What is the minimum value of the function y=e^ln(x^x) ?

We' ll differentiate the function with respect to
x

f'(x) = (e^lnx^x)(xlnx)'

We'll apply
product rule for (xlnx)':

(xlnx)' = x'*ln x +
x*(lnx)'

(xlnx)' = ln x + 1

f'(x) =
[e^(x*lnx)]*( lnx+1)

We'll put f'(x)=0 => [e^(x*lnx)]*(lnx+1)
= 0

Since the factor e^lnx is positive =>
lnx+1=0

lnx+1=0 => lnx=-1 =>
x=e^-1=1/e

The critical point is
x=1/e

To determine the minimum point, we'll substitute x by the
value of critical
point

f(1/e)=e^[ln(1/e)^1/e]

f(1/e)=e^(-1/e)

The
minimum point is represented by it's coordinates (1/e,
e^(-1/e)).