plz Help!!!!!!!?

Question

if b > or equal to 1 , Show that e^b > or equal to b^e I ried to differentiate but it didn't work

Answer 1

Let´s see
f(x) = e^x/x^e
f(1)=e>1
lim f(x) x=>+ infinity = +infinity (If needed apply L´Hôpital three times)
f´(x) = 1/x^2e *(x^e*e^x*-e*x^e-1 *e^x)=
1/x^2e *e^x *x^e-1(x*e^x-e)The expression x*e^x-e is 0 at x=1
and than positive ( the derivative is e^x(x+1))
so f´(x) >=0 and f(x) is increasing
from f(1)>1
so the numerator is always larger than the denominator