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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)`.

Text Solution

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Here, we are given `abcd = 1`.
Now, `(1+a)(1+b)(1+c)(1+d) = 1+a+b+c+d+(ab+bc+cd+ad+bd+ac)+(abc+bcd+cda+dab)+abcd`
`=1+a+b+c+d+(ab+bc+ac+1/(ab)+1/(bc)+1/(cd))+(1/a+1/b+1/c+1/d)+1`...(As abcd = 1)
`=1+1+(a+1/a)+(b+1/b)+(c+1/c)+(d+1/d)+(ab+1/(ab))+(bc+1/(bc))+(ac+1/(ac))`
Now, we know minimum value of `x+1/x` is `2`.
`:. (1+a)(1+b)(1+c)(1+d) ge 1+1+2+2+2+2+2+2+2`
`=> (1+a)(1+b)(1+c)(1+d) ge 16`
`:.` Minimum value of given expression is `16`.
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