To determine the ordered pair closest to a local minimum from the table of values, follow these steps:
1. Understand the Concept of Local Minimum:
A local minimum of a function [tex]\( f(x) \)[/tex] occurs at a point [tex]\( x \)[/tex] where the function value [tex]\( f(x) \)[/tex] is lower than that of its immediate neighbors.
2. Identify and Examine Each Point:
Let’s look at each point and compare it to its neighbors.
- Point (-2, -8): It doesn't have a left neighbor to compare, so it's not considered.
- Point (-1, -3):
[tex]\[
\text{Left neighbor} = (-2, -8), \quad \text{Right neighbor} = (0, -2)
\][/tex]
[tex]\( -3 \)[/tex] is greater than [tex]\( -8 \)[/tex] and less than [tex]\( -2 \)[/tex], so it is not a local minimum.
- Point (0, -2):
[tex]\[
\text{Left neighbor} = (-1, -3), \quad \text{Right neighbor} = (1, 4)
\][/tex]
[tex]\( -2 \)[/tex] is greater than [tex]\( -3 \)[/tex] and less than [tex]\( 4 \)[/tex], so it is not a local minimum.
- Point (1, 4):
[tex]\[
\text{Left neighbor} = (0, -2), \quad \text{Right neighbor} = (2, 1)
\][/tex]
[tex]\( 4 \)[/tex] is greater than [tex]\( -2 \)[/tex] and greater than [tex]\( 1 \)[/tex], so it is not a local minimum.
- Point (2, 1):
[tex]\[
\text{Left neighbor} = (1, 4), \quad \text{Right neighbor} = (3, 3)
\][/tex]
[tex]\( 1 \)[/tex] is less than [tex]\( 4 \)[/tex] and less than [tex]\( 3 \)[/tex], so it is a local minimum.
- Point (3, 3): It doesn't have a right neighbor to compare, so it's not considered.
3. Determine the Local Minimum:
Based on the above analysis, the point (2, 1) satisfies the condition of a local minimum because its value is less than the values of its immediate neighbors.
Therefore, the ordered pair closest to a local minimum of the function [tex]\( f(x) \)[/tex] is:
[tex]\[
(2, 1)
\][/tex]
