a: \(A=-x^2+4x+5\)

\(=-\left(x^2-4x-5\right)\)

\(=-\left(x^2-4x+4-9\right)\)

\(=-\left(x-2\right)^2+9\le9\)

Dấu '=' xảy ra khi x=2

b: \(B=-4x^2+12x-1\)

\(=-\left(4x^2-12x+1\right)\)

\(=-\left(4x^2-12x+9-8\right)\)

\(=-\left(2x-3\right)^2+8\le8\)

Dấu '=' xảy ra khi x=3/2

\(A=-\left(x^2-2x\left(y+1\right)+\left(y+1\right)^2\right)-\left(4y^2-10y-5-\left(y+1\right)^2\right)\)

\(=-\left(x-y-1\right)^2-\left(3y^2-12y-6\right)\)

\(=-\left(x-y-1\right)^2-3\left(y-2\right)^2+18\le18\)

Max A=18 khi y=2; x=3

\(B=-\left(x^2+2x\left(y-1\right)+\left(y-1\right)^2\right)-\left(2y^2+2y-\left(y-1\right)^2\right)-15\)

\(=-\left(x+y-1\right)^2-\left(y+2\right)^2-10\le-10\)

Max B=-10 khi y=-2; x= 3

A = x² + 2y² - 2xy + 2x - 2y + 1

= x² - 2xy + y² + 2 ( x - y ) + 1 + y²

= ( x - y )² + 2 ( x - y ) + 1 + y²

= ( x - y + 1 )² + y² ≥ 0

Dấu = xảy ra khi :

\(\left\{{}\begin{matrix}x-y+1=0\\y=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-1\\y=0\end{matrix}\right.\)

B = x² + 2y² - 2xy + 2x - 10y

= x² - 2xy + y² + 2x - 2y + 1 + y² - 8x + 16 - 17

= ( x - y )² + 2 ( x - y ) + 1 + ( y - 4 )² - 17

= ( x - y + 1 )² + ( y - 4 )² - 17 ≥ - 17

Dấu = xảy ra khi :

\(\left\{{}\begin{matrix}x-y+1=0\\y-4=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=3\\y=4\end{matrix}\right.\)

\(A=\left(x^2+9y^2+1-6xy+2x-6y\right)+\left(y^2+4y+4\right)+2\)

\(A=\left(x-3y+1\right)^2+\left(y+2\right)^2+2\ge2\)

\(A_{min}=2\) khi \(\left\{{}\begin{matrix}x=-7\\y=-2\end{matrix}\right.\)

\(B=\left(x^2+y^2+1-2xy+2x-2y\right)+\left(y^2+6y+9\right)+10\)

\(B=\left(x-y+1\right)^2+\left(y+3\right)^2+10\ge10\)

\(B_{min}=10\) khi \(\left\{{}\begin{matrix}x=-4\\y=-3\end{matrix}\right.\)

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