question_answer 1)

Examine the validity of L.M.V. theorem for (i) f(x) = [x]in [5, 0] (ii) f(x) = [x] in [–2, 2] (iii) f(x) = x2 – 1 in [1, 2]  

Answer:

         For (i) and (ii), f(x) is an greatest integer function which is not continuous at integral points belong to given corresponding intervals.          Since f(x) does not satisfy all the conditions of L.M.V. theorem, therefore L.M.V. theorem is not applicable for these functions.          (iii) f(x) = x2 ? 1 in [1, 2] (a)   f(x) is a polynomial function is continuous in [1, 2] (b) f?(x) exists uniquely in (1, 2)     is derivable in (1, 2)        Since f(x) satisfies all the conditions of L.M.V. theorem Therefore L.M.V. theorem is valid for this function and so there exists at least one .    Such that f?(c)          Hence the theorem is verified.