• May 14th 2009, 08:10 AM
Media_Man
For some continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$, if I know the value of $\frac{f'(x)}{f(x)}$ at some point $x=a$, does that tell me any information about the value of either $f'(a)$ or $f(a)$?
I'm doubting it, but I want to make sure. Say for example I knew that this ratio was irrational. Does that tell me anything about the rationality of $f'(a)$ or $f(a)$?
For some continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$, if I know the value of $\frac{f'(x)}{f(x)}$ at some point $x=a$, does that tell me any information about the value of either $f'(a)$ or $f(a)$?
If the ratio exists, you know $f(a) \not= 0$.
If the ratio were irrational, then at least one of $f(a)\text{ or }f'(a)$ must be irrational.