mathematics and reasoning questions

[sunny2009]

how to solve this question............need help plzzz.....
         CONSIDER the quantities e^pi and pi^e{that is...e to d power pi and pi to d power e}.
can we rank these two quantities?
if yes,then which one of them is greater??/

[ritu]

hey @sunny 2009...ur questions are intrstng ...4m where have you got them??????if u could tell me.....:))))

[ritu]

hey @sunny 2009...ur questions are intrstng ...4m where have you got them??????if u could tell me.....:))))

[Amit Goyal]

Hi Sunny, Refer the Notes (Page 4) for the answer.

[sunny2009]

@amit sir.....
............thank u so much sir.now i understood d concept behind it.:):)

[sunny2009]

hi ritu...:)
............these questions r from last year D SCHOOL entrance exams..most of them were asked before 2004!!!

[sunny2009]

how to show that.......
       log(x)< x for all x>0
where log(x) stands for natural logarithm.
   

[Amit Goyal]

for 0 < x < 1, log x < 0 < x
for x = 1, log x = 0 < 1
for x > 1, f(x) = x  - log x is an increasing function (because f'(x) = 1 - 1/x > 0) and we know that f(x) is continuous everywhere on (0, ∞). Thus, f(1) > 0  and f'(x) > 0 implies that f(x) > 0 for x > 1.