Ta có: \(2\left(x^2+y^2\right)=1+xy\)

\(\Leftrightarrow x^2+y^2=\frac{1+xy}{2}\)

\(P=7\left(x^4+y^4\right)+4x^2y^2\)

\(=7x^4+7y^4+4x^2y^2\)

\(\Rightarrow P=28x^3+28y^3+16xy\)

\(\Leftrightarrow P=0\Leftrightarrow28x^3+28y^3+16xy=0\)

\(\Leftrightarrow\orbr{\begin{cases}x=3\\y=4\end{cases}}\)

\(\Rightarrow P_{Min}=15\) và \(Max_P=\frac{12}{33}\)

B = 2\(x^2\) - 4\(x\) - 8

B = 2(\(x^2\) - 2\(x\) + 4)  - 16

B = 2(\(x-2\))² - 16 

Vì (\(x-2\))² ≥ 0 ∀ \(x\) ⇒ 2(\(x-2\))² ≥ 0 ∀ \(x\)

⇒ 2(\(x-2\))²  - 16 ≥ -16 ∀ \(x\)

Dấu bằng xảy ra khi  (\(x-2\))² = 0 ⇒ \(x-2=0\) ⇒ \(x=2\)

Vậy B_min = -16 khi \(x=2\)

Tìm min của C biết:

C = \(x^2\) - 2\(xy\) + 2y² + 2\(x\) - 10y + 17

C = (\(x^2\) - 2\(xy\) + y²) + 2(\(x\) - y) + y² - 8y + 16 + 1

C = (\(x\) - y)² + 2(\(x\) - y) + 1  + (y² - 8y + 16) 

C = (\(x-y+1\))² + (y - 4)² 

Vì (\(x\) - y + 1)² ≥ 0 ∀ \(x;y\); (y - 4)² ≥ 0 ∀ y

Dấu bằng xảy ra khi: \(\left\{{}\begin{matrix}x-y+1=0\\y-4=0\end{matrix}\right.\) ⇒ \(\left\{{}\begin{matrix}x-y+1=0\\y=4\end{matrix}\right.\)

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

Vậy C_min = 0 khi (\(x;y\)) = (3; 4)

D = \(x^2\) - \(xy\) + y² - 2\(x\) - 2y

D=[\(x^2\)-2\(x\)\(\dfrac{y}{2}\)+(\(\dfrac{y}{2}\))²]-(2\(x\)-2\(\dfrac{y}{2}\)) +1 +(\(\dfrac{3}{4}\)y²-2.\(\dfrac{\sqrt{3}}{2}\)y .\(\sqrt{3}\) +3) - 4

D = (\(x-\dfrac{y}{2}\))² - 2(\(x-\dfrac{y}{2}\))+ 1 + (\(\dfrac{\sqrt{3}}{2}\)y - \(\sqrt{3}\))² - 4

D = (\(x-\dfrac{y}{2}\) - 1)² + (\(\dfrac{\sqrt{3}}{2}\)y - \(\sqrt{3}\))² - 4

Vì (\(x-\dfrac{y}{2}\) - 1)² ≥  0 ∀ \(x\);y và (\(\dfrac{\sqrt{3}}{2}\)y - \(\sqrt{3}\))² ≥ 0 ∀ y 

Vậy (\(x-\dfrac{y}{2}\) - 1)² + (\(\dfrac{\sqrt{3}}{2}\)y - \(\sqrt{3}\))² - 4 ≥ - 4 ∀ \(x;y\)

Dấu bằng xảy ra khi: \(\left\{{}\begin{matrix}x-\dfrac{y}{2}-1=0\\\dfrac{\sqrt{3}}{2}y-\sqrt{3}=0\end{matrix}\right.\)

      ⇒ \(\left\{{}\begin{matrix}x-\dfrac{y}{2}-1=0\\\sqrt{3}.\left(\dfrac{1}{2}y-1\right)=0\end{matrix}\right.\)

  ⇒ \(\left\{{}\begin{matrix}x=1+\dfrac{1}{2}y\\\dfrac{1}{2}y=1\end{matrix}\right.\) ⇒ \(\left\{{}\begin{matrix}x=1+1\\y=1:\dfrac{1}{2}\end{matrix}\right.\) ⇒ \(\left\{{}\begin{matrix}x=2\\y=2\end{matrix}\right.\)

Vậy D_min = - 4 khi (\(x;y\)) =(2; 2)

Ta có : \(\left(x^2-y^2\right)^2+4x^2y^2+x^2-2y^2=0\)

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

\(\Leftrightarrow\left(x^2+y^2-1\right)^2=1-3x^2\le1\forall x\)

\(\Rightarrow\left(x^2+y^2-1\right)\le1\)

\(\Rightarrow-1\le x^2+y^2-1\le1\)

\(\Rightarrow0\le x^2+y^2\le2\)

\(C=x^2+y^2\) min tại \(x=y=0\)

\(C=x^2+y^2\)max tại \(x=0,y=\sqrt{2}\)

Đề bài sai, đề đúng thì phân thức đằng sau dấu chia phải là:

\(\dfrac{4x^4+4x^2y+y^2-4}{x^2+y+xy+x}\)

pt \(\Leftrightarrow\)\(\left(x+y\right)^2+7\left(x+y\right)+\frac{49}{4}=-y^2+\frac{49}{4}-10\)

\(\Leftrightarrow\)\(\left(x+y+\frac{7}{2}\right)^2=-y^2+\frac{9}{4}\le\frac{9}{4}\)

\(\Leftrightarrow\)\(\frac{-3}{2}\le x+y+\frac{7}{2}\le\frac{3}{2}\)

\(\Leftrightarrow\)\(-4\le x+y+1\le-1\)

Dấu "=" tự xét nhé 

a Tách \(M=2+\frac{4xy}{x^2+2xy+y^2}=2+\frac{4xy}{\left(x+y\right)^2}\le2+1=3\)
Dấu = xảy ra khi và chỉ khi x=y và x+y=2015 <=>x=y=2015/2
b,:\(N\ge\frac{\left(1+\frac{2015}{x}+1+\frac{2015}{y}\right)^2}{2}=\frac{\left(2+2015\left(\frac{1}{x}+\frac{1}{y}\right)\right)^2}{2}\)
áp dunngj svac =>\(N\ge\frac{\left(2+2015\left(\frac{\left(1+1\right)^2}{x+y}\right)\right)^2}{2}=\frac{\left(2+\frac{2015.4}{2015}\right)^2}{2}=18\)
dấu = xảy ra khi và chỉ khi x=y và x+y=2015 <=>x=y=2015/2

Cảm ơn bn nha :))

@ mình rep lúc 12h mà trả lời lúc 11h :))