$\eqalign{
& M = {{{a^4}} \over {{b^4}}} + {{{b^4}} \over {{a^4}}} + {a \over b} + {b \over a} - ({{{a^2}} \over {{b^2}}} + {{{b^2}} \over {{a^2}}}) \cr
& de\;thay\;M\;khong\;co\;max\;\left( {cho\;a = 1\;b \to \infty \;thi\;M \to \infty } \right) \cr
& ap\;dung\;cosi\;co\;cac\;bat\;dang\;thuc\;sau: \cr
& {{{a^4}} \over {{b^4}}} + 1 + {{{b^4}} \over {{a^4}}} + 1 \ge 2\left( {{{{a^2}} \over {{b^2}}} + {{{b^2}} \over {{a^2}}}} \right) \leftrightarrow {{{a^4}} \over {{b^4}}} + {{{b^4}} \over {{a^4}}} \ge 2\left( {{{{a^2}} \over {{b^2}}} + {{{b^2}} \over {{a^2}}}} \right) - 2 \cr
& {{{a^2}} \over {{b^2}}} + 1 + {{{b^2}} \over {{a^2}}} + 1 \ge 2\left( {\left| {{a \over b}} \right| + \left| {{b \over a}} \right|} \right) \ge - 2\left( {{a \over b} + {b \over a}} \right) \leftrightarrow {{{a^2}} \over {{b^2}}} + {{{b^2}} \over {{a^2}}} \ge - 2\left( {{a \over b} + {b \over a}} \right) - 2 \cr
& {{{a^2}} \over {{b^2}}} + {{{a^2}} \over {{b^2}}} \ge 2 \cr
& \to M \ge 2\left( {{{{a^2}} \over {{b^2}}} + {{{b^2}} \over {{a^2}}}} \right) - 2 + {a \over b} + {b \over a} - \left( {{{{a^2}} \over {{b^2}}} + {{{b^2}} \over {{a^2}}}} \right) = {1 \over 2}\left( {{{{a^2}} \over {{b^2}}} + {{{b^2}} \over {{a^2}}}} \right) + {1 \over 2}\left( {{{{a^2}} \over {{b^2}}} + {{{b^2}} \over {{a^2}}}} \right) + {a \over b} + {b \over a} - 2 \cr
& \ge {1 \over 2}*2 + {1 \over 2}\left( { - 2\left( {{a \over b} + {b \over a}} \right) - 2} \right) + {a \over b} + {b \over a} - 2 = - 2 \cr
& dau = \leftrightarrow a = - b \cr
& \cr} $
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