To determine the local minimum of the given function [tex]\(\pi(x)\)[/tex] from the table of values, we need to identify the point where the function attains the lowest value.
Here's the table again for reference:
[tex]\[
\begin{array}{|c|c|}
\hline
x & \pi(x) \\
\hline
-4 & 105 \\
\hline
-3 & 0 \\
\hline
-2 & -15 \\
\hline
-1 & 0 \\
\hline
0 & 9 \\
\hline
1 & 0 \\
\hline
2 & -15 \\
\hline
3 & 0 \\
\hline
4 & 105 \\
\hline
5 & 384 \\
\hline
\end{array}
\][/tex]
We need to:
1. Look for all the values of [tex]\(\pi(x)\)[/tex] and identify the smallest value.
2. Check that the corresponding [tex]\(x\)[/tex] value produces the global minimum or local minimum by considering neighboring values.
From the table, we observe the following values:
- [tex]\(\pi(-4) = 105\)[/tex]
- [tex]\(\pi(-3) = 0\)[/tex]
- [tex]\(\pi(-2) = -15\)[/tex]
- [tex]\(\pi(-1) = 0\)[/tex]
- [tex]\(\pi(0) = 9\)[/tex]
- [tex]\(\pi(1) = 0\)[/tex]
- [tex]\(\pi(2) = -15\)[/tex]
- [tex]\(\pi(3) = 0\)[/tex]
- [tex]\(\pi(4) = 105\)[/tex]
- [tex]\(\pi(5) = 384\)[/tex]
The smallest value in the table is [tex]\(-15\)[/tex].
We see two points where [tex]\(\pi(x) = -15\)[/tex]:
- [tex]\(\pi(-2) = -15\)[/tex]
- [tex]\(\pi(2) = -15\)[/tex]
Although both points [tex]\((-2, -15)\)[/tex] and [tex]\((2, -15)\)[/tex] share the same function value, we need to determine which one is a local minimum. In this context, we validate this by verifying adjacent values to ensure that they are higher than the minimum.
Here's the validation:
- For [tex]\(x = -2\)[/tex], the adjacent values are [tex]\(\pi(-3) = 0\)[/tex] and [tex]\(\pi(-1) = 0\)[/tex], which are both higher than [tex]\(-15\)[/tex].
- For [tex]\(x = 2\)[/tex], the adjacent values are [tex]\(\pi(1) = 0\)[/tex] and [tex]\(\pi(3) = 0\)[/tex], which are both higher than [tex]\(-15\)[/tex].
Thus, both [tex]\((-2, -15)\)[/tex] and [tex]\((2, -15)\)[/tex] can be considered local minima, but typically we report the first occurrence or the one explicitly asked for.
Therefore, the ordered pair that represents a local minimum of the function [tex]\(\pi(x)\)[/tex], according to the table, is:
[tex]\(\boxed{(-2, -15)}\)[/tex]
