To determine which statement must be true given that 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], we can use the definition of the average rate of change.
The average rate of change of a function [tex]\( g(x) \)[/tex] between [tex]\( x = a \)[/tex] and [tex]\( x = b \)[/tex] is calculated by:
[tex]\[
\frac{g(b) - g(a)}{b - a}
\][/tex]
In this problem:
- [tex]\( a = 4 \)[/tex]
- [tex]\( b = 7 \)[/tex]
- The average rate of change is [tex]\( \frac{5}{6} \)[/tex]
Substitute the given values into the formula:
[tex]\[
\frac{g(7) - g(4)}{7 - 4} = \frac{5}{6}
\][/tex]
Simplify the denominator:
[tex]\[
\frac{g(7) - g(4)}{3} = \frac{5}{6}
\][/tex]
To isolate [tex]\( g(7) - g(4) \)[/tex], multiply both sides of the equation by 3:
[tex]\[
g(7) - g(4) = 3 \times \frac{5}{6}
\][/tex]
[tex]\[
g(7) - g(4) = \frac{15}{6}
\][/tex]
[tex]\[
g(7) - g(4) = \frac{5}{2}
\][/tex]
Therefore, the statement that must be true is:
[tex]\[
\frac{g(7) - g(4)}{7 - 4} = \frac{5}{6}
\][/tex]
This corresponds to the third choice:
[tex]\[
\boxed{\frac{g(7) - g(4)}{7 - 4} = \frac{5}{6}}
\][/tex]
Hence, the correct statement from the options provided is the third one.
