Áp dụng cosi

`1/x^2+1/y^2>=2/(xy)`

`=>1/2>=2/(xy)`

`=>xy>=4`

Aps dụng cosi

`=>x+y>=2\sqrt{xy}=2.2=4`

Dấu "=" xảy ra khi `x=y=4`

Có : \(\dfrac{1}{2}=\dfrac{1}{x^2}+\dfrac{1}{y^2}\ge2\sqrt{\dfrac{1}{x^2}\cdot\dfrac{1}{y^2}}=\dfrac{2}{xy}\)

\(\Rightarrow xy\ge4\)

Ta có : \(A=x+y\ge2\sqrt{xy}=2\sqrt{4}=4\)

Dấu "=" xảy ra khi \(x=y=2\)

Vậy min A = 4 khi $x=y=2$

\(x+y\le xy\Rightarrow\dfrac{1}{x}+\dfrac{1}{y}\le1\)

\(M=\dfrac{1}{2\left(x^2+y^2\right)+y^2}+\dfrac{1}{2\left(x^2+y^2\right)+x^2}\le\dfrac{1}{4xy+y^2}+\dfrac{1}{4xy+x^2}\)

\(B\le\dfrac{1}{25}\left(\dfrac{4}{xy}+\dfrac{1}{y^2}\right)+\dfrac{1}{25}\left(\dfrac{4}{xy}+\dfrac{1}{x^2}\right)=\dfrac{1}{25}\left(\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{2}{xy}+\dfrac{6}{xy}\right)\)

\(M\le\dfrac{1}{25}\left[\left(\dfrac{1}{x}+\dfrac{1}{y}\right)^2+\dfrac{3}{2}\left(\dfrac{1}{x}+\dfrac{1}{y}\right)^2\right]=\dfrac{1}{10}\left(\dfrac{1}{x}+\dfrac{1}{y}\right)^2\le\dfrac{1}{10}\)

\(M_{max}=\dfrac{1}{10}\) khi \(x=y=2\)

\(1\ge x+\dfrac{1}{y}\ge2\sqrt{\dfrac{x}{y}}\Rightarrow\dfrac{x}{y}\le\dfrac{1}{4}\)

Đặt \(\dfrac{x}{y}=a\Rightarrow0< a\le\dfrac{1}{4}\)

\(P=\dfrac{\left(\dfrac{x}{y}\right)^2-\dfrac{2x}{y}+2}{\dfrac{x}{y}+1}=\dfrac{a^2-2a+2}{a+1}=\dfrac{4a^2-8a+8}{4\left(a+1\right)}=\dfrac{4a^2-13a+3+5\left(a+1\right)}{4\left(a+1\right)}\)

\(P=\dfrac{5}{4}+\dfrac{\left(1-4a\right)\left(3-a\right)}{4\left(a+1\right)}\ge\dfrac{5}{4}\)

Dấu "=" xảy ra khi \(a=\dfrac{1}{4}\) hay \(\left(x;y\right)=\left(\dfrac{1}{2};2\right)\)

\(K=\left(4xy+\dfrac{1}{4xy}\right)+\left(\dfrac{1}{x^2+y^2}+\dfrac{1}{2xy}\right)+\dfrac{5}{4xy}\)

\(K\ge2\sqrt{\dfrac{4xy}{4xy}}+\dfrac{4}{x^2+y^2+2xy}+\dfrac{5}{\left(x+y\right)^2}\ge2+4+5=11\)

\(K_{min}=11\) khi \(x=y=\dfrac{1}{2}\)

Do \(1\le x\le2\Rightarrow\left(x-1\right)\left(x-2\right)\le0\)

\(\Leftrightarrow x^2+2\le3x\)

Hoàn toàn tương tự ta có \(y^2+2\le3y\)

Do đó: \(P\ge\dfrac{x+2y}{3x+3y+3}+\dfrac{2x+y}{3x+3y+3}+\dfrac{1}{4\left(x+y-1\right)}\)

\(P\ge\dfrac{x+y}{x+y+1}+\dfrac{1}{4\left(x+y-1\right)}\)

Đặt \(a=x+y-1\Rightarrow1\le a\le3\)

\(\Rightarrow P\ge f\left(a\right)=\dfrac{a+1}{a+2}+\dfrac{1}{4a}\)

\(f'\left(a\right)=\dfrac{3a^2-4a-4}{4a^2\left(a+2\right)^2}=\dfrac{\left(a-2\right)\left(3a+2\right)}{4a^2\left(a+2\right)^2}=0\Rightarrow a=2\)

\(f\left(1\right)=\dfrac{11}{12}\) ; \(f\left(2\right)=\dfrac{7}{8}\) ; \(f\left(3\right)=\dfrac{53}{60}\)

\(\Rightarrow f\left(a\right)\ge\dfrac{7}{8}\Rightarrow P_{min}=\dfrac{7}{8}\) khi \(\left(x;y\right)=\left(1;2\right);\left(2;1\right)\)

Giúp mình câu này với ah. 

\(x\ge xy+1\Rightarrow1\ge y+\dfrac{1}{x}\ge2\sqrt{\dfrac{y}{x}}\Rightarrow\dfrac{y}{x}\le\dfrac{1}{4}\)

\(Q^2=\dfrac{x^2+2xy+y^2}{3x^2-xy+y^2}=\dfrac{\left(\dfrac{y}{x}\right)^2+2\left(\dfrac{y}{x}\right)+1}{\left(\dfrac{y}{x}\right)^2-\dfrac{y}{x}+3}\)

Đặt \(\dfrac{y}{x}=t\le\dfrac{1}{4}\) 

\(Q^2=\dfrac{t^2+2t+1}{t^2-t+3}=\dfrac{t^2+2t+1}{t^2-t+3}-\dfrac{5}{9}+\dfrac{5}{9}\)

\(Q^2=\dfrac{\left(4t-1\right)\left(t+6\right)}{9\left(t^2-t+3\right)}+\dfrac{5}{9}\le\dfrac{5}{9}\)

\(\Rightarrow Q_{max}=\dfrac{\sqrt{5}}{3}\) khi \(t=\dfrac{1}{4}\) hay \(\left(x;y\right)=\left(2;\dfrac{1}{2}\right)\)

\(y\ge xy+1\ge2\sqrt{xy}\Rightarrow\sqrt{\dfrac{y}{x}}\ge2\Rightarrow\dfrac{y}{x}\ge4\)

\(Q=\dfrac{1-\dfrac{2y}{x}+2\left(\dfrac{y}{x}\right)^2}{\dfrac{y}{x}+\left(\dfrac{y}{x}\right)^2}\)

Đặt \(\dfrac{y}{x}=a\ge4\)

\(Q=\dfrac{2a^2-2a+1}{a^2+a}=\dfrac{2a^2-2a+1}{a^2+a}-\dfrac{5}{4}+\dfrac{5}{4}=\dfrac{\left(a-4\right)\left(3a-1\right)}{4\left(a^2+1\right)}+\dfrac{5}{4}\ge\dfrac{5}{4}\)

\(Q_{min}=\dfrac{5}{4}\) khi \(a=4\) hay \(\left(x;y\right)=\left(\dfrac{1}{2};2\right)\)