Question

If f(x) is a differentiable function such that f : R → R and f(1/n)  = 0 ∀ n ≥ ∈ I, then

(A) f(x) = 0 ∀ x ∈(0, 1)

(B) f(0) = 0 = f′(0)

(C) f(0) = 0 but f′(0) may or may not be 0

(D) |f(x)| ≤ 1 ∀ x ∈(0, 1)

Answer 1

Answer is (B) f(0) = 0 = f′(0)

Since there are infinitely many points in x ∈ (0,1) where

And since there are infinitely many points in the neighbourhood of x = 0. Such that f(x) remains constant in the neighbourhood of x = 0. Therefore,

f′(0) = 0