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  • Solve! - Let x,y be positive real numbers and m,n positive integers. The maximum value of the expression is - Sequence and series - JEE Main-11467

Let x,y be positive real numbers and m,n positive integers. The maximum value of the expression \frac{x^{m}y^{n}}{(1+x^{2m})(1+y^{2n})}  is :

  • Option 1)

     

    1

  • Option 2)

    \frac{1}{2}

  • Option 3)

     

    \frac{1}{4}

  • Option 4)

     

    \frac{m+n}{6mn}

Answers (1)

best_answer

 

Relation between AM, GM and HM of two positive numbers -

AM\geqslant GM\geqslant HM

- wherein

Inequality of the three given means.

 

 1

\frac{x^{m}y^{n}}{(1+x^{2m})(1+y^{2n})}=\frac{1}{(x^{m}+\frac{1}{x^{m}})(y^{n}+\frac{1}{y^{n}})}\leq \frac{1}{4}


Option 1)

 

1

Option 2)

\frac{1}{2}

Option 3)

 

\frac{1}{4}

Option 4)

 

\frac{m+n}{6mn}

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