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Select the correct answer.

This table models continuous function [tex]\( f \)[/tex].

[tex]\[
\begin{tabular}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & 0 \\
\hline
-1 & -8 \\
\hline
0 & -6 \\
\hline
1 & 0 \\
\hline
2 & 4 \\
\hline
3 & 0 \\
\hline
\end{tabular}
\][/tex]

If function [tex]\( f \)[/tex] is a cubic polynomial, which statement most accurately describes the function over the interval [tex]\( (0,1) \)[/tex]?

A. The function is constant over the interval [tex]\( (0,1) \)[/tex]

B. The function is increasing over the interval [tex]\( (0,1) \)[/tex]

C. The function is decreasing over the interval [tex]\( (0,1) \)[/tex]

D. The function increases and decreases over the interval [tex]\( (0,1) \)[/tex]



Answer :

GinnyAnswer
To determine the behavior of the function [tex]\(f\)[/tex] over the interval [tex]\((0, 1)\)[/tex] from the given data, we look at the values of the function at the endpoints of the interval:
- [tex]\( f(0) = -6 \)[/tex]
- [tex]\( f(1) = 0 \)[/tex]

We observe that [tex]\( f(x) \)[/tex] changes from [tex]\( -6 \)[/tex] at [tex]\( x = 0 \)[/tex] to [tex]\( 0 \)[/tex] at [tex]\( x = 1 \)[/tex].

Since [tex]\( f(1) > f(0) \)[/tex], we can conclude that the function [tex]\( f(x) \)[/tex] increases over this interval. Therefore, the behavior of [tex]\( f \)[/tex] over the interval [tex]\((0, 1)\)[/tex] is that it is increasing.

Thus, the correct answer is:

B. The function is increasing over the interval [tex]\((0, 1)\)[/tex].

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