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A.
$a^2+4b^2+9c^2=2ab+6bc+3ac$
$\Leftrightarrow a^2+4b^2+9c^2-2ab-6bc-3ac=0$
$\Leftrightarrow 2a^2+8b^2+18c^2-4ab-12bc-6ac=0$
$\Leftrightarrow (a^2+4b^2-4ab)+(a^2+9c^2-6ac)+(4b^2+9c^2-12bc)=0$
$\Leftrightarrow (a-2b)^2+(a-3c)^2+(2b-3c)^2=0$
$\Rightarrow a-2b=a-3c=2b-3c=0$
$\Rightarrow A=(0+1)^{2022}+(0-1)^{2023}+(0+1)^{2024}=1+(-1)+1=1$
B.
$x^2+2xy+6x+6y+2y^2+8=0$
$\Leftrightarrow (x^2+2xy+y^2)+y^2+6x+6y+8=0$
$\Leftrightarrow (x+y)^2+6(x+y)+9+y^2-1=0$
$\Leftrightarrow (x+y+3)^2=1-y^2\leq 1$ (do $y^2\geq 0$ với mọi $y$)
$\Rightarrow -1\leq x+y+3\leq 1$
$\Rightarrow -4\leq x+y\leq -2$
$\Rightarrow 2020\leq x+y+2024\leq 2022$
$\Rightarrow A_{\min}=2020; A_{\max}=2022$
1) \(=\left(9x^2-25y^2\right)-\left(6x-10y\right)=\left(3x-5y\right)\left(3x+5y\right)-2\left(3x-5y\right)=\left(3x-5y\right)\left(3x+5y-2\right)\)
2) \(=9x^2y^2-\left(x^2-2xy+y^2\right)=9x^2y^2-\left(x-y\right)^2=\left(3xy-x+y\right)\left(3xy+x-y\right)\)
\(-5x^2-2xy-2y^2+14x+10y-1\\ =-\left(x^2+2xy+y^2\right)-\left(4x^2-2\cdot2\cdot\dfrac{7}{2}x+\dfrac{49}{4}\right)-\left(y^2-10y+25\right)+\dfrac{55}{4}\\ =-\left(x+y\right)^2-\left(2x-\dfrac{7}{2}\right)^2-\left(y-5\right)^2+\dfrac{55}{4}\le\dfrac{55}{4}\\ Max\Leftrightarrow\left\{{}\begin{matrix}x=-y\\2x=\dfrac{7}{2}\\y=5\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-y\\x=\dfrac{7}{4}\\y=5\end{matrix}\right.\Leftrightarrow x,y\in\varnothing\)
Vậy dấu \("="\) ko xảy ra
a: Ta có: \(-x^2+3x\)
\(=-\left(x^2-3x+\dfrac{9}{4}-\dfrac{9}{4}\right)\)
\(=-\left(x-\dfrac{3}{2}\right)^2+\dfrac{9}{4}\le\dfrac{9}{4}\forall x\)
Dấu '=' xảy ra khi \(x=\dfrac{3}{2}\)
Bài này tìm max _ giá trị lớn nhất em nhé
\(A=-\left(x^2+2xy+y^2-2x.3-2.y.3+3^2\right)+4y-2y^2+1\)
\(=-\left(x+y-3\right)^2-2\left(y^2-2y+1\right)+3\)
\(=-\left(x+y-3\right)^2-2\left(y-1\right)^2+3\le3\)
Dấu '=' xảy ra <=> \(\hept{\begin{cases}y-1=0\\x+y-3=0\end{cases}\Leftrightarrow}\hept{\begin{cases}y=1\\x=2\end{cases}}\)
max A=3 <=> x=2, y=1
đề là tìm max thì đúng nhé
\(A=-x^2-3y^2-2xy+6x+10y-8\)
\(A=-\left[x^2+2x\left(y-3\right)+\left(y-3\right)^2\right]-2\left(y^2-2y+1\right)+3\)
\(A=-\left(x+y-3\right)^2-2\left(y-1\right)^2+3\)
Ta có:
\(\hept{\begin{cases}-\left(x+y-3\right)^2\le0\forall x;y\\-2\left(y-1\right)^2\le0\forall y\end{cases}}\)
\(\Rightarrow-\left(x+y-3\right)^2-2\left(y-1\right)^2+3\le3\forall x;y\)
\(\Rightarrow A\le3\)\(\forall x;y\)
Dấu " = " xảy ra <=> \(\hept{\begin{cases}-\left(x+y-3\right)^2=0\\-2\left(y-1\right)^2=0\end{cases}\Leftrightarrow\hept{\begin{cases}x+y-3=0\\y-1=0\end{cases}\Leftrightarrow}}\hept{\begin{cases}x+1-3=0\\y=1\end{cases}}\Leftrightarrow\hept{\begin{cases}x=2\\y=1\end{cases}}\)
Vậy \(A_{max}=3\)\(\Leftrightarrow\hept{\begin{cases}x=2\\y=1\end{cases}}\)