For the function $f(x) = x^{\frac{1}{x}},x>0$, which of the following are correct?

(A) $x = 0$ is the only point where extremum may occur.
(B) The given function is maximum at $x = e$.
(C) The function has no extreme value for $x > 0$.
(D) The maximum value of the function f(x) is $e^{\frac{1}{e}}$.

Choose the correct answer from the options given below:

(B) and (D) only

The correct answer is Option (4) → (B) and (D) only

Given function: $f(x) = x^{1/x}, \; x>0$

Take natural log: $y = x^{1/x} \Rightarrow \ln y = \frac{1}{x} \ln x$

Differentiate w.r.t x:

$\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx} (\frac{\ln x}{x}) = \frac{1 - \ln x}{x^2}$

Set derivative = 0 for extrema:

$\frac{1 - \ln x}{x^2} = 0 \Rightarrow 1 - \ln x = 0 \Rightarrow \ln x = 1 \Rightarrow x = e$

Check maximum: $f(x)$ has a maximum at $x=e$

Maximum value: $f(e) = e^{1/e}$

Correct statements: (B) and (D)