Kimberly and Emily went to lunch at their favorite restaurant. Kimberly has a
$5 off coupon to use for her purchase of
$30. Emily has a 10% off coupon to use
for her purchase of $25. What is the
difference between the total amount
Kimberly and Emily will pay individually?

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.