In a multiple-choice question with four alternatives, if the null hypothesis is correct, the probability of a person choosing the correct answer is approximately

25%,

This is because there are four choices to choose from, meaning each choice has a 25% chance of being chosen at random.

Assuming there is only one correct answer, that means that there is a 25% chance of it being chosen at random.

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.