Question

Let S be the set of all continuous functions f: [-1,1]→\(\R\) satisfying the following three conditions:
(i) f is infinitely differentiable on the open interval (-1,1),
(ii) the Taylor series
\(f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+...\) of f at 0 converges to f(x) for each x ∈ (-1,1),
(iii) \(f(\frac{1}{n})=0\ \text{for all}\ n\isin\N\)
Then which of the following is/are true?

• A. f(0) = 0 for every f ∈ S.
• B. \(f'(\frac{1}{2})=0\) for every \(f\isin S\).
• C. There exists \(f\isin S\) such that \(f'(\frac{1}{2})\ne0\)
• D. There exists f ∈ S such that f (x) ≠ 0 for some x ∈ [-1,1].

Answer 1

The Correct Option is A, B

The correct option is (A): f(0) = 0 for every f ∈ S. and (B): \(f'(\frac{1}{2})=0\) for every \(f\isin S\).