giúp mình với

XIN LỖI ! MÌNH KHONG BIẾT

Ta có A = 5x² - 2xy + 2y² - 4x + 2y + 3

=> 2A = 10x² - 4xy + 4y² - 8x + 4y + 6

= (x² - 4xy + 4y²) - 2(x - 2y) + 1 + 9x² - 6x + 1 + 4

= \(\left(x-2y\right)^2-2\left(x-2y\right)+1+9\left(x^2-\frac{2}{3}x+\frac{1}{9}\right)+4\)

\(=\left(x-2y-1\right)^2+9\left(x-\frac{1}{3}\right)^2+4\)\(\ge4\)

=> A \(\ge\)2 

Dấu "=" xảy ra <=> \(\hept{\begin{cases}x-2y-=0\\x-\frac{1}{3}=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x-2y=1\\x=\frac{1}{3}\end{cases}}\Leftrightarrow\hept{\begin{cases}y=-\frac{1}{3}\\x=\frac{1}{3}\end{cases}}\)

Vậy khi x = 1/3 ; y = -1/3 thì A đạt GTNN

\(A=5x^2+2y^2-2xy-4x+2y\)\(+3\)

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

\(Tacó\)

\(A=5x^2+2y^2-2xy-4x+2y+3\)

\(\Rightarrow2A=10x^2+4y^2-4xy-8x+4y+6\)

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

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

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

Vì \(\left(x-2y-1\right)^2\ge0\), \(\left(3x-1\right)^2\ge0\forall x,y\)

\(\Rightarrow\left(x-2y-1\right)^2+\left(3x-1\right)^2\ge0\forall x,y\)

\(\Rightarrow\left(x-2y-1\right)^2+\left(3x-1\right)^2+4\ge4\forall x,y\)

\(\Rightarrow2A\ge4\)\(\Rightarrow A\ge2\)

Dấu " = " xảy ra \(\Leftrightarrow\hept{\begin{cases}x-2y-1=0\\3x-1=0\end{cases}}\Leftrightarrow\hept{\begin{cases}2y=x-1\\x=\frac{1}{3}\end{cases}}\Leftrightarrow\hept{\begin{cases}2y=\frac{-2}{3}\\x=\frac{1}{3}\end{cases}}\Leftrightarrow\hept{\begin{cases}y=\frac{-1}{3}\\x=\frac{1}{3}\end{cases}}\)

Vậy \(minA=2\)\(\Leftrightarrow\hept{\begin{cases}x=\frac{1}{3}\\y=-\frac{1}{3}\end{cases}}\)

Câu 1 :

\(E=4x^2+y^2-4x-2y+3\)

\(E=\left(2x\right)^2-2\cdot2x\cdot1+1^2+y^2-2\cdot y\cdot1+1^2+1\)

\(E=\left(2x-1\right)^2+\left(y-1\right)^2+1\ge1\forall x;y\)

Dấu "=" xảy ra \(\Leftrightarrow\hept{\begin{cases}2x-1=0\\y-1=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x=\frac{1}{2}\\y=1\end{cases}}\)

Câu 2 :

\(G=x^2+2y^2+2xy-2y\)

\(G=x^2+2xy+y^2+y^2-2.y\cdot1+1^2-1\)

\(G=\left(x+y\right)^2+\left(y-1\right)^2-1\ge-1\forall x;y\)

Dấu "=" xảy ra \(\Leftrightarrow\hept{\begin{cases}x+y=0\\y-1=0\end{cases}\Leftrightarrow\hept{\begin{cases}x+1=0\\y=1\end{cases}\Leftrightarrow}\hept{\begin{cases}x=-1\\y=1\end{cases}}}\)

Còn câu F bạn ơi. Giúp Gk vs

\(F=\frac{3}{2x^2+x+1}=\frac{3}{2\left(x^2+\frac{x}{2}+\frac{1}{2}\right)}=\frac{3}{2\left(x^2+2x\cdot\frac{1}{4}+\frac{1}{16}\right)+\frac{7}{8}}=\frac{3}{2\left(x+\frac{1}{4}\right)^2+\frac{7}{8}}\)

Vi \(2\left(x+\frac{1}{4}\right)^2\ge0\Rightarrow2\left(x+\frac{1}{4}\right)^2+\frac{7}{8}\ge8\)

\(\Rightarrow\frac{1}{2\left(x+\frac{1}{4}\right)^2+\frac{7}{8}}\le\frac{1}{\frac{7}{8}}\Rightarrow F=\frac{3}{2\left(x+\frac{1}{4}\right)^2+\frac{7}{8}}\le\frac{3}{\frac{7}{8}}=\frac{24}{7}\)

Dấu "=" xảy ra <=>x+1/4=0<=>x=-1/4

\(M=5x^2+y^2-2x+2y+2xy+2004\)

\(=\left(x^2+2x+1\right)+2y\left(x+1\right)+y^2+4x^2-4x+1+2002\)

\(=\left(x+1\right)^2+2y\left(x+1\right)+y^2+\left(2x-1\right)^2+2002\)

\(=\left(x+1+y\right)^2+\left(2x-1\right)^2+2003\ge2002\) với mọi x,y

=> \(M_{min}=2002\Leftrightarrow\left\{{}\begin{matrix}x+y+1=0\\2x-1=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}y=-\dfrac{3}{2}\\x=\dfrac{1}{2}\end{matrix}\right.\)

Vậy \(M_{min}=2002\)

Có x^2 + 2xy + 4x + 4y + 2y^2 + 3 = 0

--> (x+y)^2 + 4(x+y) + 4+ y^2 - 1 = 0

--> (x+y+2)^2 + y^2 = 1

-->(x+y+2)^2 <= 1 ( vì y^2 >=1)

--> -1 <= x+y+2 <=1

--> 2015 <= x+y+2018 <= 2017

hay 2015 <= Q , dau bang xay ra khi x+y+2=-1 --> x+y=-3

Q<=2017, dau bang xay ra khi  x+y+2=1 --> x+y=-1

Vậy giá trị nhỏ nhất của Q là 2015 khi x+y =-3

 giá trị lớn nhất của Q là 2017 khi x+y=-1

giá trị lớn nhất là 2017

\(N=2x^2+y^2+2xy-4x-2y+3\)

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

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

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

Mà  \(\left(x+y-1\right)\ge0\forall x;y\)

       \(\left(x-1\right)^2\ge0\forall x\)

\(\Rightarrow N\ge1\)

Dấu "=" xảy ra khi :

\(\hept{\begin{cases}x+y-1=0\\x-1=0\end{cases}}\Leftrightarrow\hept{\begin{cases}y=0\\x=1\end{cases}}\)

Vậy  \(N_{Min}=1\Leftrightarrow\left(x;y\right)=\left(1;0\right)\)

\(N=2x^2+y^2+2xy-4x-2y\)\(+3\)

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

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

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

\(Do\)\(\left(x+y+1\right)^2\)\(\ge\)\(0\)\(\forall\)\(x\)\(;\)\(y\)

\(x^2\)\(\ge\)\(0\)\(\forall\)\(x\)

=.>\(\left(x+y+1\right)^2+x^2+2\)\(\ge\)\(2\)\(\forall\)\(x\)\(;\)\(y\)

=>\(N\)\(\ge\)\(2\)\(\forall\)\(x\)\(;\)\(y\)

Dấu = xảy ra khi: 

\(\hept{\begin{cases}\left(x+y+1\right)^2=0\\x^2=0\end{cases}}\)

=>\(\hept{\begin{cases}x+y+1=0\\x=0\end{cases}}\)

=>\(\hept{\begin{cases}x+y=-1\\x=0\end{cases}}\)

=>\(\hept{\begin{cases}y=-1\\x=0\end{cases}}\)

Vậy \(N_{min}\)\(=\)\(2\)khi \(y=-1\)\(;\)\(x=0\)

Chúc pạn họk tốt~~~!!! :3

@jiyoonmin : bn lm như cak mk á

\(S=\left(x^2+y^2+1+2xy+2x+2y\right)+\left(y^2-4y+4\right)+2021\)

\(S=\left(x+y+1\right)^2+\left(y-2\right)^2+2021\ge2021\)

Dấu "=" xảy ra khi \(\left(x;y\right)=\left(-3;2\right)\)

a: A=(x-1)(x-3)(x²-4x+5)

\(=\left(x^2-4x+3\right)\left(x^2-4x+5\right)\)

\(=\left(x^2-4x\right)^2+8\left(x^2-4x\right)+15\)

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

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

Dấu = xảy ra khi x-2=0

=>x=2

b: \(B=x^2-2xy+2y^2-2y+1\)

\(=x^2-2xy+y^2+y^2-2y+1\)

\(=\left(x-y\right)^2+\left(y-1\right)^2>=0\)

Dấu = xảy ra khi x-y=0 và y-1=0

=>x=y=1

c: \(C=5+\left(1-x\right)\left(x+2\right)\left(x+3\right)\left(x+6\right)\)

\(=-\left(x-1\right)\left(x+6\right)\left(x+2\right)\left(x+3\right)+5\)

\(=-\left(x^2+5x-6\right)\left(x^2+5x+6\right)+5\)

\(=-\left[\left(x^2+5x\right)^2-36\right]+5\)

\(=-\left(x^2+5x\right)^2+36+5\)

\(=-\left(x^2+5x\right)^2+41< =41\)

Dấu = xảy ra khi \(x^2+5x=0\)

=>x(x+5)=0

=>\(\left[{}\begin{matrix}x=0\\x=-5\end{matrix}\right.\)