Something off 'The margin'!

Number Theory Level 3

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

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 \(x=a\).

Follow up problem:

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?


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