If the movie gives it away five dollar coupons for every four movies purchased they give away a bag of popcorn for every three movies purchased and how many movies would you have to purchase and all before receiving both five dollar coupons in a popcorn bag at the same time

1st movie nothing, 2nd movie nothing, 3rd movie popcorn, 4th movie 5$ , 5th movie nothing, 6th movie popcorn, 7th movie nothing, 8th movie 5$, 9th movie popcorn, 10th movie nothing, 11th movie nothing

12th movie popcorn and 5$

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.

the one below it is a function