To determine the [tex]\( x \)[/tex]-value on the interval [tex]\([x, 7]\)[/tex] where the average rate of change is 56, we follow these steps:
1. Understand the Formula for Average Rate of Change:
The average rate of change of a function [tex]\( f \)[/tex] between two points [tex]\( x \)[/tex] and 7 is given by:
[tex]\[
\text{Average Rate of Change} = \frac{f(7) - f(x)}{7 - x}
\][/tex]
2. Identify the Given Values:
We know [tex]\( f(7) = 192 \)[/tex] (from the table).
3. Calculate the Average Rate of Change for Each [tex]\( x \)[/tex]-Value:
We will try each [tex]\( x \)[/tex]-value in the table to find which one yields an average rate of change of 56.
- For [tex]\( x = 3 \)[/tex]:
[tex]\[
\text{Average Rate of Change} = \frac{192 - 12}{7 - 3} = \frac{180}{4} = 45
\][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[
\text{Average Rate of Change} = \frac{192 - 24}{7 - 4} = \frac{168}{3} = 56
\][/tex]
We can see that the average rate of change is 56 when [tex]\( x = 4 \)[/tex].
- For [tex]\( x = 5 \)[/tex]:
[tex]\[
\text{Average Rate of Change} = \frac{192 - 28}{7 - 5} = \frac{164}{2} = 82
\][/tex]
- For [tex]\( x = 6 \)[/tex]:
[tex]\[
\text{Average Rate of Change} = \frac{192 - 96}{7 - 6} = \frac{96}{1} = 96
\][/tex]
After evaluating each of these, we see that the correct [tex]\( x \)[/tex]-value where the average rate of change is 56 is:
[tex]\[
x = 4
\][/tex]
Thus, on the interval [tex]\([x, 7]\)[/tex], the [tex]\( x \)[/tex]-value at which the average rate of change is 56 is [tex]\( \boxed{4} \)[/tex].
