Thread: Prove pi^e < e^pi

Consider the function f(x)= ln(x)/x which is defined for all x>0.

Show that this functions is strictly increasing on the interval (0,e) [ This part is done just find f'(x) and state its increasing for all x between 0 and e]

strictly decreasing on the interval (e,infinity) [ same as the above part..]

and thus has a global maximum at x=e ... [ DONE, proving its station point is at that point].

HENCE SHOW THAT F(X) <= 1/e FOR ALL x>0.

USE THIS RESULT WITH x=pi TO SHOW THAT pi^e < e^pi.....

just that last part i dont even know where to start.....

Originally Posted by Khonics89

Consider the function f(x)= ln(x)/x which is defined for all x>0.

Show that this functions is strictly increasing on the interval (0,e) [ This part is done just find f'(x) and state its increasing for all x between 0 and e]

strictly decreasing on the interval (e,infinity) [ same as the above part..]

and thus has a global maximum at x=e ... [ DONE, proving its station point is at that point].

HENCE SHOW THAT F(X) <= 1/e FOR ALL x>0.

USE THIS RESULT WITH x=pi TO SHOW THAT pi^e < e^pi.....

just that last part i dont even know where to start.....

Hmmm ^^^^^^^^

where do I go ??

Originally Posted by Khonics89

Consider the function f(x)= ln(x)/x which is defined for all x>0.

Show that this functions is strictly increasing on the interval (0,e) [ This part is done just find f'(x) and state its increasing for all x between 0 and e]

strictly decreasing on the interval (e,infinity) [ same as the above part..]

and thus has a global maximum at x=e ... [ DONE, proving its station point is at that point].

HENCE SHOW THAT F(X) <= 1/e FOR ALL x>0.

USE THIS RESULT WITH x=pi TO SHOW THAT pi^e < e^pi.....

just that last part i dont even know where to start.....