a) Prove that there exists a differentiable function f:(0, ∞)->(0, ∞) such that f(f'(x))=x, for all x>0.
b) Prove that there is no differentiable function f:R->R such that f(f'(x))=x, for all x∈R.
The function must have an inverse equal to its derivative.
Steve Herman found the the power curve for part a. b=the golden ratio. Since b^2=b+1 the value of a can be simplified slightly to b^(-1/b).
For the function to map the rest of the reals, we'd need the function to have a negative derivative for negative values of x.
You could probably find the similar function that works for all negative numbers. If you piece the two together you get a complete function except that it isn't differentiable at zero.
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Posted by Jer
on 2021-02-19 12:59:44 |
Thought for Part b.
