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\(M=\frac{2x+y+z-15}{x}+\frac{x+2y+z-15}{y}+\frac{x+y+2z-15}{z}\)
\(M-3=\frac{x+y+z-15}{x}+\frac{x+y+z-15}{y}+\frac{x+y+z-15}{z}\)
\(M-3=\left(x+y+z-15\right)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\)
\(\Rightarrow M\ge\left(x+y+z-15\right)\cdot\frac{9}{x+y+z}+3=\frac{3}{4}\)
\("="\Leftrightarrow x=y=z=4\)
Bài 1
M=2x+y+z−15x+x+2y+z−15y+x+y+2z−15z
M=x+12−15x+y+12−15y+z+12−15z
M=x−3x+y−3y+z−3z
M=1−3x+1−3y+1−3z
M=3−(3x+3y+3z)
M=3−3(1x+1y+1z)
Áp dụng bất đẳng thức Cauchy - Schwarz dạng phân thức
⇒1x+1y+1z≥(1+1+1)2x+y+z=9x+y+z=34
⇒3(1x+1y+1z)≥94
⇒3−3(1x+1y+1z)≤34
⇔M≤34
Vậy M max=34
Dấu " = " xảy ra khi x=y=z=4
Bai nay tim GTLN moi dung nha
Ta có: \(x^2\left(y+z\right)\ge x^2.2\sqrt{yz}=2\sqrt{x^4}.\sqrt{\frac{1}{x}}=2x\sqrt{x}\)(Áp dụng BĐT Cô - si cho 2 số dương y,z và sử dụng giả thiết xyz = 1)
Hoàn toàn tương tự: \(y^2\left(z+x\right)\ge2y\sqrt{y};z^2\left(x+y\right)\ge2z\sqrt{z}\)
Do đó \(P=\frac{x^2\left(y+z\right)}{y\sqrt{y}+2z\sqrt{z}}+\frac{y^2\left(z+x\right)}{z\sqrt{z}+2x\sqrt{x}}+\frac{z^2\left(x+y\right)}{x\sqrt{x}+2y\sqrt{y}}\)
\(\ge\frac{2x\sqrt{x}}{y\sqrt{y}+2z\sqrt{z}}+\frac{2y\sqrt{y}}{z\sqrt{z}+2x\sqrt{x}}+\frac{2z\sqrt{z}}{x\sqrt{x}+2y\sqrt{y}}\)
Đặt \(a=x\sqrt{x}+2y\sqrt{y}\), \(b=y\sqrt{y}+2z\sqrt{z}\), \(c=z\sqrt{z}+2x\sqrt{x}\)
Suy ra: \(x\sqrt{x}=\frac{4c+a-2b}{9}\), \(y\sqrt{y}=\frac{4a+b-2c}{9}\), \(z\sqrt{z}=\frac{4b+c-2a}{9}\)
Do đó \(P\ge\frac{2}{9}\left(\frac{4c+a-2b}{b}+\frac{4a+b-2c}{c}+\frac{4b+c-2a}{a}\right)\)
\(=\frac{2}{9}\left[4\left(\frac{c}{b}+\frac{a}{c}+\frac{b}{a}\right)+\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)-6\right]\)
\(\ge\frac{2}{9}\left[4.3\sqrt[3]{\frac{c}{b}.\frac{a}{c}.\frac{b}{a}}+3\sqrt[3]{\frac{a}{b}.\frac{b}{c}.\frac{c}{a}}-6\right]\)(Áp dụng BĐT Cô - si cho 3 số dương)
\(=\frac{2}{9}\left[4.3+3-6\right]=2\)
Vậy \(P\ge2\)
Đẳng thức xảy ra khi x = y = z = 1
\(ĐK:x,y,z>\frac{1}{2}\)
Ta có: \(\left(x+2y\right)^2=\left(\frac{3y}{2}+\frac{y+2x}{2}\right)^2\ge4.\frac{3y}{2}.\frac{y+2x}{2}=3y\left(2x+y\right)\)\(\Rightarrow\frac{2x+y}{x+2y}\le\frac{x+2y}{3y}\Rightarrow\frac{2x+y}{x\left(x+2y\right)}\le\frac{x+2y}{3xy}=\frac{1}{3}\left(\frac{2}{x}+\frac{1}{y}\right)\)
Tương tự: \(\frac{2y+z}{y\left(y+2z\right)}\le\frac{1}{3}\left(\frac{2}{y}+\frac{1}{z}\right)\); \(\frac{2z+x}{z\left(z+2x\right)}\le\frac{1}{3}\left(\frac{2}{z}+\frac{1}{x}\right)\)
Cộng theo vế ba bất đẳng thức trên, ta được: \(VT\le\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\le\frac{1}{\sqrt{2x-1}}+\frac{1}{\sqrt{2y-1}}+\frac{1}{\sqrt{2z-1}}=3\)
Đẳng thức xảy ra khi x = y = z = 1
Lời giải:
Áp dụng BĐT Cauchy ngược dấu:
\(\frac{2y+2z-x}{x}.3\leq \left(\frac{\frac{2y+2z-x}{x}+3}{2}\right)^2=\left(\frac{y+x+z}{x}\right)^2\)
\(\Rightarrow \frac{2y+2z-x}{x}\leq \frac{1}{3}\left(\frac{x+y+z}{x}\right)^2\)
\(\Rightarrow \sqrt{\frac{x}{2y+2z-x}}\geq \frac{\sqrt{3}x}{x+y+z}\)
Hoàn toàn tương tự:
\(\sqrt{\frac{y}{2x+2z-y}}\geq \frac{\sqrt{3}y}{x+y+z}; \sqrt{\frac{z}{2x+2y-z}}\geq \frac{\sqrt{3}z}{x+y+z}\)
Cộng theo vế những BĐT trên ta có:
\(\Rightarrow P\geq \frac{\sqrt{3}(x+y+z)}{x+y+z}=\sqrt{3}\)
Vậy \(P_{\min}=\sqrt{3}\). Dấu "=" xảy ra khi \(x=y=z\)
\(ĐKXĐ:x,y,z\ge1\left(x,y,z\inℤ\right)\)
Ta có: \(\left(x+2y\right)^2=\left(\frac{2x+y}{2}+\frac{3y}{2}\right)^2\ge4.\frac{2x+y}{2}.\frac{3y}{2}=3y\left(2x+y\right)\)
\(\Rightarrow\frac{2x+y}{x+2y}\le\frac{x+2y}{3y}\Rightarrow\frac{2x+y}{x\left(x+2y\right)}\le\frac{1}{3}\left(\frac{2}{x}+\frac{1}{y}\right)\)
Tương tự: \(\frac{2y+z}{y\left(y+2x\right)}\le\frac{1}{3}\left(\frac{2}{y}+\frac{1}{z}\right)\);\(\frac{2z+x}{z\left(z+2x\right)}\le\frac{1}{3}\left(\frac{2}{z}+\frac{1}{x}\right)\)
\(\Rightarrow A\le\frac{1}{3}.3\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\)(*)
Ta có: \(\sqrt{2x-1}=\sqrt{\left(2x-1\right).1}\le\frac{2x-1+1}{2}=x\)(BĐT Cô - si)
\(\Rightarrow\frac{1}{x}\le\frac{1}{\sqrt{2x-1}}\)
Tương tự: \(\frac{1}{y}\le\frac{1}{\sqrt{2y-1}}\);\(\frac{1}{z}\le\frac{1}{\sqrt{2z-1}}\)
\(\Rightarrow\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\le\frac{1}{\sqrt{2x-1}}+\frac{1}{\sqrt{2y-1}}+\frac{1}{\sqrt{2z-1}}=3\)(**)
Từ (*) và (**) suy ra \(A=\frac{2x+y}{x\left(x+2y\right)}+\frac{2y+z}{y\left(y+2z\right)}+\frac{2z+x}{z\left(z+2x\right)}\le3\)
Đẳng thức xảy ra khi x = y = z = 1
Từ đẳng thức đã cho suy ra \(x>\frac{1}{2};y>\frac{1}{2};z>\frac{1}{2}\)
Áp dụng\(\left(a+b\right)^2\ge4ab\)ta có \(\left(x+2y\right)^2=\left(\frac{2x+y}{2}+\frac{3y}{2}\right)^2\ge4\cdot\frac{2x+y}{2}\cdot\frac{3y}{2}\)
\(\Rightarrow\left(x+2y\right)^2\ge3y\left(2x+y\right)\)(Dấu "=" xảy ra <=> x=y)
=> \(\frac{2x+y}{x+2y}\le\frac{x+2y}{3y}\Rightarrow\frac{2x+y}{x\left(x+2y\right)}\le\frac{1}{3}\left(\frac{2}{x}+\frac{1}{y}\right)\)
Tương tự \(\hept{\begin{cases}\frac{2y+z}{y\left(y+2z\right)}\le\frac{1}{3}\left(\frac{2}{y}+\frac{1}{z}\right)\\\frac{2z+x}{z\left(z+2x\right)}\le\frac{1}{3}\left(\frac{2}{z}+\frac{1}{x}\right)\end{cases}}\)
=> \(A\le\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\)(Dấu "=" xảy ra <=> x=y=z)
Ta có \(\sqrt{\left(2x-1\right)\cdot1}\le\frac{\left(2x-1\right)+1}{2}\Rightarrow\sqrt{2x-1}\le x\Rightarrow\frac{1}{x}\le\frac{1}{\sqrt{2x-1}}\)
Tương tự \(\hept{\begin{cases}\frac{1}{y}\le\frac{1}{\sqrt{2y-1}}\\\frac{1}{z}\le\frac{1}{\sqrt{2z-1}}\end{cases}}\)
Do đó \(A\le\frac{1}{\sqrt{2x-1}}+\frac{1}{\sqrt{2y-1}}+\frac{1}{\sqrt{2z-1}}=3\)(dấu "=" xảy ra <=> x=y=z=1)
Vậy MaxA=3 đạt được khi x=y=z=1
\(A=\frac{x}{x+1}+\frac{y}{y+1}+\frac{z}{z+1}\).Áp dụng BĐT Cauchy-Schwarz,ta có:
\(=\left(1-\frac{1}{x+1}\right)+\left(1-\frac{1}{y+1}\right)+\left(1-\frac{1}{z+1}\right)\)
\(=\left(1+1+1\right)-\left(\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}\right)\)
\(\ge3-\frac{9}{\left(x+y+z\right)+\left(1+1+1\right)}=\frac{3}{4}\)
Dấu "=" xảy ra khi x = y = z = 1/3
Vậy A min = 3/4 khi x=y=z=1/3
\(\frac{x^2}{2y}+\frac{y^2}{2x}+\frac{y^2}{2z}+\frac{z^2}{2y}+\frac{z^2}{2x}+\frac{x^2}{2z}\ge\frac{\left(2x+2y+2z\right)^2}{4\left(x+y+z\right)}=x+y+z\)
\(A=\frac{x^2}{xy+2xz}+\frac{y^2}{zy+2xy}+\frac{z^2}{xz+2yz}\)
\(A\ge\frac{\left(x+y+z\right)^2}{3\left(xy+yz+zx\right)}\left(cauchy-schwarz\right)\)
Sử dụng đánh giá quen thuộc:\(\left(x+y+z\right)^2\ge3\left(xy+yz+zx\right)\)
\(\Rightarrow A\ge1\)
"="<=>x=y=z