Có a^2+b^2=4. Tìm min M=2a-b

\[\begin{array}{l}
{a^2} + {b^2} = 4\,\,\,\left( * \right)\\
M = 2a – b\\
\Leftrightarrow b = 2a – M\\
\Rightarrow \left( * \right) \Leftrightarrow {a^2} + {\left( {2a – M} \right)^2} = 4\\
\Leftrightarrow {a^2} + 4{a^2} – 4Ma + {M^2} = 4\\
\Leftrightarrow 5{a^2} – 4Ma + {M^2} – 4\, = 0\,\,\,\,\,\,\left( 1 \right)\\
\Rightarrow \left( 1 \right)\,\,\,\,co\,\,\,nghiem\,\,\,\, \Leftrightarrow \Delta ‘ \ge 0\\
\Leftrightarrow 4{M^2} – 5\left( {{M^2} – 4} \right) \ge 0\\
\Leftrightarrow 4{M^2} – 5{M^2} + 20 \ge 0\\
\Leftrightarrow {M^2} \le 20\\
\Leftrightarrow – 2\sqrt 5 \le M \le 2\sqrt 5 .\\
\Rightarrow Min\,\,M = – 2\sqrt 5 .
\end{array}\]