For the function $y=\ln(x)/x$:
Show that maximum value of $y$ occurs when $x = e$.
Using this information, show that $x^e <e^x$ for all positive values of $x$.
Two positive integers, $a$ and $b$, where $a < b$, satisfy the equation $a^b = b^a$. Find $a$ and $b$ , and show that these are unique solutions.
For the first probelm, I was thinking of getting the derivative of the function (which is $(\ln(x)+1)/x^2)$ and using sign charts in order to show the max.
But I'm not sure on how I would use that for the second problem nor the third problem.