To determine at which [tex]\( x \)[/tex]-value the average rate of change on the interval [tex]\([x, 7]\)[/tex] is 56, we need to investigate each [tex]\( x \)[/tex]-value from the table and calculate the corresponding average rate of change.
The average rate of change of a function [tex]\( f \)[/tex] over the interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[
\frac{f(b) - f(a)}{b - a}
\][/tex]
Here, [tex]\( b = 7 \)[/tex] and [tex]\( f(b) = f(7) = 192 \)[/tex].
We will calculate the average rate of change for each [tex]\( x \)[/tex] from 3 to 6 (since 7 would just be the point itself and not an interval):
1. For [tex]\( x = 3 \)[/tex]:
[tex]\[
\text{Average rate of change} = \frac{f(7) - f(3)}{7 - 3} = \frac{192 - 12}{7 - 3} = \frac{180}{4} = 45
\][/tex]
2. For [tex]\( x = 4 \)[/tex]:
[tex]\[
\text{Average rate of change} = \frac{f(7) - f(4)}{7 - 4} = \frac{192 - 24}{7 - 4} = \frac{168}{3} = 56
\][/tex]
3. For [tex]\( x = 5 \)[/tex]:
[tex]\[
\text{Average rate of change} = \frac{f(7) - f(5)}{7 - 5} = \frac{192 - 28}{7 - 5} = \frac{164}{2} = 82
\][/tex]
4. For [tex]\( x = 6 \)[/tex]:
[tex]\[
\text{Average rate of change} = \frac{f(7) - f(6)}{7 - 6} = \frac{192 - 96}{7 - 6} = \frac{96}{1} = 96
\][/tex]
Among these calculations, the only [tex]\( x \)[/tex]-value for which the average rate of change is 56 is [tex]\( x = 4 \)[/tex].
Therefore, the correct location on the image where the average rate of change on the interval [tex]\([x, 7]\)[/tex] is 56 is at [tex]\( x = 4 \)[/tex].
