F(x) = lnx/x

(a): find f'(x) and show that x = e is a maximum
(b): deduce that e^x > = x^e, for x > 0

(a)
Using the quotient rule:



For maximum, f'(x) = 0;



(b) Deduce:


Soln: Since x = e is the greatest value, then f(e) ≥ f(x) > f(0)

, since ln(e) is simply equal to 1

Now, since x > 0, then we don't have to worry about flipping the signs when multiplying by x.




Taking the exponential to both sides will cancel with the natural logarithmic function in the right hand side to produce:

, as required.