To compare the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] on the interval [tex]\([0, 2]\)[/tex], let's analyze each step carefully.
Given:
[tex]\[
\begin{array}{|l|c|c|c|}
\hline
x & 0 & 1 & 2 \\
\hline
g(x) & -8 & -4 & -2 \\
\hline
\end{array}
\][/tex]
We need to determine whether [tex]\( g(x) \)[/tex] is increasing or decreasing on the interval [tex]\([0, 2]\)[/tex].
Step 1: Analyze the table for [tex]\( g(x) \)[/tex] on [tex]\([0, 2]\)[/tex]:
- At [tex]\( x = 0 \)[/tex], [tex]\( g(0) = -8 \)[/tex]
- At [tex]\( x = 1 \)[/tex], [tex]\( g(1) = -4 \)[/tex]
- At [tex]\( x = 2 \)[/tex], [tex]\( g(2) = -2 \)[/tex]
We observe that from [tex]\( x = 0 \)[/tex] to [tex]\( x = 2 \)[/tex]:
[tex]\[
-8 < -4 < -2
\][/tex]
Thus, the function [tex]\( g(x) \)[/tex] is increasing on the interval [tex]\([0, 2]\)[/tex].
Step 2: Determine if [tex]\( f(x) \)[/tex] is increasing:
Assume function [tex]\( f \)[/tex] is a generic increasing function. For instance, take [tex]\( f(x) = x^2 \)[/tex] for the interval:
[tex]\[
f(0) = 0^2 = 0 \\
f(1) = 1^2 = 1 \\
f(2) = 2^2 = 4
\][/tex]
We observe:
[tex]\[
0 < 1 < 4
\][/tex]
So, [tex]\( f(x) \)[/tex] is also increasing on the interval [tex]\([0, 2]\)[/tex].
Step 3: Compare the rates of increase of [tex]\( f \)[/tex] and [tex]\( g \)[/tex]:
- The rate of increase for [tex]\( f(x) \)[/tex] is calculated using the average change over the interval:
[tex]\[
\frac{f(2) - f(0)}{2 - 0} = \frac{4 - 0}{2} = 2
\][/tex]
- The rate of increase for [tex]\( g(x) \)[/tex] is:
[tex]\[
\frac{g(2) - g(0)}{2 - 0} = \frac{-2 - (-8)}{2} = \frac{6}{2} = 3
\][/tex]
Conclusion:
Since both functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are increasing, but the rate at which [tex]\( g \)[/tex] increases is higher than it's for [tex]\( f \)[/tex], the correct comparison is:
Both functions are increasing, but function [tex]\( g \)[/tex] is increasing faster.
Therefore, the correct statement is:
[tex]\[
\boxed{C}
\][/tex]
