On Wed, Jan 9, 2019 at 9:43 AM Andres Valloud <avalloud@smalltalk.comcastbiz.net> wrote:
Therefore, lim n->+oo df/dx^n f(t_n) > 0, absurd because by now all derivatives must be zero at x = 0.
If lim n-> inf (d^kf/dx^k) > 0 for some fixed k, that would show that the kth derivative of f at 0 was non-zero, a contradiction. But how do you get a contradiction from lim n-> inf (d^n f /dx^n) > 0? All the derivatives start at zero when x = 0, and grow as x increases, and the larger n is, the faster the nth derivative grows. But why is this a contradiction? If there was an "infiitieth" derivative, that's the limit of the nth derivative as n goes to infinity, that is also continuous, you've shown that the infinitieth derivative at 0 is non-zero, a contradiction since all the finite derivatives (and hence the infiniteth derivative) is 0 at 0. But I don't think the fact that a function is infinitely differentiable shows that it also has a continuous "infinitieth" derivative, so why is the thing you label as "absurd" impossible? Andy
Consequently, some derivative must be negative somewhere.
Andres.
On 12/31/18 18:45 , Andres Valloud wrote:
Hi, did you make progress on this? I ran into the same example you mentioned.
On 12/14/18 16:01 , Gareth McCaughan wrote:
On 14/12/2018 16:11, Victor Miller wrote:
Here's the essential idea: since f(0) = 0 and f(1)=1, if we define y = inf { x>= 0: f(x) > 0}, we have y in [0,1). Since f is infinitely differentiable, it can't hold that f(x) = 0 for all x <= y. So f must be decreasing somewhere before y, and thus its derivative is negative.
At the risk of once again showing myself to be a moron, that doesn't sound right. Suppose g(x) = 0 for x<=0 and g(x) = exp(-1/x^2) for x>0. Then g is infinitely differentiable but has the property you say f can't have on account of being infinitely differentiable.
(I am not claiming that g or anything like it is a counterexample to the theorem the question is asking for a proof of. It's easy to find places where, say, its second derivative is zero. But it's a counterexample to the I-think-not-a-theorem that says that an infinitely differentiable function can't be 0 for all x <= some x0 and strictly increasing thereafter.)
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