To determine the ordered pair closest to a local minimum of the function [tex]\( f(x) \)[/tex], we need to analyze the given points. Here we have the following points:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & -8 \\
\hline
-1 & -3 \\
\hline
0 & -2 \\
\hline
1 & 4 \\
\hline
2 & 1 \\
\hline
3 & 3 \\
\hline
\end{array}
\][/tex]
Let's examine the behavior of the function [tex]\( f(x) \)[/tex] at each point:
1. [tex]\( f(-2) = -8 \)[/tex]
2. [tex]\( f(-1) = -3 \)[/tex]
3. [tex]\( f(0) = -2 \)[/tex]
4. [tex]\( f(1) = 4 \)[/tex]
5. [tex]\( f(2) = 1 \)[/tex]
6. [tex]\( f(3) = 3 \)[/tex]
We look for a local minimum in a function by checking if the value at a certain point is lower than its neighboring points.
For the given values, let's evaluate the ordered pairs:
1. Between [tex]\( x = -2 \)[/tex] and [tex]\( x = -1 \)[/tex], [tex]\( f(x) \)[/tex] increases from -8 to -3.
2. Between [tex]\( x = -1 \)[/tex] and [tex]\( x = 0 \)[/tex], [tex]\( f(x) \)[/tex] increases from -3 to -2.
3. Between [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex], [tex]\( f(x) \)[/tex] decreases from -2 to 4.
4. Between [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex], [tex]\( f(x) \)[/tex] decreases from 4 to 1.
5. Between [tex]\( x = 2 \)[/tex] and [tex]\( x = 3 \)[/tex], [tex]\( f(x) \)[/tex] increases from 1 to 3.
The point [tex]\( (0, -2) \)[/tex] presents an interesting case where it is surrounded by higher values on both sides. This indicates a local minimum within our given range.
Thus, the ordered pair closest to a local minimum of the function [tex]\( f(x) \)[/tex] is:
[tex]\[
(0, -2)
\][/tex]
So, from the given choices, the ordered pair closest to a local minimum of the function [tex]\( f(x) \)[/tex] is:
[tex]\[
(0, -2)
\][/tex]
