\(M=\dfrac{2x+y}{xy}+\dfrac{3}{2x+y}=\dfrac{2x+y}{2}+\dfrac{3}{2x+y}=\dfrac{3\left(2x+y\right)}{16}+\dfrac{3}{2x+y}+\dfrac{5}{16}\left(2x+y\right)\ge2\sqrt{\dfrac{3}{16}.3}+\dfrac{5}{16}.2\sqrt{2xy}=\dfrac{3}{2}+\dfrac{5}{4}=\dfrac{11}{4}\).

Đẳng thức xảy ra khi x = 1; y = 2.

\(M=\dfrac{2x+y}{xy}+\dfrac{3}{2x+y}=\dfrac{2x+y}{2}+\dfrac{3}{2x+y}\)

\(M=\dfrac{3\left(2x+y\right)}{16}+\dfrac{3}{2x+y}+\dfrac{5\left(2x+y\right)}{16}\ge2\sqrt{\dfrac{9\left(2x+y\right)}{16\left(2x+y\right)}}+\dfrac{5}{16}.2\sqrt{2xy}=\dfrac{11}{4}\)

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

Ta có:

\(M=\dfrac{2x+y}{xx}+\dfrac{3}{2x+y}=\dfrac{2x+y}{2}+\dfrac{3}{2x+y}\)

\(=\left(\dfrac{3}{8}\dfrac{2x+y}{2}+\dfrac{3}{2x+y}\right)+\dfrac{5}{8}\dfrac{2x+y}{2}\)

Có: \(\dfrac{3}{8}\dfrac{2x+y}{2}+\dfrac{3}{2x+y}\ge2\sqrt{\dfrac{3}{8}\dfrac{2x+y}{2}\dfrac{3}{2x+y}}=\dfrac{3}{2}\)

Dấu '=' xảy ra \(\Leftrightarrow\dfrac{3}{8}\dfrac{2x+y}{2}=\dfrac{3}{2x+y}\)

Có: \(\dfrac{5}{8}\dfrac{2x+y}{2}\ge\dfrac{5}{8}\sqrt{2xy}=\dfrac{5}{4}\)

Dấu '=' xảy ra \(\Leftrightarrow2x=y,xy=2\)

\(\Rightarrow M\ge\dfrac{3}{2}+\dfrac{5}{4}=\dfrac{11}{4}\)

Dấu '=' xảy ra \(\Leftrightarrow x=1,y=2\)

Vậy GTNN của M là \(\dfrac{11}{4}\Leftrightarrow x=1,y=2\)

\(M=\dfrac{2x+y}{xy}\)

Lời giải:

Áp dụng BĐT Bunhiacopxky:

$(\frac{1}{x^2}+\frac{1}{y^2}+\frac{2}{xy})(x^2+y^2+2xy)\geq (1+1+2)^2=16$

$\Rightarrow \frac{1}{x^2}+\frac{1}{y^2}+\frac{2}{xy}\geq \frac{16}{(x+y)^2}=16$

Áp dụng BĐT AM-GM:

$xy\leq \frac{(x+y)^2}{4}=\frac{1}{4}$

$\Rightarrow \frac{2}{xy}\geq 8$

Cộng 2 BĐT trên lại:

$P\geq 16+8=24$

Vậy $P_{\min}=24$ khi $x=y=\frac{1}{2}$

Lời giải:

Áp dụng BĐT Bunhiacopxky:

$(\frac{1}{x^2}+\frac{1}{y^2}+\frac{2}{xy})(x^2+y^2+2xy)\geq (1+1+2)^2=16$

$\Rightarrow \frac{1}{x^2}+\frac{1}{y^2}+\frac{2}{xy}\geq \frac{16}{(x+y)^2}=16$

Áp dụng BĐT AM-GM:

$xy\leq \frac{(x+y)^2}{4}=\frac{1}{4}$

$\Rightarrow \frac{2}{xy}\geq 8$

Cộng 2 BĐT trên lại:

$P\geq 16+8=24$

Vậy $P_{\min}=24$ khi $x=y=\frac{1}{2}$

\(P=\dfrac{y}{x}+\dfrac{x}{y}+\left(\dfrac{x}{3y}+3xy+\dfrac{1}{3}+\dfrac{1}{3}\right)+12\left(xy+\dfrac{1}{9}\right)-2\)

\(P\ge2\sqrt{\dfrac{xy}{xy}}+4\sqrt[4]{\dfrac{3x^2y}{27y}}+12.2\sqrt{\dfrac{xy}{9}}-2\)

\(P\ge4\sqrt{\dfrac{x}{3}}+8\sqrt{xy}=4\left(2\sqrt{xy}+\sqrt{\dfrac{x}{3}}\right)=4\)

\(P_{min}=4\) khi \(x=y=\dfrac{1}{3}\)

Ta có: \((x + y)xy = x^2 + y^2 - xy \quad (1)\)

\(M = \frac{x^3 + y^3}{x^3 y^3} = \frac{(x + y)(x^2 - xy + y^2)}{(xy)^3}\)

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

Do đó: \(M=\frac{xy\left(x+y\right)\left(x+y\right)}{\left(xy\right)^3}=\frac{\left(x+y\right)^2}{\left(xy\right)^2}\)

Đặt \(t = \frac{x + y}{xy} = \frac{1}{x} + \frac{1}{y}\)

=>\(M=t^2\)

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

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

=>\(\frac{x + y}{xy} = \left(\frac{x + y}{xy}\right)^2 - \frac{3}{xy}\)

=>\(t=t^2-\frac{3}{xy}\iff\frac{3}{xy}=t^2-t\iff xy=\frac{3}{t^2 - t}\)

\(\left(x+y\right)^2\ge4xy\)

=>\(\left(t\cdot xy\right)^2\ge4xy\)

=>\(t^2\cdot\left(xy\right)^2-4xy\ge0\)

=>xy\(\left(t\cdot xy-4\right)\) >=0

=>\(\frac{3}{t^2 - t} \left(t^2 \cdot \frac{3}{t^2 - t} - 4\right) \ge 0\)

=>\(\frac{3}{t^2 - t}\left(\frac{3t^2 - 4(t^2 - t)}{t^2 - t}\right)\ge0\)

=>\(\frac{3(4t - t^2)}{(t^2 - t)^2}\ge0\)

=>\(4t-t^2\ge0\)

=>\(t^2-4t\le0\)

=>t(t-4)<=0

=>0<t<=4

=>t∈(0;4]

\(M = t^2 \le 4^2 = 16\)

Dấu '=' xảy ra khi t=4

=>\(xy=\frac{3}{4^2 - 4}=\frac{3}{12}=\frac{1}{4};x+y=t\cdot xy=1\)

=>x,y là các nghiệm của phương trình:

\(A^2-A+\frac14=0\)

=>\(\left(A-\frac12\right)^2=0\)

=>A-1/2=0

=>A=1/2

=>x=y=1/2

\(T=\dfrac{\left(xy\right)^2}{zx+zy}+\dfrac{\left(yz\right)^2}{xy+xz}+\dfrac{\left(zx\right)^2}{yx+yz}\ge\dfrac{xy+yz+zx}{2}\ge\dfrac{3}{2}\sqrt[3]{\left(xyz\right)^2}=\dfrac{3}{2}\)

Lời giải:

Ta có: \(A=\frac{3}{x^2+y^2}+\frac{4}{xy}=3\left(\frac{1}{x^2+y^2}+\frac{1}{2xy}\right)+\frac{5}{2xy}\)

Áp dụng BĐT Cauchy-Schwarz:

\(\frac{1}{x^2+y^2}+\frac{1}{2xy}\geq \frac{4}{x^2+y^2+2xy}=\frac{4}{(x+y)^2}=4\)

Áp dụng BĐT Am-Gm: \(xy\leq \frac{(x+y)^2}{4}=\frac{1}{4}\Rightarrow \frac{5}{2xy}\geq 10\)

Do đó: \(A\geq 3.4+10\Leftrightarrow A\geq 22\)

Vậy \(A_{\min}=22\Leftrightarrow x=y=\frac{1}{2}\)

còn cách làm khác không ạ?