Let y = x^(1/x)

Take log on both side, we get

log y = (1/x) * log x

Now, differentiate w.r.t. x, we get

      (1/y) * dy/dx = -(1/x² ) * log x + 1/x²

=> dy/dx = y{-(1/x² ) * log x + 1/x² }

Now put dy/dx = 0 for extreme points, we get

       y{-(1/x² ) * log x + 1/x² } = 0

=>  x^(1/x) * {-(1/x² ) * log x + 1/x² } = 0

=> x^(1/x) * {-(1/x² ) * log x + 1/x² } = 0

=> x^(1/x) * (1/x² ) * (1 - log x) = 0

=> x^{(1/x - 2)} * (1 - log x) = 0

=> 1 - log x = 0

=> log x = 1

=> log x = log_e e

=> x = e

Hence, the maximum point is x = e

So, the maximum value = e^(1/e)