question_answer 118)

Examine if Rolle’s theorem is applicable to any of the following functions. Can you say something about the converse of Rolle’s theorem from these examples ? (i) f(x) = [x] on [5, 9] (ii) f(x) = [x] on [–2, 2] (iii) f(x) = x2 – 1 on [1, 2]  

Answer:

(i) f(x) = [x] on [5, 9].          Here f(x) is a greatest integer function, which is discontinuous at integral points 5, 6, 7, 8 and 9  [5,9].          Since f(x) does not satisfy all the conditions of Roll?s theorem therefore Rolle?s theorem is not valid for the given function.          Although f?(x) = 0 for non integral points  does not satisfies all the conditions of Rolle?s theorem.          Hence converse of Roll?es theorem, is not true.          (ii) f(x) = [x] on [?2, 2]          Here f(x) is a greatest integer function which is not continuous at points ?2, ?1, 0, 1, 2          Since f(x) does not satisfy all the conditions of Rolle?s theorem. Therefore. Therefore Rolle?s theorem is not valid for the given function.          In this case, the converse of Rolle?s therorem is not true.          (iii) f(x) = x2 ? 1 on [1, 2]          f(1) = 1 ? 1 = 0, f(2) = 4 ? 1 = 3          Since f(1)  theorem f(x) does not satisfy all the conditions of Rolle?s theorem. Hence Rolle?s theorem is not applicable for the given function.                    Converse of Rolle?s theorem is not true here.