# Find f(0) and f'(0)

Gold Member

## 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.

## 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?

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Office_Shredder
Staff Emeritus
Gold Member
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
Gold Member
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.

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HallsofIvy