b ) \(x^2+9y^2-4xy=2xy-\left|x-3\right|\)
\(\Leftrightarrow x^2+9y^2-4xy-2xy+\left|x-3\right|=0\)
\(\Leftrightarrow x^2-6xy+9y^2+\left|x-3\right|=0\)
\(\Leftrightarrow\left(x-3y\right)^2+\left|x-3\right|=0\)
Do \(\left(x-3y\right)^2\ge0;\left|x-3\right|\ge0\forall x;y\)
\(\Rightarrow\left(x-3y\right)^2+\left|x-3\right|\ge0\forall x;y\)
Dấu " = " xảy ra
\(\Leftrightarrow\left\{{}\begin{matrix}x-3y=0\\x-3=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=3y\\x=3\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}y=1\\x=3\end{matrix}\right.\)
Mà \(M=\left(x-4\right)^{2013}+\left(y-1\right)^{2014}\)
\(\Leftrightarrow M=\left(3-4\right)^{2013}+\left(1-1\right)^{2014}\)
\(\Leftrightarrow M=-1^{2013}+0^{2014}\)
\(\Leftrightarrow M=-1+0\)
\(\Leftrightarrow M=-1\)
Vậy \(M=-1\)
\(a+b+c+d=0\)
\(\Leftrightarrow a+b=-\left(c+d\right)\)
\(\Leftrightarrow\left(a+b\right)^3=-\left(c+d\right)^3\)
\(\Leftrightarrow a^3+b^3+3a^2b+3b^2a=-c^3-d^3-3c^2d-3d^2c\)
\(\Leftrightarrow a^3+b^3+3a^2b+3b^2a+c^3+d^3+3c^2d+3d^2c=0\)
\(\Leftrightarrow a^3+b^3+c^3+d^3=-3a^2b-3b^2a-3c^2d-3d^2c\)
\(\Leftrightarrow a^3+b^3+c^3+d^3=3\left(-a^2b-b^2a-c^2d-d^2c\right)\)
\(\Leftrightarrow a^3+b^3+c^3+d^3=3\left[-ab\left(a+b\right)-cd\left(c+d\right)\right]\)
\(\Leftrightarrow a^3+b^3+c^3+d^3=3\left[-ab\left(a+b\right)+cd\left(a+b\right)\right]\)
\(\Leftrightarrow a^3+b^3+c^3+d^3=3\left(dc-ab\right)\left(a+b\right)\left(đpcm\right)\)
