calculate the value the maximum value of 1 by x to the power x is
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f(x) = (1/x)^x
F(x) = ln(f(x)) = x ln(1/x) = - xln(x)
F'(x) = - ln(x) - 1
F''(x) = - 1/x
For extreme values of F(x), F'(x) =0
=> ln() = - 1 => x = 1/e
F''(1/e) = - e <0
So f(x) is maximum at x = 1/e
Maximum value = e^(1/e)
F(x) = ln(f(x)) = x ln(1/x) = - xln(x)
F'(x) = - ln(x) - 1
F''(x) = - 1/x
For extreme values of F(x), F'(x) =0
=> ln() = - 1 => x = 1/e
F''(1/e) = - e <0
So f(x) is maximum at x = 1/e
Maximum value = e^(1/e)
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