If the function $$f(x) = \left(\frac{1}{x}\right)^{2x}; x \gt 0$$ attains the maximum value at $$x = \frac{1}{e}$$ then :

A

$$e^{\pi} \lt \pi^e$$

B

$$e^{\pi} \gt \pi^e$$

C

$$(2e)^{\pi} \gt \pi^{(2e)}$$

D

$$e^{2\pi} \lt (2\pi)^e$$