Question

[Hà thúy anh]

Tìm max:

a) P = - x² - 4x +16

b) Q = - x² + 2xy - 4y² + 2x + 10y - 2017

Answer 1

[Lightning Farron]

a)\(P=-x^2-4x+16\)

\(=-x^2-4x-4-12\)

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

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

Đẳng thức xảy ra khi \(x=-2\)

b)\(-x^2+2xy-4y^2+2x+10y-2017\)

\(=\left(-x^2+2xy-y^2+2x-2y-1\right)+\left(-3y^2+12y-12\right)-2004\)

\(=-\left(x^2-2xy+y^2-2x+2y+1\right)-3\left(y^2-4y+4\right)-2004\)

\(=-\left[\left(x-y\right)^2-2\left(x-y\right)+1\right]-3\left(y-2\right)^2-2004\)

\(=-\left(x-y-1\right)^2-3\left(y-2\right)^2-2004\le-2004\)

Đẳng thức xảy ra khi \(\left\{{}\begin{matrix}x-y-1=0\\y-2=0\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}x=3\\y=2\end{matrix}\right.\)