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HF
14 tháng 8 2020
\(P=\frac{\sqrt{1+x^2+y^2}}{xy}+\frac{\sqrt{1+y^2+z^2}}{yz}+\frac{\sqrt{1+z^2+x^2}}{zx}\)
\(\ge\text{Σ}\frac{\sqrt{\frac{\left(1+x+y\right)^2}{3}}}{xy}\text{=}\frac{1+x+y}{xy\sqrt{3}}\)
\(=\frac{\sqrt{3}}{3}\left(\frac{1+x+y}{xy}+\frac{1+y+z}{yz}+\frac{1+z+x}{zx}\right)\)
\(=\frac{\sqrt{3}}{3}\left(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{xz}+\frac{1}{x}+\frac{1}{y}+\frac{1}{y}+\frac{1}{z}+\frac{1}{z}+\frac{1}{x}\right)\)
\(=\frac{\sqrt{3}}{3}\left(x+y+z+2xy+2yz+2zx\right)\)\(\ge\frac{\sqrt{3}}{3}\left(3\sqrt[3]{xyz}+2\cdot3\sqrt[3]{x^2y^2z^2}\right)=\frac{\sqrt{3}}{3}\left(3+6\right)=3\sqrt{3}\)
Dấu = xảy ra khi \(x=y=z=1\)
HF
18 tháng 8 2020
+) \(P=\frac{x^2}{y^2+yz+z^2}+\frac{y^2}{x^2+xz+z^2}+\frac{z^2}{x^2+xy+y^2}\)
\(\ge\text{Σ}\frac{x^2}{y^2+\frac{y^2+z^2}{2}+z^2}=\frac{2}{3}\text{Σ}\frac{x^2}{y^2+z^2}\)
+) Đặt \(a=x^2;b=y^2;c=z^2\)
Ta có: \(A=\text{Σ}\frac{x^2}{y^2+z^2}=\text{Σ}\frac{a}{b+c}=\text{Σ}\frac{a^2}{ab+ac}\ge\frac{\left(a+b+c\right)^2}{2\left(ab+bc+ac\right)}\ge\frac{3}{2}\)(BDT Nesbitt)
Vậy \(P=\frac{2}{3}A\ge1\)
Dấu = xảy ra khi x = y = z
TH
2
26 tháng 4 2020
Ta có \(P=\frac{x\left(yz+1\right)^2}{z^2\left(zx+1\right)}+\frac{y\left(zx+1\right)^2}{x^2\left(xy+1\right)}+\frac{z\left(xy+1\right)^2}{y^2\left(yz+1\right)}\)
\(=\frac{\frac{\left(yz+1\right)^2}{z^2}}{\frac{zx+1}{x}}+\frac{\frac{\left(zx+1\right)^2}{x^2}}{\frac{xy+1}{y}}+\frac{\frac{\left(xy+1\right)^2}{y^2}}{\frac{yz+1}{z}}\)
\(=\frac{\left(y+\frac{1}{z}\right)^2}{z+\frac{1}{x}}+\frac{\left(z+\frac{1}{x}\right)^2}{x+\frac{1}{y}}+\frac{\left(x+\frac{1}{y}\right)^2}{y+\frac{1}{z}}\)
Áp dụng BĐT \(\frac{a_1^2}{b_1}+\frac{a_2^2}{b_2}+\frac{a_3^2}{b_3}\ge\frac{\left(a_1+a_2+a_3\right)^2}{b_1+b_2+b_3}\)
Dấu "=" xảy ra khi \(\frac{a_1}{b_1}=\frac{a_2}{b_2}=\frac{a_3}{c_3}\)
\(P=\frac{\left(y+\frac{1}{z}\right)^2}{z+\frac{1}{x}}+\frac{\left(z+\frac{1}{x}\right)^2}{x+\frac{1}{y}}+\frac{\left(x+\frac{1}{y}\right)^2}{y+\frac{1}{z}}\ge\frac{\left(x+y+z+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)^2}{\left(x+y+z+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)}\)
\(P\ge a+y+z+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\)
Áp dụng BĐT: \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\ge\frac{9}{x+y+z}\)
=> \(P\ge x+y+z+\frac{9}{x+y+z}=\left[x+y+z+\frac{9}{4\left(x+y+z\right)}\right]+\frac{27}{4\left(x+y+z\right)}\)
Ta có: \(x+y+z+\frac{9}{4\left(x+y+z\right)}\ge2\sqrt{\frac{9}{4}}=3;\frac{27}{4\left(x+y+z\right)}=\frac{27}{4\cdot\frac{3}{2}}=\frac{9}{2}\)
=> \(P\ge3+\frac{9}{2}=\frac{15}{2}\).
Dấu "=" xảy ra <=> x=y=z=\(\frac{1}{2}\)
Vậy MinP=\(\frac{15}{2}\)đạt được khi x=y=z=\(\frac{1}{2}\)
KN
26 tháng 4 2020
Ta có:
\(P=\frac{x\left(yz+1\right)^2}{z^2\left(zx+1\right)}+\frac{y\left(zx+1\right)^2}{x^2\left(xy+1\right)}+\frac{z\left(xy+1\right)^2}{y^2\left(yz+1\right)}\)
\(=\frac{\left(\frac{yz+1}{z}\right)^2}{\left(\frac{zx+1}{x}\right)}+\frac{\left(\frac{zx+1}{x}\right)^2}{\left(\frac{xy+1}{y}\right)}+\frac{\left(\frac{xy+1}{y}\right)^2}{\left(\frac{yz+1}{z}\right)}\)
\(=\frac{\left(y+\frac{1}{z}\right)^2}{z+\frac{1}{x}}+\frac{\left(z+\frac{1}{x}\right)^2}{x+\frac{1}{y}}+\frac{\left(x+\frac{1}{y}\right)^2}{y+\frac{1}{z}}\)
Áp dụng BĐT Bunhiacopxki dạng phân thức, ta có:
\(\frac{\left(y+\frac{1}{z}\right)^2}{z+\frac{1}{x}}+\frac{\left(z+\frac{1}{x}\right)^2}{x+\frac{1}{y}}+\frac{\left(x+\frac{1}{y}\right)^2}{y+\frac{1}{z}}\)\(\ge\frac{\left(x+y+z+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)^2}{x+y+z+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}}=x+y+z+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\)
\(\ge\left(x+y+z\right)+\frac{9}{x+y+z}=\left(x+y+z\right)+\frac{9}{4\left(x+y+z\right)}\)
\(+\frac{27}{4\left(x+y+z\right)}\ge2\sqrt{\left(x+y+z\right).\frac{9}{4\left(x+y+z\right)}}+\frac{27}{4.\frac{3}{2}}=\frac{15}{2}\)(Áp dụng BĐT Cô - si cho 2 số không âm)
Đẳng thức xảy ra khi \(x=y=z=\frac{1}{2}\)
IT
cho số thực:x, y, z thỏa mãn: \(y^2+yz+z^2=1-\frac{3x^2}{2}\). tìm Max và Min của biểu thức: A=x+y+z
1
AH
Akai Haruma
Giáo viên
16 tháng 1 2017
Lời giải:
ĐKĐB \(\Leftrightarrow \frac{3x^2}{2}+y^2+yz+z^2=1\)
Áp dụng BĐT Am-Gm ta có \(yz\leq \left (\frac{y+z}{2}\right)^2\)
\(\Rightarrow 1=\frac{3x^2}{2}+y^2+yz+z^2=\frac{3x^2}{2}+(y+z)^2-yz\geq \frac{3x^2}{2}+\frac{3(y+z)^2}{4}\)
\(\Leftrightarrow \frac{2}{3}\geq x^2+\frac{(y+z)^2}{2}\)
Áp dụng BĐT Cauchy- Schwarz: \(3\left [x^2+\frac{(y+z)^2}{2}\right]=\left [x^2+\frac{(y+z)^2}{2}\right](1+2)\geq (x+y+z)^2\)
\(\Rightarrow 2\geq 3\left [x^2+\frac{(y+z)^2}{2}\right]\geq (x+y+z)^2\Rightarrow -\sqrt{2}\leq x+y+z\leq \sqrt{2}\)
Vậy
\(x+y+z (\max)=\sqrt{2}\Leftrightarrow (x,y,z)=\left (\frac{\sqrt{2}}{3},\frac{\sqrt{2}}{3},\frac{\sqrt{2}}{3}\right)\)
\(x+y+z(\min)=-\sqrt{2}\Leftrightarrow (x,y,z)=\left(\frac{-\sqrt{2}}{3},\frac{-\sqrt{2}}{3},\frac{-\sqrt{2}}{3}\right)\)
NT
4
\(GT\Leftrightarrow3x^2+y^2+z^2+\left(y+z\right)^2=2\)
Áp dụng BĐT bunyakovsky:\(y^2+z^2\ge\frac{1}{2}\left(y+z\right)^2\)
\(2\ge\frac{3}{2}\left(y+z\right)^2+3x^2\Leftrightarrow4\ge3\left(y+z\right)^2+6x^2=3\left[\left(y+z\right)^2+2x^2\right]\)
\(\left(2+1\right)\left[\left(y+z\right)^2+2x^2\right]\ge2\left(x+y+z\right)^2\)
\(\left(x+y+z\right)^2\le2\Leftrightarrow-\sqrt{2}\le x+y+z\le\sqrt{2}\)