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Which ordered pair is closest to a local minimum of the function [tex]\( f(x) \)[/tex]?

[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]

A. [tex]\((-1, -3)\)[/tex]
B. [tex]\((0, -2)\)[/tex]
C. [tex]\((1, 4)\)[/tex]
D. [tex]\((2, 1)\)[/tex]


Sagot :

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]