Application Of Derivatives

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