Let \(f:[2,4] \rightarrow \mathbb{R}\) be a differentiable function such that \(\left(x \log _e x\right)\) \(f^{\prime}(…

Let \(f:[2,4] \rightarrow \mathbb{R}\) be a differentiable function such that \(\left(x \log _e x\right)\) \(f^{\prime}(x)+\log _e x f(x)+f(x) \geq 1, x \in[2,4]\) 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

[JEE Main 2023, 11 Apr (Shift 1)]