26 Jan '04 22:06>
Suppose f: R -> R is a smooth (=infinitely differentiable) function such that f(x) = 0 for all x < 0. Does it follow that f(x) = 0 for all x?
Originally posted by Acolytei don't think it follows thus.
Suppose f: R -> R is a smooth (=infinitely differentiable) function such that f(x) = 0 for all x < 0. Does it follow that f(x) = 0 for all x?
Originally posted by BarefootChessPlayerWell, there are either an uncountable number of suitable functions, or there are none. The function you've given (x^n for x non-negative) would be an example, had I said n-1 times differentiable. But its (n-1)th derivative is n!x for x > 0, and 0 for x < 0, so it's not n times differentiable at x = 0. I haven't thought of the answer to this myself yet, but it would have to be a function whose every derivative is 0 at x = 0.
i don't think it follows thus.
how about f(x) = x^n for n in [b]Z+? or is that not smooth by your definition?
i know that f(x) couldn't be soemthing like sin x since f'(x) is not differentiable at x=0, nor exp x for the same reason.
here is how i think it works for f(x) = x^n:
start with n=1. then f'(x) = 0 for x<0 and 1 elsewi ...[text shortened]...
it thus appears that there are infinitely many functions that meet the criterion.
am i right?[/b]
Originally posted by Acolytesomething liek that, i'd say.
Well, there are either an uncountable number of suitable functions, or there are none. The function you've given (x^n for x non-negative) would be an example, had I said n-1 times differentiable. But its (n-1)th derivative is n!x for x > ...[text shortened]... would have to be a function whose every derivative is 0 at x = 0.
Originally posted by FiathahelActually, BCP is right. The function he gave, exp(-1/x^2), is an example of a smooth function for which a Taylor series isn't very useful, since the error term in its Taylor series about zero is equal to the function itself.
It cant be polynomial. cause the value of the nth derivitive in 0 equals (the coefficient of x^n)*n! . If that equals 0 then all coefficients equal 0.
Secondly it cant be any other function. Cause every everywhere-infinite-differentiable function can be writen as a infinite-Taylor polynomial.