A calculus problem by Parth Thakkar

Calculus Level 3

Let \( f(x) = (x-1)(x-2)\ldots(x-n) \) where \( n \in \mathbb{N} \) be a function from \( \mathbb{R} \) to \( \mathbb{R} \).

Let \( \int \dfrac{ f(x)f''(x) - (f'(x))^2 } { (f(x))^2 } dx = u(x) + C \) where \( C \) is some arbitrary constant.

If the number of (real) solutions to the equation \( u(x) = 5 \) is \( \alpha_n \) where \( \alpha_n \in \mathbb{N} \), then the minimum possible value of \( n \) (that is, for the equation to have \(\alpha_n\) roots) can be written as \( k \alpha_n \), where \( k \in \mathbb{N} \) and is independent of \( n \) or \(\alpha_n\). What is the value of \( k \)?

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