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Prove that [(a^(2) + b^(2))/(a + b)]^(...

Prove that
`[(a^(2) + b^(2))/(a + b)]^(a + b) gt a^(a) b^(b) gt {(a + b)/(2)}^(a + b)`, where a,b `gt` 0

Text Solution

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`([a+a+...a "times"]+[b+b+....b" times"])/(a+b)ge [a^ab^b]^((1)/(a+b))`
`ge (a+b)/(((1)/(a)+(1)/(a)+....a " times")+((1)/(b)+(1)/(b)+.....b" times"))`
or ` (a^2+b^2)/(a+b)ge[a^ab^b]^((1)/(a+b))ge (a+b)/(1+1)`
or ` ((a^2+b^2)/(a+b))^(a+b) ge a^a b^b ge ((a+b)/(2))^(a+b)`
or ` [(a^2+b^2)/(a+b)]^(a+b) gt a^a b^b gt {(a+b)/(2)}^(a+b)`
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Knowledge Check

  • What is agr (bi) where b gt0 ?

    A
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    B
    `(pi)/(2)`
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    D
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    A
    A.P.
    B
    G.P.
    C
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    D
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  • Square root of 2a - sqrt(3a^2 - 2ab - b^2) , (a gt b gt 0) is

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    B
    `1/(sqrt(2)) (sqrt(3a - b) + sqrt(a + b))`
    C
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    D
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