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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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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-INEQUALITIES INVOLVING MEANS -Examples
  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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  3. Prove that 2/(b+c)+2/(c+a)+2/(a+b)<<1/a+1/b+1/c ,w h e r ea ,b ,c>>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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  5. Find the minimum value of 4^sin^(2x)+4^cos^(2x) .

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  6. If (log)2(a+b)+(log)2(c+d)geq4. Then find the minimum value of the exp...

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

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  8. Find all positive real solutions to 4x+(18)/(y)=14,2y+(9)/(z)=15,9z+(1...

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  9. Let A;G;H be the arithmetic; geometric and harmonic means between thre...

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  10. .Let a1, a2,............ be positive real numbers in geometric progr...

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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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  12. If y z+z x+x y=12 ,w h e r ex ,y ,z are positive values, find the grea...

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