11Ans ) (a), (c), (d) ???????????
Q. Which of the following statements is true?
(a) e3 > 3e
(b) e2 > 2e
(c) epi > pie
(d) √epi > pie/2
not my doubt :)
-
UP 0 DOWN 0 0 8
8 Answers
1why anger on b) ? any reason !!!!
If u r trying to solve it by taking exact values , it might lead to wrong conclusions.
21assume f(x)=x1/x
f'(x)=x1/x(1-lnxx2)
f'(x)>0 for x belonging to (0,e)
f'(x)<0 for x belonging to (e,∞)
hence f(x) is monotonically increasing for x (0,e)
and monotonically decreasing for x(e,∞)
therfore
1)
3>e
e1/e>31/3
raising to power 3e
e3 >3e
2)
e>2
e1/e>21/2
raising to power 2e
e2>2e
3)
pi >e
e1/e>pie1/pi
raising to power pi e
epi>pie
4)
epi>pie
raising to power 1/2
epi/2>pie/2
11Ans) Consider f(x) = e x
(a) Taking log both sides, we get
3 log e > e log 3
log ee > log 33 ,
Let g(x) = log x / x
g'(x) = (1 - log x)x 2 = 0 at x =e
Now for x > e,
log x > 1 .................................. g(x) is dec
And for x < e,
log x < 1 ................................... g(x) is inc
Since 3 > e, therefore,
g(x) < g(e) .............................bcoz g(x) is dec
Therefore, g(3) < g(e)
log 33 < log ee
Therefore, (a) is CORRECT
(B) Since e > 2
Therefore, g(e) < g(x) ..........................bcoz g(x) is inc
g(e) < g(2)
Therefore, logee < log 22
Therefore, (b) is INCORRECT
(c) Since pi > e,
therefore, g(x) < g(e) ........................bcoz g(x) is dec
Therefore, g(pi) < g(e)
Therefore, log pipi < log ee
Therefore, (c) is CORRECT
(d) Frm (c), e pi > pi e
e pi / 2 > pi e /2
Therefore, (d) is CORRECT
106@karna. no ur graph is not correct.
ex > xe for all x >0 equality only at x=e
Alternative approach:
Consider f(x) = ex/xe
f(0+) = 1
further f'(x) < 0 till x=e and then f'(x) >0 when x>e
so, x=e is local min. and since both functions are increasing in nature,
f(e) < f(x)
=> ex > xe