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Let alpha,beta,gamma and delta are four ...

Let `alpha,beta,gamma and delta` are four positive real numbers such that their product is unity, then the least value of `(1+alpha)(1+beta)(1+gamma)(1+delta)` is :

A

6

B

16

C

0

D

32

Text Solution

Verified by Experts

The correct Answer is:
b
Since , `alpha betagammadelta=1" "`…(i)
As, we know `AM geGM`
` rArr (1+alpha)/2 ge sqrt(alpha) rArr 1+ alpha ge 2 sqrt(2) " "` ..(ii)
Similarly , ` 1 + beta ge 2sqrt(beta)" "` …(iii)
` 1 + gamma ge 2 sqrt(gamma)" "` ...(iv)
and `1+delta ge 2 sqrt(delta)" " ` ...(v)
On multiplying Eqs. (ii),(iii) ,(iv) and (v) ,we get
` (1+alpha )(1+beta)(1+gamma)(1+delta)ge 16 sqrt(alpha beta gamma delta)`
` :. " Least value of" (1+alpha) (1+beta)(1+gamma)(1+delta)=16`
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