Let f:[2,4] → R be a differentiable function such that (x log_e x) f^′(x)+( log_e x) f(x)+f(x) ≥ 1, x ∈[2,4] with f(2)=1/2 and f(4)=1/4 Consider the…

Let $\displaystyle f:[2,4] \rightarrow \mathbb{R}$ be a differentiable function such that

$$\begin{aligned} & \left(x \log _e x\right) f^{\prime}(x)+\left(\log _e x\right) f(x)+f(x) \geq $\displaystyle 1$, x \in[2,4] \text { with } f($\displaystyle 2$)=\frac{1}{2} \text { and } & f($\displaystyle 4$)=\frac{1}{4} \end{aligned}

$$Consider the following two statements :

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

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

Then,