calculate the number of messages per day that each user receives if the number of users is 2586 and the messages per day is 205.6

This question makes no sense but, the way that's worded, each user receives 12.57 messages a day. If you meant how many messages total its 531681.6 messages.

step-by-step :

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.