Question

If f (x) is continuous and differentiable function and f (1/n) = 0 $\forall$ n $\ge$ 1and n $\in$ I, then

• A. f(x) = 0, x $\in$ (0, 1]
• B. f(0) = 0, f '(0) = 0
• C. f(0) = 0 = f '(0), x $\in$ (0, 1]
• D. f(0) = 0 and f '(0) need not to be zero

Answer 1

Answer: (B)

f(0) = 0, f '(0) = 0

Answer 2

Given that f (x) is a continuous and differentiable function and $f \left( \frac{1}{x} \right) = 0 , x = n \in I$
$\therefore \, f(0^{+} ) = f \left( \frac{ 1}{\infty}\right) = 0$
Since R.H.L. = 0,
$\therefore \, f(0) = 0$ for $f(x)$ to be continuous.
Also $f'(0) = \displaystyle \lim_{h \to 0} \frac{f(h) - f(0)}{h - 0} = \displaystyle \lim_{h \to 0} \frac{f(h)}{h} = 0$
$= 0$ [Using f (0) = 0 and $f (0^+) = 0$]
Hence $f (0) = 0, f ' (0) = 0$