\(a)M=5-4x-x^2\\=-(x^2+4x-5)\\=-(x^2+4x+4)+9\\=-(x+2)^2+9\)
Ta thấy: \(\left(x+2\right)^2\ge0\forall x\)
\(\Rightarrow-\left(x+2\right)^2\le0\forall x\)
\(\Rightarrow-\left(x+2\right)^2+9\le9\forall x\)
Dấu \("="\) xảy ra \(\Leftrightarrow x+2=0\Leftrightarrow x=-2\)
Vậy \(Max_{M}=9\) khi \(x=-2\)
\(---\)
\(b)N=4x-2x^2+3\\=-2(x^2-2x)+3\\=-2(x^2-2x+1)+5\\=-2(x-1)^2+5\)
Ta thấy: \(\left(x-1\right)^2\ge0\forall x\)
\(\Rightarrow-2\left(x-1\right)^2\le0\forall x\)
\(\Rightarrow-2\left(x-1\right)^2+5\le5\forall x\)
Dấu \("="\) xảy ra \(\Leftrightarrow x-1=0\Leftrightarrow x=1\)
Vậy \(Max_{N}=5\) khi \(x=1\)
\(---\)
\(c)P=x(4-x)\\=4x-x^2\\=-(x^2-4x)\\=-(x^2-4x+4)+4\\=-(x-2)^2+4\)
Ta thấy: \(\left(x-2\right)^2\ge0\forall x\)
\(\Rightarrow-\left(x-2\right)^2\le0\forall x\)
\(\Rightarrow-\left(x-2\right)^2+4\le4\forall x\)
Dấu \("="\) xảy ra \(\Leftrightarrow x-2=0\Leftrightarrow x=2\)
Vậy \(Max_{P}=4\) khi \(x=2\)
\(---\)
\(d)Q=2021+4y-2x-x^2-y^2\\=-(x^2+2x)-(y^2-4y)+2021\\=-(x^2+2x+1)-(y^2-4y+4)+2026\\=-(x+1)^2-(y-2)^2+2026\)
Ta thấy: \(\left\{{}\begin{matrix}\left(x+1\right)^2\ge0\forall x\\\left(y-2\right)^2\ge0\forall y\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}-\left(x+1\right)^2\le0\forall x\\-\left(y-2\right)^2\le0\forall y\end{matrix}\right.\)
\(\Rightarrow-\left(x+1\right)^2-\left(y-2\right)^2\le0\forall x,y\)
\(\Rightarrow-\left(x+1\right)^2-\left(y-2\right)^2+2026\le2026\forall x,y\)
Dấu \("="\) xảy ra \(\Leftrightarrow\left\{{}\begin{matrix}x+1=0\\y-2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-1\\y=2\end{matrix}\right.\)
Vậy \(Max_{Q}=2026\) khi \(x=-1;y=2\)
#\(Toru\)