If f(x) is continuous and differentiable function `f((1)/(n))=0 AA n le 1 and n in Z.` then prove that `f(0) =0 and f'(0)=0`

Options

A. `f(x)=0"for all "x in N uu (0,1]`
B. `f(0)=0,"f'(0)=0`
C. `f'(0)=0,f''(0)=0`
D. f(0) and f'(0) may or may not be zero

Answer

B

Explanation

We have, `f((1)/(n))=0"for all "n in N`
`Rightarrow f(1)=f((1)/(2))=f((1)/(3))=...=underset(n to oo)lim f((1)/(n))=0`
`Rightarrow f(1)=f((1)/(2))=f((1)/(3))=...f(0)=0`
Now,
`f'(0)=underset(h to 0)lim (f(h)-f(0))/(h-0)`
`Rightarrow f'(0)=underset(h to 0)lim (f((1)/n))/((1)/(n))=underset(n to oo)lim (0)/((1)/(n))" "[therefore f((1)/(n))=0]`
`Rightarrow f'(0)=underset( n to oo)lim 0=0`