To analyze the table of values for the function [tex]\( f(x) \)[/tex] and determine the intervals where local maxima and minima occur, follow the steps below:
1. Identify Critical Points:
Examine the function values to identify changes in the direction of [tex]\( f(x) \)[/tex]. A local maximum occurs when [tex]\( f(x) \)[/tex] changes from increasing to decreasing, and a local minimum occurs when [tex]\( f(x) \)[/tex] changes from decreasing to increasing.
2. Observations from the Table:
- As [tex]\( x \)[/tex] increases from [tex]\(-3\)[/tex] to [tex]\(-2\)[/tex] to [tex]\(-1\)[/tex], [tex]\( f(x) \)[/tex] increases from [tex]\(-16\)[/tex] to [tex]\(-1\)[/tex] to [tex]\( 2 \)[/tex].
- As [tex]\( x \)[/tex] increases from [tex]\(-1\)[/tex] to [tex]\( 0 \)[/tex], [tex]\( f(x) \)[/tex] decreases from [tex]\( 2 \)[/tex] to [tex]\(-1 \)[/tex].
- As [tex]\( x \)[/tex] increases from [tex]\( 0 \)[/tex] to [tex]\( 1 \)[/tex] to [tex]\( 2 \)[/tex], [tex]\( f(x) \)[/tex] decreases from [tex]\(-1 \)[/tex] to [tex]\(-4 \)[/tex] to [tex]\(-1\)[/tex].
3. Determine the Intervals:
- Local Maximum:
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 2 \)[/tex], where the function achieves its highest value in the interval [tex]\([-2, 0]\)[/tex].
- Hence, there is a local maximum over the interval [tex]\((-2, -1, 0)\)[/tex].
- Local Minimum:
- At [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -4 \)[/tex], where the function achieves its lowest value in the interval [tex]\((0, 2)\)[/tex].
- Hence, there is a local minimum over the interval [tex]\((0, 1, 2)\)[/tex].
Therefore, to complete the statements:
- A local maximum occurs over the interval [tex]\((-2, -1, 0)\)[/tex].
- A local minimum occurs over the interval [tex]\((0, 1, 2)\)[/tex].
