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,