To complete the statements, let's analyze the table provided for the given cubic function.
The table shows the values of [tex]\( y \)[/tex] for different values of [tex]\( x \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
x & -7 & -6 & -5 & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
y & 216 & 110 & 40 & 0 & -16 & -14 & 0 & 20 & 40 & 54 & 56 & 40 & 0 & -70 & -176 \\
\hline
\end{array}
\][/tex]
### Finding the Relative Maximum
A relative maximum occurs where the function reaches a peak before starting to decrease.
- As [tex]\(x\)[/tex] increases from -7 to 3, the function values increase to a maximum value of 56 at [tex]\(x = 3\)[/tex] before decreasing again at [tex]\(x = 4\)[/tex].
Therefore, the relative maximum occurs when [tex]\(x\)[/tex] is near 3.
### Behavior as x Approaches Positive Infinity
For cubic functions of the form [tex]\(f(x) = ax^3 + bx^2 + cx + d\)[/tex] with [tex]\(a > 0\)[/tex], as [tex]\(x\)[/tex] approaches positive infinity, the [tex]\(x^3\)[/tex] term will dominate, causing the function to approach positive infinity.
Therefore, as [tex]\(x\)[/tex] approaches positive infinity, the value of the function approaches infinity.
### Completed Statements
- The function has a relative maximum when [tex]\(x\)[/tex] is near 3.
- As [tex]\(x\)[/tex] approaches positive infinity, the value of the function approaches infinity.
