Question

x | f(x)
-2 | -8
-1 | -3
0 | -2
1 | 4
2 | 1
3 | 3
what ordered pair is closest to a local minimum of the function, f(x)?
○ (-1, -3)
○ (0, -2)
○ (1, 4)
○ (2, 1)

Explanation:

Step1: Understand local minimum

A local minimum is a point where the function changes from decreasing to increasing (or has the lowest value in a local region). We analyze the function values as \( x \) increases.

Step2: Analyze the table

• For \( x = -2 \), \( f(x) = -8 \); \( x = -1 \), \( f(x) = -3 \) (function is increasing here, since \(-3 > -8\)).
• \( x = 0 \), \( f(x) = -2 \) (still increasing from \( x=-1 \) as \(-2 > -3\)).
• \( x = 1 \), \( f(x) = 4 \) (increasing from \( x=0 \) as \( 4 > -2 \)).
• \( x = 2 \), \( f(x) = 1 \) (now decreasing from \( x=1 \) as \( 1 < 4 \)).
• \( x = 3 \), \( f(x) = 3 \) (increasing from \( x=2 \) as \( 3 > 1 \)).

The function decreases from \( x = 1 \) to \( x = 2 \) (since \( f(2)=1 < f(1)=4 \)) and then increases from \( x = 2 \) to \( x = 3 \) (since \( f(3)=3 > f(2)=1 \)). So around \( x = 2 \), the function has a local minimum. The ordered pair at \( x = 2 \) is \( (2, 1) \), which is the point closest to the local minimum (as it is the local minimum point in this table's context).

Answer:

(2, 1)