无难度,由 CS 有
\begin{align*}
f(x,y)&\leqslant \sqrt {(1+1)(x-y+1+2x+y-2)}+\sqrt {2-x}\\
&=\sqrt {6x-2}+\sqrt {2-x}\\
&\leqslant \sqrt {(6+1)\left( {x-\frac 13+2-x} \right)}\\
&=\sqrt {\frac {35}3},
\end{align*}
由 `\sqrt a+\sqrt b\geqslant\sqrt{a+b}` 有
\begin{align*}
f(x,y)&\geqslant \sqrt {x-y+1+2x+y-2}+\sqrt {2-x}\\
&=\sqrt {3x-1}+\sqrt {2-x}\\
&\geqslant \sqrt {x-\frac 13}+\sqrt {2-x}\\
&\geqslant \sqrt {x-\frac 13+2-x}\\
&=\sqrt {\frac 53},
\end{align*}
取等略。 |