -12 + 2 + 4 - 18 = -24
\][/tex]
Comparing the total changes, we get:
Therefore, based on the total changes:
- [tex]\( f(x) \)[/tex] increased by 30.
- [tex]\( g(x) \)[/tex] decreased by 24.
From this, the only true statement is:
- [tex]\( f(x) \)[/tex] is increasing at the same rate that [tex]\( g(x) is decreasing.
Thus, the correct answer is:
C. Over the interval \([-2, 2]\)[/tex], function [tex]\( f \)[/tex] and function [tex]\( g \)[/tex] are decreasing at the same rate.
Therefore, the correct statement is:
D. Over the interval [tex]\([-2,2]\)[/tex], function [tex]\( f \)[/tex] is increasing at the same rate that function [tex]\( g \)[/tex] is decreasing.
