Với mọi số thực ta luôn có:

`(x-y)^2>=0`

`<=>x^2-2xy+y^2>=0`

`<=>x^2+y^2>=2xy`

`<=>(x+y)^2>=4xy`

`<=>(x+y)^2>=16`

`<=>x+y>=4(đpcm)`

\(\dfrac{1}{x+3}+\dfrac{1}{y+3}=\dfrac{x+3+y+3}{\left(x+3\right)\left(y+3\right)}\)

\(=\dfrac{x+y+6}{3x+3y+13}\)(vì \(xy=4\))

=> \(\dfrac{x+y+6}{3x+3y+13}\)≤\(\dfrac{2}{5}\)

<=> \(5\left(x+y+6\right)\)≤\(2\left(3x+3y+13\right)\)

<=>\(6x+6y+26-5x-5y-30\)≥\(0\)

<=> \(x+y-4\)≥\(0\)

Áp dụng BĐT AM-GM \(\dfrac{a+b}{2}\)≥\(\sqrt{ab}\)

Ta có \(\dfrac{x+y}{2}\)≥\(\sqrt{xy}\)

<=>\(x+y\) ≥ 2\(\sqrt{xy}\)

=>2\(\sqrt{xy}-4\)≥\(0\)

<=> \(4-4\)≥0

<=>0≥0 ( Luôn đúng )

Vậy \(\dfrac{1}{x+3}+\dfrac{1}{y+3}\)≤\(\dfrac{2}{5}\)

c1: phân tích từng cái

c2, nhân x cho (1) y cho 2

sau đs dùng bunhia 

từ x+y=1

=> x^2-xy+y^2...

\(VT-VP=\frac{\left(3x^2+7xy+3y^2\right)\left(x-y\right)^2}{3\left(1-x^2\right)\left(1-y^2\right)}\ge0\)

Áp dụng giả thiết x + y = 1, ta được:\(\frac{x}{1-x^2}+\frac{y}{1-y^2}=\frac{x}{\left(1+x\right)\left(1-x\right)}+\frac{y}{\left(1+y\right)\left(1-y\right)}=\frac{x}{y\left(1+x\right)}+\frac{y}{x\left(1+y\right)}\)

Theo bất đẳng thức AM - GM:\(\frac{x}{y\left(1+x\right)}+\frac{y}{x\left(1+y\right)}\ge2\sqrt{\frac{x}{y\left(1+x\right)}.\frac{y}{x\left(1+y\right)}}=\frac{2}{\sqrt{xy+x+y+1}}=\frac{2}{\sqrt{xy+2}}\ge\frac{2}{\sqrt{\frac{\left(x+y\right)^2}{4}+2}}=\frac{4}{3}\)Vậy bất đẳng thức được chứng minh

Đẳng thức xảy ra khi x = y = ¹/₂

\(\left(x^2+\frac{1}{x^2}\right)\left(2^2+\frac{1}{2^2}\right)\ge\left(2x+\frac{1}{2x}\right)^2\) 

\(\Leftrightarrow x^2+\frac{1}{x^2}\ge\frac{4}{17}\left(2x+\frac{1}{2x}\right)^2\)Rồi tương tự các kiểu...

Suy ra \(M\ge\sqrt{\frac{4}{17}}\left[2\left(x+y\right)+\frac{1}{2}\left(\frac{1}{x}+\frac{1}{y}\right)\right]\ge\sqrt{\frac{4}{17}}\left(2.4+\frac{1}{2}.\frac{4}{x+y}\right)=\sqrt{17}\)

"=" <=> x = y = 2

Is that true?

different way

Áp dụng min-cop-xki ta có:

\(M=\sqrt{x^2+\frac{1}{x^2}}+\sqrt{y^2+\frac{1}{y^2}}\ge\sqrt{\left(x+y\right)^2+\left(\frac{1}{x}+\frac{1}{y}\right)^2}\ge\sqrt{16+\frac{16}{\left(x+y\right)^2}}=\sqrt{17}\)

Dau '=' xay ra khi \(x=y=2\)

\(S=\dfrac{x}{2}+\dfrac{1}{2x}+\dfrac{y}{2}+\dfrac{2}{y}+\dfrac{1}{2}\left(x+y\right)\)

\(S\ge2\sqrt{\dfrac{x}{4x}}+2\sqrt{\dfrac{2y}{2y}}+\dfrac{1}{2}.3=\dfrac{9}{2}\)

Dấu "=" xảy ra khi \(\left(x;y\right)=\left(1;2\right)\)

a/ \(M=\left(x^2+\frac{1}{y^2}\right)\left(y^2+\frac{1}{x^2}\right)=x^2y^2+\frac{1}{x^2y^2}+2=\left(xy-\frac{1}{xy}\right)^2+4\ge4\)

Suy ra Min M = 4 . Dấu "=" xảy ra khi x=y=1/2

b/ Đề đúng phải là \(\frac{1}{3x+3y+2z}+\frac{1}{3x+2y+3z}+\frac{1}{2x+3y+3z}\ge\frac{3}{2}\)

Ta có \(6=\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x}\ge\frac{9}{2\left(x+y+z\right)}\Rightarrow x+y+z\ge\frac{3}{4}\)

Lại có \(\frac{1}{3x+3y+2z}+\frac{1}{3x+2y+3z}+\frac{1}{2x+3y+3z}\ge\frac{9}{8\left(x+y+z\right)}\ge\frac{9}{8.\frac{3}{4}}=\frac{3}{2}\)

\(x+y+\frac{1}{2x}+\frac{2}{y}=\left(\frac{x}{2}+\frac{1}{2x}\right)+\left(\frac{y}{2}+\frac{2}{y}\right)+\left(\frac{x}{2}+\frac{y}{2}\right)\ge2\sqrt{\frac{x}{2}.\frac{1}{2x}}+2\sqrt{\frac{y}{2}.\frac{2}{y}}+\frac{3}{2}=1+2+\frac{3}{2}=\frac{9}{2}\)Đẳng thức xảy ra khi và chỉ khi :

\(\frac{x}{2}=\frac{1}{2x}\Leftrightarrow2x^2=2\Rightarrow x=1\)(vì x>0)

\(\frac{y}{2}=\frac{2}{y}\Leftrightarrow y^2=4\Rightarrow y=2\)(vì y>0)

\(x+y=3\)

\(\Rightarrow x=1;y=2\)

tưởng ngon ăn dùng cô-si ai dè @@

\(F=\frac{x^2}{x+x^3}+\frac{y^2}{y+y^3}\ge\frac{\left(x+y\right)^2}{\left(x+y\right)\left(x^2+y^2-xy+1\right)}=\frac{1}{1+\left(x+y\right)^2-3xy}=\frac{1}{2-3xy}\)\(\ge\frac{1}{2-\frac{3}{4}}=\frac{4}{5}\)

Dấu bằng xảy ra khi x=y=\(\frac{1}{2}\)