Let f(x) = \(\frac{log\,x}{x}\)
Clearly,
f(x) only defined for x > 0.
f'(x) = \(\frac{(1-log\,x)}{x^2}\) and
f"(x) = \(\frac{(2xlog\,x-3x)}{x^4}\) . (f"(x) < 0 for all x)
So,
f"(x) = 0
⇒ x= e and f''(e) < 0
So,
x = e is a point of maxima.
Therefore,
f(e) = \(\frac{1}{e}\)