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 $$k$$?

×