 Hi and welcome to the session. Let us discuss following question. Question says, Examining if roles theorem is applicable to any of the following functions. Can you say something about the converse of roles theorem? From these examples, first part is fx is equal to greatest integer of x for x belonging to closed interval 59. First of all, let us understand what is roles theorem if we are given a function f from closed interval ab to r. This function is continuous on closed interval ab. Function is differentiable on open interval ab such that f a is equal to fb. Then, there exists some c belonging to open interval ab such that f dash c is equal to 0. This is known as roles theorem. Now we will use it as our key idea to solve the given question. Let us now start with the solution. We are given fx is equal to greatest integer of x for x belonging to closed interval 59. Now we know given function that is fx is equal to greatest integer of x is not continuous at all integer points. So we can write function f given by fx is equal to greatest integer of x is not continuous for x belonging to closed interval 59. Now clearly we can see the given function does not satisfy the first condition of roles theorem. So we can say roles theorem is not applicable to the given function. Now let us discuss converse of roles theorem. Converse of roles theorem states that if we are given a function f from closed interval ab to r and it is continuous on closed interval ab and differentiable on open interval ab and if there is some c belonging to open interval ab such that f dash c is equal to 0. Then f a is equal to fb. Now clearly we can see given function does not satisfy the first condition given in the converse of roles theorem. So we can say converse of roles theorem is not applicable to the given function. Now clearly we can see if roles theorem is not applicable to the given function then converse of roles theorem is also not applicable to it. So this is our required answer. This completes the session. Hope you understood the session. Take care and have a nice day.