 Hi and welcome to the session. I am Shashi and I am going to help you with the following question. Question says verify Rolls theorem for the following function. Given function is fx is equal to x square minus 5x plus 4 on closed interval 1 comma 4. First of all let us recall Rolls theorem. Rolls theorem states that if function f from closed interval a comma b to r is continuous on closed interval a comma b and differentiable on open interval a comma b such that f a is equal to f b then there exists some c in open interval a comma b such that f dash c is equal to 0. This theorem is the key idea to solve the given question. Let us now start with the solution. Now we have fx is equal to x square minus 5x plus 4. Clearly we can see this is a polynomial function and we know every polynomial function is continuous. So we can write fx being a polynomial function is continuous in closed interval 1 comma 4. Also given polynomial function is differentiable in open interval 1 comma 4. Now using key idea we will find values of f1 and f4 and we will check if f1 is equal to f4. So first of all we will find f1 it is equal to 1 minus 5 plus 4. Substituting 1 for x in this expression we get 1 minus 5 plus 4 which is further equal to 0. Similarly we can find f4 substituting 4 for x in this expression we get 4 square minus 5 multiplied by 4 plus 4 it is equal to 0 only. Now clearly we can see f1 is equal to f4 so we can write f1 is equal to f4. So clearly we can see all the conditions of Roll's theorem are satisfied. So we can say there exists some c in open interval 1 comma 4 such that f dash c is equal to 0. Now we can write so there exists some c in open interval 1 comma 4 such that f dash c is equal to 0. Now f dash c is equal to 0 implies 2c minus 5 is equal to 0. We know fx is equal to x square minus 5x plus 4. Now differentiating both the sides with respect to x we get 2x minus 5 is equal to f dash x. So we can find f dash c by replacing x by c in this expression and we get f dash c is equal to 2c minus 5. Now this further implies 2c is equal to 5 adding 5 on both the sides of this equation we get 2c is equal to 5. Now dividing both the sides of this equation by 2 we get c is equal to 5 upon 2. Now 5 upon 2 belongs to open interval 1 comma 4. So we get value of c is equal to 5 upon 2 and this value belongs to the open interval 1 comma 4. So we can say hence the Roll's theorem is verified. This completes the session hope you understood the solution take care and have a nice day.