Derivative =0

x^2 = 0, x = 0

or

2xlnxe - lnx = 0

lnx(2xe-1) = 0

lnx = 0 or 2ex - 1 - 0

x = 1 or x = 1/(2e)

So, horizontal tangents occur at

x = 1

x = 1/(2e)

x = 0

Double check that the second derivative is never equal to 0 for each of those points... when it does, it isn't an extrema