# Something off 'The margin'!

$\large f(x) = \frac{a}{1+ax} + \frac{b}{1+bx} - \frac{c}{1+cx}$

Let $$f(x)$$ be a function as defined above where $$a, \ b$$ and $$c$$ are arbitrary non-negative integral constants and $$abc \neq 0$$. Which of the following results are possible?

I. $$f(0)$$ can be zero.

II. $$f^{(1)} (0)$$ can be zero.

III. $$f^{(2)} (0)$$ can be zero.

IV. $$f^{(3)} (0)$$ can be zero.

Notations: $$f^{(n)}(a)$$ denotes the $$n^{th}$$ derivative of $$f(x)$$ at $$x=a$$.

What conclusion do you deduce for $$f^{(n)}(0)$$ for non-negative integral values of $$n$$ (note that $$f^{(0)}(0)$$ is the same as $$f(0)$$ )? Does this problem have trivial solutions? If yes, then what are they?