Daily questions _191118

Known \ (A, B \ in \ R & lt mathbb {} \) , \ (A + B =. 4 \) , then \ (\ dfrac {1} { a ^ 2 + 1} + \ dfrac {1} {b ^ 2 + 1} \) the maximum value of \ (\ underline {\ qquad \ qquad} \) .
Analysis:
note expression is required to be \ (M \) , set \ [t = 1-ab = 1-a . (4-a), a \ in \ mathbb {R} \] is \ (T \) ranges \ ([- 3, + \ infty) \) is. \ [\ the begin {Split} M & = \ dfrac {b ^ 2 + 1 + a ^ 2 + 1} {(a ^ 2 + 1) (b ^ 2 + 1)} \\ & = \ dfrac {(a + b) ^ 2 + 2 (1 -ab)} {(a + b ) ^ 2 + (ab-1) ^ 2} \\ & = \ dfrac {16 + 2t} {16 + t ^ 2} \\ & = 2 \ cdot \ dfrac {8 + t} {16+ \ left [ (8 + t) -8 \ right] ^ 2} \\ & = \ dfrac {2} {(8 + t) + \ dfrac {80} {8 + t} -16 } \\ & \ leqslant \ dfrac { 2} {2 \ sqrt {80} -16} \\ & = \ dfrac {\ sqrt {5}} {4} + \ dfrac {1} {2}. \ end { split} \]
above inequality iff \ (. 8 + T = \ dfrac {80} + {T}. 8 \) , i.e.\ (t = 4 \ sqrt { 5} -8 \) when taking the like, so as to evaluate the expression \ (M \) maximum value of \ (\ dfrac {\ sqrt { 5}} {4} + \ dfrac { 1} {2} \) .