 Hello friends welcome to the session I am Malka, let us verify mean value theorem if fx equal to x cube minus pi by x square minus 3x in the interval a, b, where a equal to 1 and b equal to 3 find all c being the element of open interval 1, 3 for which f dash c equal to 0. So let us start with the solution we are given fx equal to x cube minus pi by x square minus 3x. Now being a polynomial function it is continuous, it is continuous in close interval 1, 3 hence differentiable open interval 1, 3. Now we will find f1 which is equal to 1 minus 5 minus 3 equal to minus 7, f3 equal to 27 minus 45 minus 9 equal to minus 27 this implies f1 is not equal to f3 thus we see that all the three conditions are satisfied all three conditions value theorem are satisfied therefore f dash c is not equal to 0 in close interval 1, 3. Now we are given fx equal to x cube minus 5x square minus 3x and we will find f dash x which will be equal to 3x square minus 10x minus 3 therefore f dash c will be equal to 3c square minus 10c minus 3. Now but we are given that f dash c equal to 0 therefore 3c square minus 10c minus 3 equal to 0 this implies c equal to minus b which is minus of minus 10 plus minus square root of v square which is minus 10 minus 4ac upon 2a this is equal to 10 plus minus square root of 100 plus 36 upon 6 this implies c equal to 10 plus minus square root of 4 into 34 by 6 this implies c equal to 5 plus minus root 34 upon 3 this implies c equal to 5 plus minus 5.83 upon 3 this is approximate value of square root of 34 now this implies c equal to 10.83 by 3 or minus 0.83 by 3. There is no value of c being the element of open interval 1, 3. There no value c being the element of open interval 1, 3. So hope you understood the solution and enjoyed the session. Goodbye and take care.