Why are the solutions to the proportions StartFraction 40 over 8 EndFraction = StartFraction x over 10 EndFraction and StartFraction x over 40 EndFraction = StartFraction 10 over 8 EndFraction the same? because both result in the equation 8 x = 400, which simplifies to x = 5 because both result in the equation 8 x = 400, which simplifies to x = 50 because both result in the equation 40 x = 80, which simplifies to x = 2 because both result in the equation 40 x = 80, which simplifies to x = 20

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.