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If a,b,c and d are four positive real nu...

If a,b,c and d are four positive real numbers such that abcd=1 , what is the minimum value of `(1+a)(1+b)(1+c)(1+d)`.

A

1

B

4

C

16

D

64

Text Solution

Verified by Experts

The correct Answer is:
C
`:.(1+a)(1+b)(1+c)(1+d)` ,brgt `=1+a+b+c+d+ab+ac+ad+bc+bd+cd+abc+abd+cda+cdb+abcd " " [" 16 terms "]`
`:.AMgeGM`
`((1+a)(1+b)(1+c)(1+d))/(16)ge (a^(8)b^(8)c^(8)d^(8))^((1)/(16))`
`=(abcd)^((1)/(2))=(1)^((1)/(2))=1 " " [:. Abcd=1]`
`((1+a)(1+b)(1+c)(1+d))/(16)ge1`
`(1+a)(1+b)(1+c)(1+d)ge16`
`:.` Minimum value of `(1+a)(1+b)(1+c)(1+d)` is 16.
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