Raul is paid $75 per week plus $5 for each new gym membership he sells. He may switch to a gym that pays $50 per week and $7.50 for each new membership. How many memberships per week does Raul have to sell for the new gym to be a better deal for him?

47 - 90 = 43

43 - 180 = x

x = 137

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.