Question

Having this function,`f:(0,oo)->RR;f(x)=xlogp-plogx;p inRR,p>0` how to demonstrate that `e^x>=x^e forallx in(0,oo)`?

Answer 1

From

https://socratic.org/questions/having-function-f-0-oo-rr-f-x-xlnp-plnx-pinrr-p-0-how-to-find-p-0-sush-that-f-x-?answerEditSuccess=1

we know that

`xlogp-plogx ge 0-> x^(1/x) le e^(1/e)` and also

`x^(1/x) le e^(1/e)->x^e le e^x`