Answer :
To solve this problem, we need to understand what the term "average rate of change" means, and how it applies to the function [tex]\( T(d) \)[/tex].
The average rate of change of a function over an interval [tex]\([a, b]\)[/tex] is given by the formula:
[tex]\[ \text{Average rate of change} = \frac{T(b) - T(a)}{b - a} \][/tex]
Here, [tex]\( T(d) \)[/tex] represents the number of tickets sold, and [tex]\( d \)[/tex] is the number of days since the movie was released. We are given the interval [tex]\([d_1, d_2]\)[/tex] where [tex]\( d_1 = 4 \)[/tex] and [tex]\( d_2 = 10 \)[/tex].
We are also given that the average rate of change of [tex]\( T(d) \)[/tex] over this interval is 0. Substituting the values into the formula, we have:
[tex]\[ 0 = \frac{T(10) - T(4)}{10 - 4} \][/tex]
Simplify the denominator:
[tex]\[ 0 = \frac{T(10) - T(4)}{6} \][/tex]
This equation means that the numerator must be zero for the overall fraction to be zero, since dividing zero by any non-zero number is still zero. So:
[tex]\[ T(10) - T(4) = 0 \][/tex]
Adding [tex]\( T(4) \)[/tex] to both sides:
[tex]\[ T(10) = T(4) \][/tex]
This tells us that the number of tickets sold on the tenth day is equal to the number of tickets sold on the fourth day.
Therefore, the statement that must be true is:
The same number of tickets was sold on the fourth day and tenth day.
The average rate of change of a function over an interval [tex]\([a, b]\)[/tex] is given by the formula:
[tex]\[ \text{Average rate of change} = \frac{T(b) - T(a)}{b - a} \][/tex]
Here, [tex]\( T(d) \)[/tex] represents the number of tickets sold, and [tex]\( d \)[/tex] is the number of days since the movie was released. We are given the interval [tex]\([d_1, d_2]\)[/tex] where [tex]\( d_1 = 4 \)[/tex] and [tex]\( d_2 = 10 \)[/tex].
We are also given that the average rate of change of [tex]\( T(d) \)[/tex] over this interval is 0. Substituting the values into the formula, we have:
[tex]\[ 0 = \frac{T(10) - T(4)}{10 - 4} \][/tex]
Simplify the denominator:
[tex]\[ 0 = \frac{T(10) - T(4)}{6} \][/tex]
This equation means that the numerator must be zero for the overall fraction to be zero, since dividing zero by any non-zero number is still zero. So:
[tex]\[ T(10) - T(4) = 0 \][/tex]
Adding [tex]\( T(4) \)[/tex] to both sides:
[tex]\[ T(10) = T(4) \][/tex]
This tells us that the number of tickets sold on the tenth day is equal to the number of tickets sold on the fourth day.
Therefore, the statement that must be true is:
The same number of tickets was sold on the fourth day and tenth day.
