IIT JEE , AIEEE Forum - Mathematics , Algebra - To prove root 3 + root 2 > pi ( without using directly the values)

ramyani

^{One of my classmates gave me the following problem. Can it be solved}without putting directly the values of root 3 and root 2 ?
^{Q.Prove That ( without putting directly the values of root 3 and root 2)

[2 ]}

3 + ^{[2 ]}

2 > Pi

dream_iit

(3)^1/2 + (2)^1/2 > 3.14

LHS
sq. both sides

9 + 4 + 2(6)1/2 is always greater than 9.85

is that rite????

debabratanag99

By squaring left hand side where do you get 9 and 4 from........?????????

ramyani

no no.

SMARTY

in other words you need to prove
^{[ ]}

3+^{[ ]}

2 - 3.14>0
squaring both sides
you get
something like,
8.14+1.14^{[ ]}

6 which is greater(>) than zero
rate me for my efforts...

Mr.IITIAN007

Hey Ramyani ....you know that ... 2 log_e (

- 1) > log_e 3 - log_e2
or , log_e (

- 1) > (1/2) [ log_e3 - log_e2 ]
or, log_e (

- 1) > log_e

3 - log_e

2
or, log_e (

- 1) > log_e (

3 /

2 )
or,

- 1 >

3 /

2
or,

> (

3 /

2) + 1
or,

<

3 +

2
Hence Proved.

pkg1960

smarty, just a small flaw in your method. u cannot square both sides of an inequality like u do for an equality.

u proved the question in the following manner

sqrt(2) + sqrt(3) - pi > 0 , then squared both sides, rite?

that just proves [sqrt(2) + sqrt(3) - pi] ^ 2 > 0

but does not prove

sqrt(2) + sqrt(3) - pi > 0

this is just like doing something like the following

prove -1 > 0

simple

-1 > 0

sqaring both side

(-1)^2 > 0^2

1 > 0

hence proved!!!!!

thats not how it works, you know!!!

pkg1960

mr iitan2007,
can u pls explain this particualr step to me
log_e (2

- 2) > log_e 3 - log_e2
log_e (

- 1) > (1/2) [ log_e3 - log_e2 ]
from what i know abt logarithms
log_e (2

- 2) cannot be written as 2log_e (

- 1)
and this inequality that u have used, wud u pls explain that too, i havent come across such an inequality, can wait for its proof!!!!

saarika

@ mr.iitian

u said that log (2pi -2) > log3 -log2

then log 2(pi -1) >log3 - log2

=> log2 +log(pi-1) >log3 -log2 ......

....but how did u get log (pi -1)>1/2 (log3 -log2)

nd in the last step pi> (rt3/rt2) + 1

that doesnot imply that pi < rt3 + rt2

for eg ---- 10 > 5/2 + 1

but doesnot imply that 10< 5+2

pls correct me if i am wrong

pkg1960

ya saarika absolutely right, something terribly wrong with mr IITian 2007...!!!

Mr.IITIAN007

Sorry guys actually I was out of my mind at that time. In fact I typed something else . The current had gone and I was very very irritated .
In fact the first statement is wrong .I am sorry. Well I am trying for the correct answer.

Vincent

Ya Ya..something was really distubing with you mr iitian....just like
the Chairman of the board Vincent Kennedy Mcmahon.You only told me to intake good qualities of Mr.Mcmahon.....and now you are intaking the bads of our dear chairman.

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