Please please help me with these problems :’) (It’d be helpful if you can put down how you got the answer )

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.

the furthest the dog can be is 508 meters away from the house, and the closest he can be is 492 meters away from the house.

answer: i think it is c