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\(A^2=\dfrac{x^2}{y}+\dfrac{y^2}{z}+\dfrac{z^2}{x}+2\left(\dfrac{xy}{\sqrt{yz}}+\dfrac{yz}{\sqrt{xz}}+\dfrac{xz}{\sqrt{xy}}\right)\)
Áp dụng BĐT cosi:
\(\dfrac{x^2}{y}+\dfrac{xy}{\sqrt{yz}}+\dfrac{xy}{\sqrt{yz}}+z\ge4\sqrt[4]{\dfrac{x^4y^2z}{y^2z}}=4x\)
\(\dfrac{y^2}{z}+\dfrac{yz}{\sqrt{xz}}+\dfrac{yz}{\sqrt{xz}}+x\ge4\sqrt[4]{\dfrac{y^4z^2x}{z^2x}}=4y\)
\(\dfrac{z^2}{x}+\dfrac{xz}{\sqrt{xy}}+\dfrac{xz}{\sqrt{xy}}+y\ge4\sqrt[4]{\dfrac{z^4x^2y}{x^2z}}=4z\)
Cộng VTV 3 BĐT trên:
\(\Leftrightarrow A^2+\left(x+y+z\right)\ge4\left(x+y+z\right)\\ \Leftrightarrow A^2\ge3\left(x+y+z\right)\ge3\cdot12=36\\ \Leftrightarrow A\ge6\)
Dấu \("="\Leftrightarrow x=y=z=\dfrac{12}{3}=4\)
\(P^2=\dfrac{x^2}{y}+\dfrac{y^2}{z}+\dfrac{z^2}{x}+\dfrac{2xy}{\sqrt{yz}}+\dfrac{2yz}{\sqrt{zx}}+\dfrac{2zx}{\sqrt{xy}}\)
\(P^2=\left(\dfrac{x^2}{y}+\dfrac{xy}{\sqrt{yz}}+\dfrac{xy}{\sqrt{yz}}+z\right)+\left(\dfrac{y^2}{z}+\dfrac{yz}{\sqrt{zx}}+\dfrac{yz}{\sqrt{zx}}+x\right)+\left(\dfrac{z^2}{x}+\dfrac{zx}{\sqrt{xy}}+\dfrac{zx}{\sqrt{xy}}+y\right)-\left(x+y+z\right)\)
\(P^2\ge4\sqrt[4]{\dfrac{x^4y^2z}{y^2z}}+4\sqrt[4]{\dfrac{y^4z^2x}{z^2x}}+4\sqrt[4]{\dfrac{z^4x^2y}{x^2y}}-\left(x+y+z\right)=3\left(x+y+z\right)\ge36\)
\(\Rightarrow P\ge6\)
\(P_{min}=6\) khi \(x=y=z=4\)
\(2=4\sqrt{xy}+2\sqrt{xz}\le2x+2y+x+z=3x+2y+z\)
Ta có:
\(VT=\dfrac{3yz}{x}+\dfrac{4zx}{y}+\dfrac{5xy}{z}=2\left(\dfrac{xy}{z}+\dfrac{zx}{y}+\dfrac{yz}{x}\right)+\left(\dfrac{yz}{x}+\dfrac{xy}{z}\right)+2\left(\dfrac{zx}{y}+\dfrac{xy}{z}\right)\)
\(VT\ge2\left(x+y+z\right)+2y+4x\)
\(VT\ge2\left(3x+2y+z\right)\ge4\)
Dấu "=" xảy ra khi \(x=y=z=\dfrac{1}{3}\)
Thử nhé
Vì P là bất đẳng thức đối xứng nên dự đoán điểm rơi \(x=y=z=\dfrac{\sqrt{2021}}{3}\)
Thay vo P ta duoc \(P=4.\sqrt{2021}\)
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\(P=\sum\dfrac{\left(x+y\right)\sqrt{\left(y+z\right)\left(z+x\right)}}{z}\)
Cauchy-Schwarz:
\(\Rightarrow\left(y+z\right)\left(z+x\right)\ge\left(z+\sqrt{xy}\right)^2\Rightarrow\sqrt{\left(y+z\right)\left(z+x\right)}\ge z+\sqrt{xy}\)
\(\Rightarrow P\ge\sum\dfrac{\left(x+y\right)\left(z+\sqrt{xy}\right)}{z}\ge\sum\dfrac{xz+yz+x\sqrt{y}+y\sqrt{x}}{z}=\sum x+y+\dfrac{\left(x+y\right)\sqrt{xy}}{z}\ge\sum x+y+\dfrac{2xy}{z}\)
\(\Rightarrow P\ge2(x+y+z)+2\left(\dfrac{xy}{z}+\dfrac{yz}{x}+\dfrac{zx}{y}\right)\)
Cauchy-Schwarz: \(\left(\dfrac{xy}{z}+\dfrac{yz}{x}+\dfrac{zx}{y}\right)\left(\dfrac{xy}{z}+\dfrac{yz}{x}+\dfrac{zx}{y}\right)\ge\left(\sqrt{\dfrac{xy}{z}.\dfrac{yz}{z}}+\sqrt{\dfrac{yz}{x}.\dfrac{zx}{y}}+\sqrt{\dfrac{zx}{y}.\dfrac{xy}{z}}\right)^2=\left(x+y+z\right)^2\)
\(\Rightarrow P\ge2(x+y+z)+2\left(x+y+z\right)=4\left(x+y+z\right)=4\sqrt{2021}\)
\("="\Leftrightarrow x=y=z=\dfrac{\sqrt{2021}}{3}\)
Giả thiết thiếu rồi em, chỗ \(\dfrac{1}{x+1}+...\) thiếu đoạn sau nữa
Đặt \(\left(\dfrac{1}{\sqrt{x}};\dfrac{1}{\sqrt{y}};\dfrac{1}{\sqrt{z}}\right)=\left(a;b;c\right)\Rightarrow\dfrac{a^2}{a^2+1}+\dfrac{b^2}{b^2+1}+\dfrac{c^2}{c^2+1}=1\)
Ta cần chứng minh: \(ab+bc+ca\le\dfrac{3}{2}\)
Thật vậy, ta có:
\(1=\dfrac{a^2}{a^2+1}+\dfrac{b^2}{b^2+1}+\dfrac{c^2}{c^2+1}\ge\dfrac{\left(a+b+c\right)^2}{a^2+b^2+c^2+3}\)
\(\Rightarrow a^2+b^2+c^2+3\ge a^2+b^2+c^2+2\left(ab+bc+ca\right)\)
\(\Rightarrow ab+bc+ca\le\dfrac{3}{2}\) (đpcm)
Đặt \(\left\{{}\begin{matrix}\sqrt{y+z-4}=a>0\\\sqrt{z+x-4}=b>0\\\sqrt{x+y-4}=c>0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=\dfrac{b^2+c^2-a^2+4}{2}\\y=\dfrac{c^2+a^2-b^2+4}{2}\\z=\dfrac{a^2+b^2-c^2+4}{2}\end{matrix}\right.\).
\(2P=\dfrac{b^2+c^2-a^2+4}{a}+\dfrac{c^2+a^2-b^2+4}{b}+\dfrac{a^2+b^2-c^2+4}{c}=\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{a}+\dfrac{b^2}{a}+\dfrac{c^2}{b}+\dfrac{a^2}{c}+\dfrac{4}{a}+\dfrac{4}{b}+\dfrac{4}{c}-a-b-c\).
Áp dụng bất đẳng thức AM - GM:
\(\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{a}=\left(\dfrac{a^2}{b}+b\right)+\left(\dfrac{b^2}{c}+c\right)+\left(\dfrac{c^2}{a}+a\right)-\left(a+b+c\right)\ge2a+2b+2c-a-b-c=a+b+c\).
Tương tự, \(\dfrac{b^2}{a}+\dfrac{c^2}{b}+\dfrac{a^2}{c}\ge a+b+c\).
Do đó \(2P\ge a+b+c+\dfrac{4}{a}+\dfrac{4}{b}+\dfrac{4}{c}=\left(a+\dfrac{4}{a}\right)+\left(b+\dfrac{4}{b}\right)+\left(c+\dfrac{4}{c}\right)\ge4+4+4=12\Rightarrow P\ge6\).
Đẳng thức xảy ra khi a = b = c = 2 hay x = y = z = 4.
Vậy Min P = 6 khi x = y = z = 4.
\(P=\dfrac{4x}{2.2.\sqrt{y+z-4}}+\dfrac{4y}{2.2.\sqrt{x+z-4}}+\dfrac{4z}{2.2.\sqrt{x+y-4}}\)
\(P\ge4\left(\dfrac{x}{y+z}+\dfrac{y}{z+x}+\dfrac{z}{x+y}\right)\ge4.\dfrac{3}{2}=6\)
Dấu "=" xảy ra khi \(x=y=z=4\)
TA có \(P^2=\dfrac{x^2}{y}+\dfrac{y^2}{z}+\dfrac{z^2}{x}+2\left(\dfrac{xy}{\sqrt{yz}}+\dfrac{yz}{\sqrt{zx}}+\dfrac{zx}{\sqrt{xy}}\right)\)
Áp dụng BĐt AM-GM, ta có \(\dfrac{x^2}{y}+\dfrac{xy}{\sqrt{yz}}+\dfrac{xy}{\sqrt{yz}}+z\ge4x\)
tương tự rồi cộng lại, ta có \(P^2+\left(x+y+z\right)\ge4\left(x+y+z\right)\Rightarrow P^2\ge3\left(x+y+z\right)\ge36\Rightarrow P\ge6\)
dấu = xảy ra <=> x=y=z=4