Let f(x)=(x)/( x) ;(x > 1) Consider the following statements: 1. f^...

$$\begin{aligned} f^{\prime}(x) & =\frac{(\ln x)^2 \times \frac{1}{x}-(\ln x-1) \times 2 \frac{\ln x}{x}}{(\ln x)^4} \\ & =\frac{(\ln x)^2-2(\ln x)^2 \times 2 \ln x}{x(\ln x)^4}\end{aligned}$$\Rightarrow \quad f^{\prime \prime}(e)=\frac{1-2+2}{e \times 1}=\frac{1}{e}( > 0)\Rightarrow$Statement (1) is correct. Since$f^{\prime \prime}(e) > 0$(true)$f(x)$attains local minima at$x=e\Rightarrow$Statement (2) is correct. A local minimum value occurs at$x=ef(x)$at$x=e$, is$f(e)=\frac{e}{\ln e}=e\Rightarrow$ Statement (3) is correct. Hence, Statements (1), (2) and (3) are correct.