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Prove that (a b+x y)(a x+b y)>4a b x y(a...

Prove that `(a b+x y)(a x+b y)>4a b x y(a , b ,x ,y >0)dot`

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Verified by Experts

Using A.M `ge` G.M we have
`(ab + xy)/(2) gt sqrt(abxy)`
`implies ax + xy gt 2 sqrt(abxy)`
similarly, ax + by `gt 2 sqrt(abxy)`
Multiplying (1) and (2) we get
`(ab + xy) (ax + by) gt 4abxy`
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CENGAGE PUBLICATION-INEQUALITIES INVOLVING MEANS -Illustration
  1. Prove that (a b+x y)(a x+b y)>4a b x y(a , b ,x ,y >0)dot

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  2. Prove that b^2c^2+c^2a^2+a^2b^2gtabc(a+b+c), where a,b,c gt 0 .

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  4. If a ,b , a n dc are distinct positive real numbers such that a+b+c=1,...

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  7. Find all real solutions to 2^x+x^2=2-(1)/(2^x).

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  11. If a+b+c=1, then prove that 8/(27a b c)>{1/a-1}{1/b-1}{1/c-1}> 8.

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