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a) Ta có BĐT:
\(a^3+b^3=\left(a+b\right)\left(a^2+b^2-ab\right)\ge\left(a+b\right)ab\)
\(\Rightarrow a^3+b^3+abc\ge ab\left(a+b+c\right)\)
\(\Rightarrow\frac{1}{a^3+b^3+abc}\le\frac{1}{ab\left(a+b+c\right)}\)
Tương tự cho 2 bất đẳng thức còn lại rồi cộng theo vế:
\(VT\le\frac{1}{ab\left(a+b+c\right)}+\frac{1}{bc\left(a+b+c\right)}+\frac{1}{ca\left(a+b+c\right)}\)
\(=\frac{a+b+c}{abc\left(a+b+c\right)}=\frac{1}{abc}=VP\)
Khi \(a=b=c\)
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Áp dụng bđt Cauchy, ta có:
\(\sqrt{\frac{a}{bc}}\)+\(\sqrt{\frac{b}{ca}}\)≥ \(2\sqrt{\sqrt{\frac{ab}{abc^2}}}\)= \(2\sqrt{\sqrt{\frac{1}{c^2}}}\)= \(2\sqrt{\frac{1}{c}}\) (vì c>0)
Tương tự: \(\sqrt{\frac{b}{ca}}\)+\(\sqrt{\frac{c}{ab}}\)≥ \(2\sqrt{\frac{1}{a}}\)
\(\sqrt{\frac{c}{ab}}\)+\(\sqrt{\frac{a}{bc}}\)≥ \(2\sqrt{\frac{1}{b}}\)
Cộng vế theo vế của các bđt với nhau, ta có: \(2\)\(\left(\sqrt{\frac{a}{bc}}+\sqrt{\frac{b}{ca}}+\sqrt{\frac{c}{ab}}\right)\text{≥}\)\(2\left(\sqrt{\frac{1}{a}}+\sqrt{\frac{1}{b}}+\sqrt{\frac{1}{c}}\right)\)
<=> \(\sqrt{\frac{a}{bc}}+\sqrt{\frac{b}{ca}}+\sqrt{\frac{c}{ab}}\text{≥}\)\(\sqrt{\frac{1}{a}}+\sqrt{\frac{1}{b}}+\sqrt{\frac{1}{c}}\)(đpcm)
Dấu "=" xảy ra <=> a = b = c
gt <=> \(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}=1\)
Đặt: \(\frac{1}{a}=x;\frac{1}{b}=y;\frac{1}{c}=z\)
=> Thay vào thì \(VT=\frac{\frac{1}{xy}}{\frac{1}{z}\left(1+\frac{1}{xy}\right)}+\frac{1}{\frac{yz}{\frac{1}{x}\left(1+\frac{1}{yz}\right)}}+\frac{1}{\frac{zx}{\frac{1}{y}\left(1+\frac{1}{zx}\right)}}\)
\(VT=\frac{z}{xy+1}+\frac{x}{yz+1}+\frac{y}{zx+1}=\frac{x^2}{xyz+x}+\frac{y^2}{xyz+y}+\frac{z^2}{xyz+z}\ge\frac{\left(x+y+z\right)^2}{x+y+z+3xyz}\)
Có BĐT x, y, z > 0 thì \(\left(x+y+z\right)\left(xy+yz+zx\right)\ge9xyz\)Ta thay \(xy+yz+zx=1\)vào
=> \(x+y+z\ge9xyz=>\frac{x+y+z}{3}\ge3xyz\)
=> Từ đây thì \(VT\ge\frac{\left(x+y+z\right)^2}{x+y+z+\frac{x+y+z}{3}}=\frac{3}{4}\left(x+y+z\right)\ge\frac{3}{4}.\sqrt{3\left(xy+yz+zx\right)}=\frac{3}{4}.\sqrt{3}=\frac{3\sqrt{3}}{4}\)
=> Ta có ĐPCM . "=" xảy ra <=> x=y=z <=> \(a=b=c=\sqrt{3}\)
\(\left(a+b\right)\left(b+c\right)\left(c+a\right)+abc\)
\(=abc+a^2b+ab^2+a^2c+ac^2+b^2c+bc^2+abc+abc\)
\(=\left(a+b+c\right)\left(ab+bc+ca\right)\)( phân tích nhân tử các kiểu )
\(\Rightarrow\left(a+b\right)\left(b+c\right)\left(c+a\right)\ge\left(a+b+c\right)\left(ab+bc+ca\right)-abc\left(1\right)\)
\(a+b+c\ge3\sqrt[3]{abc};ab+bc+ca\ge3\sqrt[3]{a^2b^2c^2}\Rightarrow\left(a+b+c\right)\left(ab+bc+ca\right)\ge9abc\)
\(\Rightarrow-abc\ge\frac{-\left(a+b+c\right)\left(ab+bc+ca\right)}{9}\)
Khi đó:\(\left(a+b+c\right)\left(ab+bc+ca\right)-abc\)
\(\ge\left(a+b+c\right)\left(ab+bc+ca\right)-\frac{\left(a+b+c\right)\left(ab+bc+ca\right)}{9}\)
\(=\frac{8\left(a+b+c\right)\left(ab+bc+ca\right)}{9}\left(2\right)\)
Từ ( 1 ) và ( 2 ) có đpcm
Bài 1:
\(BDT\Leftrightarrow\sqrt{\frac{3}{a+2b}}+\sqrt{\frac{3}{b+2c}}+\sqrt{\frac{3}{c+2a}}\le\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}+\frac{1}{\sqrt{c}}\)
\(\Leftrightarrow\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}+\frac{1}{\sqrt{c}}\ge\sqrt{3}\left(\frac{1}{\sqrt{a+2b}}+\frac{1}{\sqrt{b+2c}}+\frac{1}{\sqrt{c+2a}}\right)\)
Áp dụng BĐT Cauchy-Schwarz và BĐT AM-GM ta có:
\(\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}+\frac{1}{\sqrt{b}}\ge\frac{9}{\sqrt{a}+\sqrt{2}\cdot\sqrt{2b}}\ge\frac{9}{\sqrt{\left(1+2\right)\left(a+2b\right)}}=\frac{3\sqrt{3}}{\sqrt{a+2b}}\)
Tương tự cho 2 BĐT còn lại ta cũng có:
\(\frac{1}{\sqrt{b}}+\frac{1}{\sqrt{c}}+\frac{1}{\sqrt{c}}\ge\frac{3\sqrt{3}}{\sqrt{b+2c}};\frac{1}{\sqrt{c}}+\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{a}}\ge\frac{3\sqrt{3}}{\sqrt{c+2a}}\)
Cộng theo vế 3 BĐT trên ta có:
\(3\left(\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}+\frac{1}{\sqrt{c}}\right)\ge3\sqrt{3}\left(\frac{1}{\sqrt{a+2b}}+\frac{1}{\sqrt{b+2c}}+\frac{1}{\sqrt{c+2a}}\right)\)
\(\Leftrightarrow\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}+\frac{1}{\sqrt{c}}\ge\sqrt{3}\left(\frac{1}{\sqrt{a+2b}}+\frac{1}{\sqrt{b+2c}}+\frac{1}{\sqrt{c+2a}}\right)\)
Đẳng thức xảy ra khi \(a=b=c\)
Bài 2: làm mãi ko ra hình như đề sai, thử a=1/2;b=4;c=1/2
Bài 2/
\(\frac{bc}{a^2b+a^2c}+\frac{ca}{b^2c+b^2a}+\frac{ab}{c^2a+c^2b}\)
\(=\frac{b^2c^2}{a^2b^2c+a^2c^2b}+\frac{c^2a^2}{b^2c^2a+b^2a^2c}+\frac{a^2b^2}{c^2a^2b+c^2b^2a}\)
\(=\frac{b^2c^2}{ab+ac}+\frac{c^2a^2}{bc+ba}+\frac{a^2b^2}{ca+cb}\)
\(\ge\frac{\left(bc+ca+ab\right)^2}{2\left(ab+bc+ca\right)}=\frac{ab+bc+ca}{2}\)
\(\ge\frac{3\sqrt[3]{ab.bc.ca}}{2}=\frac{3}{2}\)
Dấu = xảy ra khi \(a=b=c=1\)
Đặt đẳng thức là A. Áp dụng bất đẳng thức AM-GM ta có:
\(\sqrt{2b\left(a-b\right)}\le\frac{2b+\left(a+b\right)}{2}=\frac{a+3b}{2}\)
Từ đó: \(A\ge\frac{2a\sqrt{2}}{a+3b}+\frac{2b\sqrt{2}}{b+3c}+\frac{2c\sqrt{2}}{c+3a}\)
Ta sẽ chứng minh: \(M=\frac{a}{a+3b}+\frac{b}{b+3c}+\frac{c}{c+3a}\ge\frac{3}{4}\)
Thật vậy, ta có: \(M=\frac{a^2}{a^2+3ab}+\frac{b^2}{b^2+3bc}+\frac{c^2}{c^2+3ca}\)
Theo BĐT AM-GM ta có:
\(ab+bc+ca\le a^2+b^2+c^2\)
Áp dụng BĐT cauchy ta được:
\(M\ge\frac{\left(a+b+c\right)^2}{\frac{4}{3}\left(a^2+b^2+c^2\right)+\frac{8}{3}\left(ab+bc+ca\right)}\)\(=\frac{\left(a+b+c\right)^2}{\frac{4}{3}\left(a+b+c\right)^2}=\frac{3}{4}\)
Vì vậy: \(\frac{a}{a+3b}+\frac{b}{b+3c}+\frac{c}{c+3a}\ge\frac{3}{4}\)
Từ đó ta có: \(A\ge\frac{2a\sqrt{2}}{a+3b}+\frac{2b\sqrt{2}}{b+3c}+\frac{2c\sqrt{2}}{c+3a}\ge2\sqrt{2}.\frac{3}{4}=\frac{3\sqrt{2}}{2}\)
Vậy đẳng thức xảy xa khi và chỉ khi a=b=c
Đặt \(a^2=x;b^2=y;c^2=z\)
Ta có:
\(VT=\sqrt{\frac{x}{x+y}}+\sqrt{\frac{y}{y+z}}+\sqrt{\frac{z}{z+x}}\)
Mặt khác:
\(\sqrt{\frac{x}{x+y}}=\sqrt{\frac{x}{\left(x+y\right)\left(x+z\right)}\cdot\sqrt{x+z}}\)
Áp dụng Bđt Cauchy-Schwarz ta có:
\(VT^2\le2\left[\frac{x}{\left(x+y\right)\left(x+z\right)}+\frac{y}{\left(y+z\right)\left(y+x\right)}+\frac{z}{\left(z+x\right)\left(z+y\right)}\right]\left(x+y+z\right)\)
\(\Leftrightarrow VT^2\le\frac{4\left(x+y+z\right)\left(xy+yz+zx\right)}{\left(x+y\right)\left(y+z\right)\left(z+x\right)}\)
Vì \(VP^2=\frac{9}{2}\) nên cần chứng minh \(VT^2\le\frac{9}{2}\)
\(\Leftrightarrow9\left(x+y\right)\left(y+z\right)\left(z+x\right)\ge8\left(x+y+z\right)\left(xy+yz+zx\right)\)
bn tự lm tiếp