This table represents values of a cubic polynomial function.
\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
-2 & -12 \\
\hline
-1 & 0 \\
\hline
0 & 6 \\
\hline
1 & 7.5 \\
\hline
2 & 6 \\
\hline
3 & 3 \\
\hline
\end{tabular}

Based on the information in the table, which sentence best describes the interval [tex]$[-2,1]$[/tex]?

A. The function is both increasing and decreasing on the interval [tex]$[-2,1]$[/tex].

B. The function is constant on the interval [tex]$[-2,1]$[/tex].

C. The function is decreasing on the interval [tex]$[-2,1]$[/tex].

D. The function is increasing on the interval [tex]$[-2,1]$[/tex].



Answer :

Let's analyze the given values of the function on the interval [tex]\([-2, 1]\)[/tex]:

[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -2 & -12 \\ \hline -1 & 0 \\ \hline 0 & 6 \\ \hline 1 & 7.5 \\ \hline \end{array} \][/tex]

We will look at how the [tex]\( y \)[/tex]-values change as [tex]\( x \)[/tex] increases from [tex]\(-2\)[/tex] to [tex]\(1\)[/tex]:

1. From [tex]\(-2\)[/tex] to [tex]\(-1\)[/tex]:
- When [tex]\( x = -2 \)[/tex], [tex]\( y = -12 \)[/tex]
- When [tex]\( x = -1 \)[/tex], [tex]\( y = 0 \)[/tex]
- The function value increases from [tex]\(-12\)[/tex] to [tex]\(0\)[/tex].

2. From [tex]\(-1\)[/tex] to [tex]\(0\)[/tex]:
- When [tex]\( x = -1 \)[/tex], [tex]\( y = 0 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 6 \)[/tex]
- The function value increases from [tex]\(0\)[/tex] to [tex]\(6\)[/tex].

3. From [tex]\(0\)[/tex] to [tex]\(1\)[/tex]:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 6 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 7.5 \)[/tex]
- The function value increases from [tex]\(6\)[/tex] to [tex]\(7.5\)[/tex].

In each of these intervals, the [tex]\( y \)[/tex]-value increases as the [tex]\( x \)[/tex]-value increases. Thus, the function is consistently increasing throughout the entire interval [tex]\([-2, 1]\)[/tex].

Therefore, the sentence that best describes the interval [tex]\([-2, 1]\)[/tex] is:

D. The function is increasing on the interval [tex]\([-2, 1]\)[/tex].