Option : (B)
f(x) = \(x^{\frac{1}{x}}\)
Let y = \(x^{\frac{1}{x}}\)
Therefore,
logy = \(log_ex\)
Differentiating w.r.t x,
So,
Now,
lets put y’ = 0
So,
\(x^{\frac{1-2x}{x}}\) = 0
Or,
(1 - log x) = 0
Therefore,
x = 1/e and x = 0
Hence by second derivative test
f’’(1/e) < 0 so it’s a point of maximum and maximum value is \((\frac{1}{e})^e\).