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Ta có: \(\sqrt{2a+bc}=\sqrt{a^2+ab+ac+bc}=\sqrt{\left(a+b\right)\left(a+c\right)}\le\frac{a+b+a+c}{2}\)
C/m tương tự \(\sqrt{2b+ac}\le\frac{b+a+b+c}{2}\)
\(\sqrt{2c+ab}\le\frac{c+a+c+b}{2}\)
\(\Rightarrow Q\le\frac{a+b+a+c+b+a+b+c+c+a+c+b}{2}=\frac{4\left(a+b+c\right)}{2}=4\)
Dấu "=" khi a = b = c = 2/3
Ta có: \(P=\frac{ab}{\sqrt{ab+2c}}+\frac{bc}{\sqrt{bc+2a}}+\frac{ca}{\sqrt{ca+2b}}\)
\(P=\frac{ab}{\sqrt{ab+\left(a+b+c\right)c}}+\frac{bc}{\sqrt{bc+\left(a+b+c\right)a}}+\frac{ca}{\sqrt{ca+\left(a+b+c\right)b}}\)
\(P=\frac{ab}{\sqrt{\left(a+c\right)\left(b+c\right)}}+\frac{bc}{\sqrt{\left(b+a\right)\left(c+a\right)}}+\frac{ca}{\sqrt{\left(c+b\right)\left(a+b\right)}}\)
\(P=\sqrt{\frac{ab}{\left(a+c\right)}.\frac{ab}{\left(b+c\right)}}+\sqrt{\frac{bc}{b+a}.\frac{bc}{c+a}}+\sqrt{\frac{ca}{c+b}.\frac{ca}{a+b}}\le\frac{1}{2}\left(\frac{ab}{a+c}+\frac{ab}{b+c}+\frac{bc}{b+a}+\frac{bc}{c+a}+\frac{ca}{c+b}+\frac{ca}{a+b}\right)=\frac{\left(a+b+c\right)}{2}=1\)
Vậy Max P=1 khi \(a=b=c=\frac{2}{3}\)
\(P=\Sigma\dfrac{ab}{\sqrt{ab+2c}}=\Sigma\dfrac{ab}{\sqrt{ab+\left(a+b+c\right)c}}=\Sigma\dfrac{\sqrt{ab}.\sqrt{ab}}{\sqrt{\left(a+c\right)\left(b+c\right)}}\le\dfrac{1}{2}.\Sigma\left(\dfrac{ab}{a+c}+\dfrac{ab}{b+c}\right)\) \(=\dfrac{1}{2}.\left(a+b+c\right)=1\)
Đặt \(\left(a,b,c\right)\rightarrow\left(\frac{x}{y},\frac{y}{z},\frac{z}{x}\right)\)
\(VT=\Sigma_{cyc}\frac{1}{\sqrt{\frac{x}{z}+\frac{x}{y}+2}}=\Sigma_{cyc}\frac{\sqrt{yz}}{\sqrt{xy+xz+2yz}}\)
\(\Rightarrow VT^2\le\left(1+1+1\right)\left(\Sigma_{cyc}\frac{yz}{xy+xz+2yz}\right)\)\(\le\frac{3}{4}\left[\Sigma_{cyc}yz\left(\frac{1}{xy+yz}+\frac{1}{xz+yz}\right)\right]=\frac{9}{4}\)
Đẳng thức xảy ra khi a = b = c = 1
Bài 1: Bổ đề: \(x^2+y^2\ge\frac{\left(x+y\right)^2}{2}\)
\(P=\frac{1}{\sqrt{2}}\left(\sqrt{4a^2+2ab+4b^2}+\sqrt{4b^2+2bc+4c^2}+\sqrt{4c^2+2ca+4a^2}\right)\)
\(=\frac{1}{\sqrt{2}}\left(\sqrt{3\left(a^2+b^2\right)+\left(a+b\right)^2}+\sqrt{3\left(b^2+c^2\right)+\left(b+c\right)^2}+\sqrt{3\left(c^2+a^2\right)+\left(c+a\right)^2}\right)\)
\(\ge\frac{1}{\sqrt{2}}\left(\sqrt{\frac{3}{2}\left(a+b\right)^2+\left(a+b\right)^2}+\sqrt{\frac{3}{2}\left(b+c\right)^2+\left(b+c\right)^2}+\sqrt{\frac{3}{2}\left(c+a\right)^2+\left(c+a\right)^2}\right)\)
\(=\frac{1}{\sqrt{2}}\left(\sqrt{\frac{5}{2}\left(a+b\right)^2}+\sqrt{\frac{5}{2}\left(b+c\right)^2}+\sqrt{\frac{5}{2}\left(c+a\right)^2}\right)\)
\(=\frac{1}{\sqrt{2}}.\frac{\sqrt{5}}{\sqrt{2}}+\left[\left(a+b\right)+\left(b+c\right)+\left(c+a\right)\right]\)\(=\frac{\sqrt{5}}{2}.2\left(a+b+c\right)=\sqrt{5}.2020\)
Dấu "=" xảy ra khi \(a=b=c=\frac{2020}{3}\)
Sử dụng BĐT Bunhiacopxki ta có:
\(\sqrt{a^2+b^2c^2}=\sqrt{a^2\left(a^2+b^2+c^2\right)+b^2c^2}=\sqrt{\left(a^2+b^2\right)\left(a^2+c^2\right)}\ge\sqrt{\left(a^2+bc\right)^2}=a^2+bc\)
Tương tự: \(\sqrt{b^2+c^2a^2}\ge b^2+ca\)
\(\sqrt{c^2+a^2b^2}\ge c^2+ab\)
Cộng mại ta có: \(VT\ge ab+bc+ca+1\)
bài này ko khác gì câu 921427 nhé bạn, có điều bạn tìm cách tách a + 3b + 2c = (a + b) + (b + c) + (b + c)
Thêm nữa, áp dụng BĐT \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge\frac{9}{a+b+c}\) với a, b, c > 0
Đẳng thức xảy ra khi và chỉ khi a = b = c.
EZ!!!Sau khi sử dụng 1 số bđt đơn giản, ta sẽ được:
\(\text{Σ}_{cyc}\frac{ab}{a+3b+2c}\le\frac{1}{9}\text{Σ}_{cyc}\left(\frac{ab}{a+c}+\frac{ab}{b+c}+\frac{a}{2}\right)=K\)
\(P\le K=\frac{1}{9}\left[\text{Σ}_{cyc}\left(\frac{ab}{a+c}+\frac{bc}{a+c}\right)+\frac{a+b+c}{2}\right]\)
\(=\frac{1}{9}\left(b+a+c+\frac{a+b+c}{2}\right)=\frac{a+b+c}{6}\le1\)
Dấu "=" xảy ra khi và chỉ khi a = b = c = 2
Ta có : \(p=\frac{bc}{a^2\left(b+c\right)}+\frac{ca}{b^2\left(a+c\right)}+\frac{ab}{c^2\left(a+b\right)}\)
Áp dụng bất đẳng thức AM - GM ta có :
\(\frac{bc}{a^2\left(b+c\right)}+\frac{b+c}{4bc}\ge2\sqrt{\frac{bc}{a^2\left(b+c\right)}.\frac{b+c}{4ab}}=\frac{1}{a}\)
\(\frac{ac}{b^2\left(a+c\right)}+\frac{a+c}{4ac}\ge4\sqrt{\frac{ac}{b^2\left(a+c\right)}.\frac{a+c}{4ac}}=\frac{1}{b}\)
\(\frac{ab}{c^2\left(a+b\right)}+\frac{a+b}{4ab}\ge2\sqrt{\frac{ab}{c^2\left(a+b\right)}.\frac{a+b}{4ab}}=\frac{1}{c}\)
Cộng vế với vế ta được \(p+\frac{1}{4c}+\frac{1}{4a}+\frac{1}{4b}+\frac{1}{4a}+\frac{1}{4c}+\frac{1}{4b}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\)
\(\Leftrightarrow p+\frac{1}{2a}+\frac{1}{2b}+\frac{1}{2c}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\)
\(\Rightarrow p\ge\frac{1}{2a}+\frac{1}{2b}+\frac{1}{2c}\ge3\sqrt[3]{\frac{1}{2a.2b.2c}}=\frac{3}{\sqrt[3]{8abc}}=\frac{3}{2}\)
Dấu "=" xảy ra \(\Leftrightarrow a=b=c=1\)
Xét: \(\frac{bc}{a^2b+ca^2}=\frac{bc}{a\cdot abc\cdot\frac{1}{c}+a\cdot abc\cdot\frac{1}{b}}=\frac{b^2c^2}{ab+ca}\)(*)
Tương tự với (*) ta có: \(\hept{\begin{cases}\frac{ca}{b^2c+ab^2}=\frac{c^2a^2}{ab+bc}\\\frac{ab}{c^2a+bc^2}=\frac{a^2b^2}{ca+bc}\end{cases}}\)
\(\Rightarrow\Sigma_{cyc}\frac{bc}{a^2b+ca^2}=\Sigma_{cyc}\frac{b^2c^2}{ab+ca}\)
Ta thấy\(\Sigma_{cyc}\frac{b^2c^2}{ab+ca}\) có dạng: \(\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\ge\frac{1}{2}\left(a+b+c\right)\)
Bước cuối Cô-si ba số và kết hợp điều kiện abc=1 là xong
\(P=\frac{\frac{1}{a^2}}{\frac{1}{b}+\frac{1}{c}}+\frac{\frac{1}{b^2}}{\frac{1}{a}+\frac{1}{c}}+\frac{\frac{1}{c^2}}{\frac{1}{a}+\frac{1}{b}}\)
Đặt \(\hept{\begin{cases}x=\frac{1}{a}\\y=\frac{1}{b}\\z=\frac{1}{c}\end{cases}}\Rightarrow xyz=1\Rightarrow P=\frac{x^2}{y+z}+\frac{y^2}{x+z}+\frac{z^2}{x+y}\)
Áp dụng BĐT Cauchy-Schwarz dạng Engel ta có:
\(P\ge\frac{\left(x+y+z\right)^2}{y+z+x+z+x+y}=\frac{x+y+z}{2}\ge\frac{3\sqrt[3]{xyz}}{2}=\frac{3}{2}\)
Dấu "=" xảy ra khi \(x=y=z\Leftrightarrow a=b=c=1\)
Cần cách khác thì nhắn cái
\(Q\le\sqrt{3\left(2a+2b+2c+ab+bc+ca\right)}\)
\(Q\le\sqrt{3\left(4+\frac{\left(a+b+c\right)^2}{3}\right)}=4\)
\(Q_{max}=4\) khi \(a=b=c=\frac{2}{3}\)