To solve these statements, let's look into the given information.
1. Finding the relative maximum:
A relative maximum of a function occurs at points where the first derivative changes from positive to negative. The first derivative indicates the rate of change of the function's value with respect to [tex]\( x \)[/tex].
From the given values:
[tex]\[
y = 216, 110, 40, 0, -16, -14, 0, 20, 40, 54, 56, 40, 0, -70, -176
\][/tex]
The calculated first derivatives are:
[tex]\[
[-106, -70, -40, -16, 2, 14, 20, 20, 14, 2, -16, -40, -70, -106]
\][/tex]
Observing the change of signs in the first derivatives, it is positive before [tex]\( x = 3 \)[/tex] and negative after [tex]\( x = 3 \)[/tex].
Thus, the function has a relative maximum near [tex]\( x = 3 \)[/tex].
2. Behavior as [tex]\( x \)[/tex] approaches positive infinity:
A cubic function tends to [tex]\(\infty\)[/tex] or [tex]\(-\infty\)[/tex] as [tex]\( x \)[/tex] approaches [tex]\(\infty\)[/tex], depending on the leading coefficient of the cubic term. Given the values and the nature of the function, it suggests the polynomial has a positive leading coefficient.
Consequently, as [tex]\( x \)[/tex] approaches positive infinity, the value of the function approaches [tex]\(\infty\)[/tex].
Therefore, the completed statements are:
- 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]\( \boxed{\infty} \)[/tex].
