question_answer 121)

Verify L.M.V. theorem for f(x) ? x3 ? 5x2 ? 3x in [1, 3]. Also find all  for which f?(c) = 0  

Answer:

f(x) = x3 ? 5x2 ? 3x in [1, 3]                   (i) Being a polynomial function, f(x) is continuous in             [1, 3]                                                 (ii) f?(x) exists uniquely in (1, 3),              is derivable in (1, 3).          Since f(x) satisfies both conditions of L.M.V. theorem.          Therefore there exists at least one  such that                                     3c2 ? 10c + 7 = 0           3c2 ? 3c + 7c + 7 = 0           3c(c ? 1) ? 7 (c ? 1) = 0           (c ? 1) (3c ?1) = 0                   Here          Hence the theorem verified.          Now f?(c) = 0                            Hence there is no  that f?(c) = 0