To determine the [tex]\( x \)[/tex]-value where the average rate of change of the function [tex]\( f \)[/tex] on the interval [tex]\([x, 7]\)[/tex] is 56, we use the given points:
- [tex]\(x\)[/tex]-values: [tex]\(3, 4, 5, 6, 7\)[/tex]
- [tex]\(f(x)\)[/tex]-values: [tex]\(12, 24, 28, 96, 192\)[/tex]
The average rate of change of a function between two points [tex]\((x_1, f(x_1))\)[/tex] and [tex]\((x_2, f(x_2))\)[/tex] is calculated using the formula:
[tex]\[
\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
Here, [tex]\( x_2 \)[/tex] is 7 and [tex]\( f(x_2) \)[/tex] is 192. We need to find the [tex]\( x \)[/tex] such that:
[tex]\[
\frac{192 - f(x)}{7 - x} = 56
\][/tex]
We'll test each [tex]\(x\)[/tex]-value to find the correct one:
1. For [tex]\(x = 3\)[/tex]:
[tex]\[
\frac{192 - 12}{7 - 3} = \frac{180}{4} = 45
\][/tex]
2. For [tex]\(x = 4\)[/tex]:
[tex]\[
\frac{192 - 24}{7 - 4} = \frac{168}{3} = 56
\][/tex]
3. For [tex]\(x = 5\)[/tex]:
[tex]\[
\frac{192 - 28}{7 - 5} = \frac{164}{2} = 82
\][/tex]
4. For [tex]\(x = 6\)[/tex]:
[tex]\[
\frac{192 - 96}{7 - 6} = \frac{96}{1} = 96
\][/tex]
Among these calculations, the correct [tex]\(x\)[/tex]-value is 4 since the average rate of change is 56 only when [tex]\( x = 4 \)[/tex].
Thus, the [tex]\( x \)[/tex]-value is [tex]\( \boxed{4} \)[/tex].
