Let y = xee-x, where x ≥ 0. Find the stationary points by finding the derivative and setting it equal to zero.
dy/dx = (exe-1 - xe)e-x
The derivative is zero at x = 0 and x = e, and nowhere else. The value of y at x = 0 is y = 0, which is clearly a minimum; the value at x = e is y = 1. The latter is in fact a maximum and can be shown to be so by finding the second derivative and showing that its value is negative at x = e (left to the reader).
Therefore, xee-x ≤ 1 for all x ≥ 0, with equality only at the unique maximum x = e. Multiply both sides by ex to get
xe ≤ ex
for all x ≥ 0, with equality only when x = e. QED.
Note: The reader can plot the graph of the function y = xee-x with this handy Java applet: Graph plotter and calculator and/or find the derivatives at the Wolfram/Alpha site: Derivatives.