Why must f'(0)=0 if f(0)=0 and f(1/n)=0 for all n?

[utkarshakash]

Homework Statement

If f(x) is a continuous and differentiable function and f(1/n)=0, for all nεN, then
a)f(x)=0 for all x ε (0,1]
b) f(0)=f'(0)=0
c) f'(0)=f"(0)
d)f(0)=0 and f'(0) need not be zero.

Homework Equations

The Attempt at a Solution

I would say d but the correct answer is b. Why should I believe that f'(0)=0 if f(0)=0?

 

[Office_Shredder]

If I instead told you that f(1/n) = 1/n, then f(0) = 0 still but the actual values of the sequence probably suggest the function looks a bit different near zero (can you guess what f'(0) will be?). You need to use the actual values of the sequence to show that the derivative is zero.

 

[UltrafastPED]

Clearly f(0) = 0 because the limit as n->infinity is zero ...

Also it is zero at an unlimited number of points near zero; for every finite interval specified, no matter how small, there are an infinite number of zeros in that interval.

But it was given that the function was differentiable as well as continuous; so form the derivative as
f'(0) = lim n-> inf [f(0+1/n) - f(0)]/[1/n] ... but the numerator is zero, and so is f'(0).

Hence (b) is correct.

 

[HallsofIvy]

Another way of looking at it: 1/n is a sequence of points converging to 0. Since f is continuous, f(0)= lim f(1/n)= 0.

As UltrafastPED said, [f(0+ 1/n)- f(0)]/(1/n)= 0. Either the limit, as n goes to 0, does not exist or it is 0. Since we are told that the function is differentiable, the limit exists, so is 0, so f'(0)= 0.