To determine which statement must be true given the average rate of change of [tex]\( g(x) \)[/tex] between [tex]\( x=4 \)[/tex] and [tex]\( x=7 \)[/tex] is [tex]\(\frac{5}{6}\)[/tex], let's start by understanding the formula for the average rate of change of a function between two points.
The average rate of change of a function [tex]\( g(x) \)[/tex] between two points [tex]\( x=a \)[/tex] and [tex]\( x=b \)[/tex] is given by:
[tex]\[
\frac{g(b) - g(a)}{b - a}
\][/tex]
Given [tex]\( a = 4 \)[/tex] and [tex]\( b = 7 \)[/tex], the average rate of change can be written as:
[tex]\[
\frac{g(7) - g(4)}{7 - 4}
\][/tex]
From the problem, we know:
[tex]\[
\frac{g(7) - g(4)}{7 - 4} = \frac{5}{6}
\][/tex]
Simplifying the denominator, we have:
[tex]\[
\frac{g(7) - g(4)}{3} = \frac{5}{6}
\][/tex]
Next, we solve for [tex]\( g(7) - g(4) \)[/tex] by eliminating the denominator on the left-hand side:
[tex]\[
g(7) - g(4) = \frac{5}{6} \times 3
\][/tex]
[tex]\[
g(7) - g(4) = \frac{5 \times 3}{6}
\][/tex]
[tex]\[
g(7) - g(4) = \frac{15}{6}
\][/tex]
[tex]\[
g(7) - g(4) = 2.5
\][/tex]
Therefore, the statement that must be true is that [tex]\( g(7) - g(4) = 2.5 \)[/tex]. Among the provided statements, let's evaluate each one:
1. [tex]\( g(7) - g(4) = \frac{5}{6} \)[/tex]
- This is not correct because we found [tex]\( g(7) - g(4) = 2.5 \)[/tex].
2. [tex]\( \frac{g(7-4)}{7-4} = \frac{5}{6} \)[/tex]
- This is incorrect because it does not properly apply the average rate of change formula.
3. [tex]\( \frac{\alpha(7) - q(4)}{7 - 4} = \frac{5}{6} \)[/tex]
- This is incorrect because it refers to different functions, [tex]\( \alpha(x) \)[/tex] and [tex]\( q(x) \)[/tex].
4. [tex]\( \frac{g(7)}{g(4)} = \frac{5}{6} \)[/tex]
- This is not correct because it suggests a different kind of ratio that isn't related to the average rate of change.
None of the given statements are perfectly correct as per our initial detailed solution. It appears there was an intended correct statement that should reflect the [tex]\( g(7) - g(4) = 2.5 \)[/tex] result, but that option isn't listed perfectly in the provided choices. However, based on our calculations, the correct value is [tex]\( g(7) - g(4) = 2.5 \)[/tex].
