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C/m BĐT : \(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\ge\dfrac{9}{x+y+z}\)
Áp dụng BĐT Sơ-vác-sơ:
\(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\ge\dfrac{\left(1+1+1\right)^2}{x+y+z}\ge\dfrac{9}{x+y+z}\)
Ta có: \(9\dfrac{ab}{a+3b+2c}=\dfrac{9ab}{\left(a+c\right)+\left(b+c\right)+2b}\le\dfrac{ab}{a+c}+\dfrac{ab}{b+c}+\dfrac{a}{2}\left(1\right)\)
CM tương tự
\(\dfrac{9bc}{b+3c+2a}\le\dfrac{bc}{a+c}+\dfrac{bc}{a+b}+\dfrac{b}{2}\left(2\right)\)
\(\dfrac{9ca}{c+3a+2b}\le\dfrac{ca}{b+c}+\dfrac{ca}{a+b}+\dfrac{c}{2}\left(3\right)\)
Cộng vế (1), (2), (3) => đpcm
Áp dụng bất đẳng thức \(\dfrac{1}{a+b}\le\dfrac{1}{4}\left(\dfrac{1}{a}+\dfrac{1}{b}\right)\) với a , b > 0
\(\Rightarrow\left\{{}\begin{matrix}\dfrac{bc}{2a+b+c}=\dfrac{bc}{a+b+a+c}\le\dfrac{bc}{4}\left(\dfrac{1}{a+b}+\dfrac{1}{a+c}\right)\\\dfrac{ca}{a+2b+c}=\dfrac{ca}{a+b+b+c}\le\dfrac{ca}{4}\left(\dfrac{1}{a+b}+\dfrac{1}{b+c}\right)\\\dfrac{ab}{a+b+2c}=\dfrac{ab}{a+c+b+c}\le\dfrac{ab}{4}\left(\dfrac{1}{a+c}+\dfrac{1}{b+c}\right)\end{matrix}\right.\)
\(\Rightarrow VT\le\dfrac{bc}{4}\left(\dfrac{1}{a+b}+\dfrac{1}{a+c}\right)+\dfrac{ca}{4}\left(\dfrac{1}{a+b}+\dfrac{1}{b+c}\right)+\dfrac{ab}{4}\left(\dfrac{1}{a+c}+\dfrac{1}{b+c}\right)\)
\(\Rightarrow VT\le\dfrac{bc}{4\left(a+b\right)}+\dfrac{bc}{4\left(a+c\right)}+\dfrac{ca}{4\left(a+b\right)}+\dfrac{ca}{4\left(b+c\right)}+\dfrac{ab}{4\left(a+c\right)}+\dfrac{ab}{4\left(b+c\right)}\)
\(\Rightarrow VT\le\left[\dfrac{bc}{4\left(a+b\right)}+\dfrac{ca}{4\left(a+b\right)}\right]+\left[\dfrac{bc}{4\left(a+c\right)}+\dfrac{ab}{4\left(a+c\right)}\right]+\left[\dfrac{ca}{4\left(b+c\right)}+\dfrac{ab}{4\left(b+c\right)}\right]\)
\(\Rightarrow VT\le\dfrac{bc+ca}{4\left(a+b\right)}+\dfrac{bc+ab}{4\left(a+c\right)}+\dfrac{ca+ab}{4\left(b+c\right)}\)
\(\Rightarrow VT\le\dfrac{c\left(a+b\right)}{4\left(a+b\right)}+\dfrac{b\left(c+a\right)}{4\left(a+c\right)}+\dfrac{a\left(b+c\right)}{4\left(b+c\right)}\)
\(\Rightarrow VT\le\dfrac{a+b+c}{4}\)
\(\Leftrightarrow\dfrac{bc}{2a+b+c}+\dfrac{ca}{a+2b+c}+\dfrac{ab}{a+b+2c}\le\dfrac{a+b+c}{4}\) ( đpcm )
a.
\(\sum\dfrac{ab}{a+c+b+c}\le\dfrac{1}{4}\sum\left(\dfrac{ab}{a+c}+\dfrac{ab}{b+c}\right)=\dfrac{a+b+c}{4}\)
2.
\(\dfrac{ab}{a+3b+2c}=\dfrac{ab}{a+b+2c+2b}\le\dfrac{ab}{9}\left(\dfrac{4}{a+b+2c}+\dfrac{1}{2b}\right)=4.\dfrac{ab}{a+b+2c}+\dfrac{a}{18}\)
Quay lại câu a
Bổ đề :\(\left(x+y+z\right)\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)\ge9\)
Áp dụng bất đẳng thức Cô-si ta có:
\(x+y+z\ge3\sqrt[3]{xyz};\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\ge3\sqrt[3]{\dfrac{1}{x}.\dfrac{1}{y}.\dfrac{1}{z}}\)
\(\Rightarrow\left(x+y+z\right)\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)\ge3\sqrt[3]{xyz}.3\sqrt[3]{\dfrac{1}{x}\dfrac{1}{y}\dfrac{1}{z}}=9\)
Dấu "=" xảy ra ⇔ x=y=z
Ta có:\(\dfrac{ab}{a+3b+2c}=\dfrac{ab}{9}.\dfrac{9}{a+3b+2c}\le\dfrac{ab}{9}.\left(\dfrac{1}{a+c}+\dfrac{1}{b+c}+\dfrac{1}{2b}\right)\)
Tương tự ta có:\(\dfrac{bc}{b+3c+2a}\le\dfrac{bc}{9}\left(\dfrac{1}{b+a}+\dfrac{1}{c+a}+\dfrac{1}{2c}\right)\)
\(\dfrac{ca}{c+3a+2b}\le\dfrac{ca}{9}.\left(\dfrac{1}{c+b}+\dfrac{1}{a+b}+\dfrac{1}{2a}\right)\)
Cộng vế với vế ta có:
\(A\le\dfrac{1}{9}.\left(\dfrac{ab+bc}{a+c}+\dfrac{cb+ac}{a+b}+\dfrac{ca+ab}{b+c}+\dfrac{a+b+c}{2}\right)\)
\(=\dfrac{1}{9}.\left(a+b+c+\dfrac{a+b+c}{2}\right)=\dfrac{1}{9}.\left(6+\dfrac{6}{3}\right)=1\)
Dấu "=" xảy ra ⇔ a=b=c=2
Vậy Max A=1⇔ a=b=c=2
Lời giải:
Ta có: \(a^2b+b^2c+c^2a\geq \frac{9a^2b^2c^2}{1+2a^2b^2c^2}\)
\(\Leftrightarrow (a^2b+b^2c+c^2a)(1+2a^2b^2c^2)\geq 9a^2b^2c^2\)
\(\Leftrightarrow a^2b+b^2c+c^2a+2a^4b^3c^2+2a^2b^4c^3+2a^3b^2c^4\geq 3a^2b^2c^2(a+b+c)(*)\)
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Áp dụng BĐT AM-GM ta có:
\(a^2b+a^4b^3c^2+a^3b^2c^4\geq 3\sqrt[3]{a^9b^6c^6}=3a^3b^2c^2\)
\(b^2c+a^2b^4c^3+a^4b^3c^2\geq 3a^2b^3c^2\)
\(c^2a+a^3b^2c^4+a^2b^4c^3\geq 3a^2b^2c^3\)
Cộng theo vế:
\(\Rightarrow a^2b+b^2c+c^2a+2a^4b^3c^2+2a^2b^4c^3+2a^3b^2c^4\geq 3a^2b^2c^2(a+b+c)\)
Vậy $(*)$ đúng
Do đó ta có đpcm
Dấu bằng xảy ra khi $a=b=c=1$
Áp dụng BĐT cauchy-schwarz:
\(\sum\dfrac{a^4b}{2a+b}=\sum\dfrac{a^4b^2}{2ab+b^2}\ge\dfrac{\left(a^2b+b^2c+c^2a\right)^2}{\left(a+b+c\right)^2}\)
giờ ta chỉ cần có:\(a^2b+b^2c+c^2a\ge a+b+c\)
Áp dụng AM-GM:
\(a^2b+\dfrac{1}{b}\ge2a\)..tương tự ,ta suy ra:
\(a^2b+b^2c+c^2a\ge2\left(a+b+c\right)-\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\)(*)
Theo giả thiết: \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\le3\)
Dễ dàng suy ra được \(a+b+c\ge3\) ( từ BĐT \(\left(a+b+c\right)\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\ge9\))
theo đó thì \(a+b+c\ge\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\)
Nên từ (*) ta có đpcm.
Dấu = xảy ra khi a=b=c=1
\(\dfrac{ab}{6+2b+c}=\dfrac{ab}{a+b+c+2b+c}=\dfrac{ab}{\left(a+c\right)+\left(b+c\right)+2b}\le\dfrac{1}{9}\left(\dfrac{ab}{a+c}+\dfrac{ab}{b+c}+\dfrac{ab}{2b}\right)\)
Tương tự:
\(\dfrac{bc}{6+2c+a}\le\dfrac{1}{9}\left(\dfrac{bc}{a+b}+\dfrac{bc}{a+c}+\dfrac{bc}{2c}\right)\)
\(\dfrac{ac}{6+2a+b}\le\dfrac{1}{9}\left(\dfrac{ac}{a+b}+\dfrac{ac}{b+c}+\dfrac{ac}{2a}\right)\)
Cộng vế:
\(P\le\dfrac{1}{9}\left(\dfrac{ac+bc}{a+b}+\dfrac{ab+ac}{b+c}+\dfrac{ab+bc}{a+c}+\dfrac{a+b+c}{2}\right)=\dfrac{1}{6}\left(a+b+c\right)=1\)