Consider the theorem: If f and f’ are both strictly increasing real-valued functions on the interval (0, infinite) then lim f(x) = infinite (x -> infinite). The following argument is suggested as a proof of this theorem.
(1) By the Mean Value Theorem, there is a c1 in the interval (1, 2) such that
f’(c1) = {f(2)-f(1)}/ (2-1) = f(2) – f(1) > 0.
(2) For each x > 2, there is a c(x) in (2, x) such that {f(x) – f(2)}/(x-2) = f’{c(x)}
(3) For each x > 2, {f(x)-f(2)}/(x-2) = f’(c(x)) > f’(c1) since f’ strictly incrasing
(4) For each x > 2, f(x) > f(2) + (x-2) f’(c1)
(5) lim f(x) = infinite (x -> infinite)
Which of the following statements is true?
(A) The argument is valid.
(B) The argument is not valid since the hypotheses of the Mean Value Theorem are not satisfied in (1) and (2).
(C) The argument is not valid since (3) is not valid.
(D) The argument is not valid since (4) cannot be deduced from the previous steps.
(E) The argument is not valid since (4) does not imply (5).