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\[\sqrt {{a^2} + {b^2}} + \sqrt {ab} = \sqrt {{a^2} + {b^2}} + \frac{1}{{\sqrt 2 }}\sqrt {2ab} \leqslant \sqrt {\left( {1{\text{ + }}\frac{1}{2}} \right)\left( {{a^2} + {b^2}{\text{ + }}2ab} \right)} {\text{ = }}\sqrt {\frac{3}{2}{{\left( {a + b} \right)}^2}} {\text{ = }}\frac{{\sqrt 6 }}{2}\left( {a + b} \right)\]
\[\frac{{ab}}{{a + b}} + \sqrt {ab} \leqslant \frac{{a + b}}{4} + \frac{{a + b}}{2}{\text{ = }}\frac{{3\left( {a + b} \right)}}{4}\]
\[\sqrt {{a^2} + {b^2}} + \frac{{ab}}{{a + b}} \leqslant a + b\]
\[\sqrt {{a^2} + {b^2}} + \sqrt {ab} + \frac{{ab}}{{a + b}} \leqslant \frac{{2\sqrt 2 + 3}}{4}\left( {a + b} \right)\] |
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