analysis question: Let f : R → (0,∞) have the property...?
Let f : R → (0,∞) have the property that f′(x) = f(x) for all x. Show
that f is an increasing function for all x. Show also that (f−1)′(x) = 1
x .
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Clearly, f'=f>0, so f is increasing. Also, (f-1)'(x)=1/f'(f-1(x))=1/f(f-1(x))=1/x.