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Question: If f(x) is continuous and differentiable function and f(1/n) = 0 for n ≥ 1 and n ∈ I, then:...

If f(x) is continuous and differentiable function and f(1/n) = 0 for n ≥ 1 and n ∈ I, then:

A

f(x) = 0, x ∈ (0, 1]

B

f(0) = 0, f’(0) = 0

C

f(0) = 0 = f’(0), x ∈ (0, 1]

D

f(0) = 0 and f’(0) need not to be zero

Answer

f(0) = 0, f’(0) = 0

Explanation

Solution

Given that f(x) is a continuous and differentiable function and f(1x)=0,f\left( \frac{1}{x} \right) = 0, x = n, n ∈ I

∴ f(0)+=f(1∞)=0f(0)^{+} = f\left( \frac{1}{\infty} \right) = 0

Since R.H.L. = 0, ∴ f(0) = 0 for f(x) to be continuous.

Also f’(0) = lim⁡h→0f(h)−f(0)h−0=lim⁡h→0f(h)h=0\lim_{h \rightarrow 0}\frac{f(h) - f(0)}{h - 0} = \lim_{h \rightarrow 0}\frac{f(h)}{h} = 0

[Using f(0) = 0] = 0 [∴ f(0+) = 0]

Hence f(0) = 0, f’(0) = 0.