Let ln x denote the logarithm of x with respect to the base e Let S ⊂ R be the set of all points where the function ln x 2 - 1 is well-defined. Then, the number of functions f : S → R that are differentiable, satisfy f ' x = ln x 2 - 1 for all x ∈ S and f 2 = 0 , is

Step-by-step solution

We have, f ' x = ln x 2 + 1 f x = ∫ ln x 2 - 1 d x f x = x ln x 2 - 1 - ∫ 2 x 2 x 2 - 1 d x f x = x ln x 2 - 1 - 2 ∫ x 2 - 1 x 2 - 1 + 1 x 2 - 1 d x f x = x ln x 2 - 1 - 2 x - ln x - 1 x + 1 + C f 2 = 2 ln 3 - 4 - ln 1 3 + C = 0 ∵ f 2 = 0 ⇒ C = 4 - 3 ln 3 ∴ f x = x ln x 2 - 1 - 2 n ln x - 1 x + 1 + 4 - 3 ln 3 defined for S infinite C values possible in set S such that f ' x = ln x 2 - 1