QJEEM22Ma51d8: JEE Main 2022 Mathematics - Class 11 JEEM22Ma51d8 Solution: Let and be three positive real numbers. Let and be such that…

$f\left( {g\left( {{1 \over n}\sum\limits_{i = 1}^n {f({a_i})} } \right)} \right)$

${{{a_1} + {a_2} + {a_3}\, + \,......\, + \,{a_n}} \over n} = 0$

$\therefore$ First and last term, second and second last and so on are equal in magnitude but opposite in sign.

$f(x) = \alpha {x^5} + \beta {x^3} + \gamma x$

$\sum\limits_{i = 1}^n {f({a_i}) = \alpha \left( {a_1^5 + a_2^5 + a_3^5\, + \,.....\, + \,a_n^5} \right) + \beta \left( {a_1^3 + a_2^3\, + \,....\, + \,a_n^3} \right) + \gamma \left( {{a_1} + {a_2}\, + \,....\, + \,{a_n}} \right)}$

$= 0\alpha + 0\beta + 0\gamma$

$= 0$

$\therefore$ $f\left( {g\left( {{1 \over n}\sum\limits_{i = 1}^n {f({a_i})} } \right)} \right) = {1 \over n}\sum\limits_{i = 1}^n {f({a_i}) = 0}$