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

Answer: (A), (B)

f(0) = 0 for every f ∈ S.

Answer 2

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