Arrange in descending order 1/2, 3/4, 2/3

The question is asking which numbers are larger with respect to one another. An easy way to do this is to convert to decimals, or just memorize it. 1/2=0.5, 3/4=0.75, 2/3=0.666666. So descending order (largest to smallest) 3/4, 2/3, 1/2.

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.