To determine which ordered pair is a local maximum of the function [tex]\( f(x) \)[/tex], we need to identify points in the table where the function value is greater than the values of its immediate neighbors on both sides.
Let's consider each point and compare its value with its neighboring points:
1. For [tex]\( x = -2 \)[/tex], [tex]\( f(-2) = 0 \)[/tex]. It doesn't have a left neighbor, but the right neighbor is [tex]\( f(-1) = 45 \)[/tex]. Since [tex]\( 0 < 45 \)[/tex], it is not a local maximum.
2. For [tex]\( x = -1 \)[/tex], [tex]\( f(-1) = 45 \)[/tex]. The left neighbor is [tex]\( f(-2) = 0 \)[/tex] and the right neighbor is [tex]\( f(0) = 64 \)[/tex]. Since [tex]\( 45 < 64 \)[/tex], it is not a local maximum.
3. For [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 64 \)[/tex]. The left neighbor is [tex]\( f(-1) = 45 \)[/tex] and the right neighbor is [tex]\( f(1) = 45 \)[/tex]. Since [tex]\( 64 > 45 \)[/tex] and [tex]\( 64 > 45 \)[/tex], this is a local maximum.
4. For [tex]\( x = 1 \)[/tex], [tex]\( f(1) = 45 \)[/tex]. The left neighbor is [tex]\( f(0) = 64 \)[/tex] and the right neighbor is [tex]\( f(2) = 0 \)[/tex]. Since [tex]\( 45 < 64 \)[/tex], it is not a local maximum.
5. For [tex]\( x = 2 \)[/tex], [tex]\( f(2) = 0 \)[/tex]. The left neighbor is [tex]\( f(1) = 45 \)[/tex] and the right neighbor is [tex]\( f(3) = -35 \)[/tex]. Since [tex]\( 0 < 45 \)[/tex], it is not a local maximum.
6. For [tex]\( x = 3 \)[/tex], [tex]\( f(3) = -35 \)[/tex]. The left neighbor is [tex]\( f(2) = 0 \)[/tex] and the right neighbor is [tex]\( f(4) = 0 \)[/tex]. Since [tex]\( -35 < 0 \)[/tex], it is not a local maximum.
7. For [tex]\( x = 4 \)[/tex], [tex]\( f(4) = 0 \)[/tex]. The left neighbor is [tex]\( f(3) = -35 \)[/tex] and the right neighbor is [tex]\( f(5) = 189 \)[/tex]. Since [tex]\( 0 < 189 \)[/tex], it is not a local maximum.
8. For [tex]\( x = 5 \)[/tex], [tex]\( f(5) = 189 \)[/tex]. The left neighbor is [tex]\( f(4) = 0 \)[/tex] and the right neighbor is [tex]\( f(6) = 640 \)[/tex]. Since [tex]\( 189 < 640 \)[/tex], it is not a local maximum.
9. For [tex]\( x = 6 \)[/tex], [tex]\( f(6) = 640 \)[/tex]. It doesn't have a right neighbor, but the left neighbor is [tex]\( f(5) = 189 \)[/tex]. Since [tex]\( 640 > 189 \)[/tex], it is not a local maximum.
Based on the comparisons, the only ordered pair that is a local maximum is [tex]\((0, 64)\)[/tex]. Thus, the ordered pair that is a local maximum of the function [tex]\( f(x) \)[/tex] according to the table is:
[tex]\[
(0, 64)
\][/tex]
