Mathematics Question from JEE Main 2023 · 11 April, Shift 1

Let $f: [2, 4] \to \mathbb{R}$ be a differentiable function such that $(x \log_e x) f'(x) + (\log_e x) f(x) + f(x) \geq 1$ for all $x \in [2, 4]$, with $f(2) = \frac{1}{2}$ and $f(4) = \frac{1}{4}$. Consider the following two statements:

$(A)$ $f(x) \leq 1$ for all $x \in [2, 4]$

$(B)$ $f(x) \geq \frac{1}{8}$ for all $x \in [2, 4]$

Then,