Let's address the two statements step-by-step using the given data from the table:
### Analyzing Relative Maximum
1. Finding Relative Maximum:
- To find a relative maximum, we need to identify where the values of [tex]\( y \)[/tex] increase to a peak point and then start decreasing. This occurs when [tex]\( y \)[/tex] is greater than its neighboring values on both sides.
- Observing the given table:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
x & -7 & -6 & -5 & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
y & 216 & 110 & 40 & 0 & -16 & -14 & 0 & 20 & 40 & 54 & 56 & 40 & 0 & -70 & -176 \\
\hline
\end{tabular}
\][/tex]
- Focus on the [tex]\( y \)[/tex]-values, we note that at [tex]\( x = 3 \)[/tex], the value [tex]\( y = 56 \)[/tex] is greater than both its neighboring values [tex]\( y = 54 \)[/tex] (at [tex]\( x = 2 \)[/tex]) and [tex]\( y = 40 \)[/tex] (at [tex]\( x = 4 \)[/tex]).
- Thus, the function has a relative maximum when [tex]\( x \)[/tex] is near 3.
### Behavior as [tex]\( x \)[/tex] Approaches Positive Infinity
2. Analyzing the Behavior at Infinity:
- Cubic functions, in general, tend to [tex]\( \pm \infty \)[/tex] as [tex]\( x \)[/tex] tends to [tex]\( \pm \infty \)[/tex].
- Since the coefficient of the leading term (the term with the highest power of [tex]\( x \)[/tex]) of the cubic function is positive (implied by the general increase to positive values in the data), the function will approach positive infinity as [tex]\( x \)[/tex] becomes very large.
- Therefore, as [tex]\( x \)[/tex] approaches positive infinity, the value of the function approaches positive infinity.
### Summary
Combining all deductions from the analysis:
- The function has a relative maximum when [tex]\( x \)[/tex] is near [tex]\( \boxed{3} \)[/tex].
- As [tex]\( x \)[/tex] approaches positive infinity, the value of the function approaches [tex]\( \boxed{\text{positive infinity}} \)[/tex].
