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Lời giải:
Từ \(ab+bc+ac=3abc\Rightarrow \frac{1}{c}+\frac{1}{a}+\frac{1}{b}=3\)
Áp dụng BĐT Bunhiacopxky:
\(\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{b}+\frac{1}{c}+\frac{1}{c}+\frac{1}{c}\right)(a+b+b+c+c+c)\geq (1+1+1+1+1+1)^2\)
\(\Leftrightarrow \frac{1}{a}+\frac{2}{b}+\frac{3}{c}\geq \frac{36}{a+2b+3c}\)
Hoàn toàn tương tự:
\(\frac{1}{b}+\frac{2}{c}+\frac{3}{a}\geq \frac{36}{b+2c+3a}\)
\(\frac{1}{c}+\frac{2}{a}+\frac{3}{b}\geq \frac{36}{c+2a+3b}\)
Cộng các BĐT vừa thu được ở trên theo vế và rút gọn:
\(\frac{6}{a}+\frac{6}{b}+\frac{6}{c}\geq \frac{36}{a+2b+3c}+\frac{36}{b+2c+3a}+\frac{36}{c+2a+3b}\)
\(\Leftrightarrow 6\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq 36F\)
\(\Leftrightarrow 18\geq 36F\Leftrightarrow F\leq \frac{1}{2}\)
Vậy \(F_{\max}=\frac{1}{2}\)
Dấu bằng xảy ra khi \(a=b=c=1\)
Tính ra a+b+c<=4 nhé (dùng Bu-nhi-a cop-xki)
Phần còn lại tự xử nhé)
Áp dụng BĐT \(\dfrac{1}{x}+\dfrac{1}{y}\ge\dfrac{4}{x+y}\) ta có:
\(\dfrac{1}{a+3b}+\dfrac{1}{a+b+2c}\ge\dfrac{4}{2a+4b+2c}=\dfrac{2}{a+2b+c}\)
Tương tự cho 2 BĐT còn lại ta cũng có:
\(\dfrac{1}{b+3c}+\dfrac{1}{2a+b+c}\ge\dfrac{2}{a+b+2c};\dfrac{1}{c+3a}+\dfrac{1}{a+2b+c}\ge\dfrac{2}{2a+b+c}\)
Cộng theo vế 3 BĐT trên ta có:
\(VT=\dfrac{1}{b+3c}+\dfrac{1}{c+3a}+\dfrac{1}{a+3b}\)
\(\ge\dfrac{1}{a+b+2c}+\dfrac{1}{2a+b+c}+\dfrac{1}{a+2b+c}=VP\)
ÁP DỤNG BĐT COSI TA CÓ :\(\sqrt{\frac{a}{b+c+2a}}\le\frac{a}{b+c+2a}+\frac{1}{4}\)
\(\sqrt[]{\frac{b}{a+c+2b}}\le\frac{b}{a+c+2b}+\frac{1}{4}\)
\(\sqrt[]{\frac{c}{a+b+2c}}\le\frac{c}{a+b+2c}+\frac{1}{4}\)
ĐẶT A=\(\sqrt[]{\frac{a}{b+c+2a}}+\sqrt[]{\frac{b}{a+c+2b}}+\sqrt[]{\frac{c}{a+b+2c}}\)
\(\le\frac{a}{b+c+2a}+\frac{b}{a+c+2b}+\frac{c}{a+b+2c}+\frac{3}{4}\)
ÁP DỤNG BĐT :\(\frac{1}{a}+\frac{1}{b}\ge\frac{4}{a+b}\)
\(\Rightarrow\frac{a}{b+c+2a}\le\frac{1}{4}\left(\frac{a}{a+b}+\frac{a}{a+c}\right)\)
\(\Rightarrow\frac{b}{a+c+2b}\le\frac{1}{4}\left(\frac{b}{a+b}+\frac{b}{b+c}\right)\)
\(\Rightarrow\frac{c}{a+b+2c}\le\frac{1}{4}\left(\frac{c}{a+c}+\frac{c}{c+b}\right)\)
\(\Rightarrow A\le\frac{1}{4}\left(\frac{a}{a+b}+\frac{a}{a+c}+\frac{b}{b+c}+\frac{b}{a+b}+\frac{c}{a+c}+\frac{c}{b+c}\right)+\frac{3}{4}\)
\(\Rightarrow A\le\frac{1}{4}\left(\frac{a+b}{a+b}+\frac{b+c}{b+c}+\frac{c+a}{c+a}\right)+\frac{3}{4}\)
\(\Rightarrow A\le\frac{1}{4}\left(1+1+1\right)+\frac{3}{4}\)
\(\Rightarrow A\le\frac{3}{2}\)
DẤU = XẢY RA\(\Leftrightarrow a=b=c\)
Một lời giải khác:
\(\left(\Sigma\sqrt{\frac{a}{b+c+2a}}\right)^2=\left(\Sigma\sqrt{\frac{a\left(a+2c+b\right)}{\left(a+2c+b\right)\left(b+c+2a\right)}}\right)^2\)
\(\le\left[\Sigma a\left(a+2c+b\right)\right]\left[\Sigma\frac{1}{\left(a+2c+b\right)\left(b+c+2a\right)}\right]=\Sigma\frac{a^2+3ab}{\left(a+2c+b\right)\left(b+c+2a\right)}\)
\(=\frac{4\left(\Sigma a^2+3\Sigma ab\right)\left(\Sigma a\right)}{\Pi\left(a+2c+b\right)}\)
Cần chứng minh \(\frac{4\left(\Sigma a^2+3\Sigma ab\right)\left(\Sigma a\right)}{\Pi\left(a+2c+b\right)}\le\frac{9}{4}\)
Chịu khó quy đồng :V
Đặt \(\left(a+1;b+1;c+1\right)\rightarrow\left(x;y;z\right)\).Giả thiết trở thành:\(xyz=x+y+z\) và cần tìm max của \(P=\sum\dfrac{x}{x^2+1}\)
Ta có: \(P=\sum\dfrac{x}{x^2+1}=\sum\dfrac{xyz}{x\left(x+y+z\right)+yz}=xyz.\sum\dfrac{1}{\left(x+y\right)\left(x+z\right)}\)
\(=\dfrac{2xyz\left(x+y+z\right)}{\left(x+y\right)\left(y+z\right)\left(x+z\right)}\)
Do \(\left(x+y\right)\left(y+z\right)\left(x+z\right)\ge\dfrac{8}{9}\left(x+y+z\right)\left(xy+yz+xz\right)\) nên \(P\le\dfrac{2xyz}{\dfrac{8}{9}\left(xy+yz+xz\right)}=\dfrac{9}{4\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)}\)(*)
Mặt khác , từ giả thiết ta có : \(1=\dfrac{1}{xy}+\dfrac{1}{yz}+\dfrac{1}{xz}\le\dfrac{1}{3}\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)^2\)( theo AM-GM)
\(\Rightarrow\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\ge\sqrt{3}\)
Kết hợp với (*) , ta suy ra \(P\le\dfrac{9}{4\sqrt{3}}=\dfrac{3\sqrt{3}}{4}\)
Dấu = xảy ra khi \(x=y=z=\sqrt{3}\) hay \(a=b=c=\sqrt{3}-1\)
P/s: Chứng minh \(\left(a+b\right)\left(b+c\right)\left(c+a\right)\ge\dfrac{8}{9}\left(a+b+c\right)\left(ab+bc+ca\right)\)
khai triển ra ta có: \(\sum ab\left(a+b\right)\ge6abc\)hay \(\left(a+b\right)\left(b+c\right)\left(c+a\right)\ge8abc\)( đúng)
Lời giải:
Áp dụng BĐT Cauchy-Schwarz:
\(\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{c+a}\geq \frac{9}{b+c+c+a+c+a}=\frac{9}{3c+2a+b}\)
\(\frac{1}{a+c}+\frac{1}{a+b}+\frac{1}{a+b}\geq \frac{9}{a+c+a+b+a+b}=\frac{9}{3a+2b+c}\)
\(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{b+c}\geq \frac{9}{a+b+b+c+b+c}=\frac{9}{3b+2c+a}\)
Cộng theo vế rồi rút gọn ta thu được
\(\frac{1}{b+c}+\frac{1}{a+c}+\frac{1}{a+b}\geq 3\left(\frac{1}{3a+2b+c}+\frac{1}{3b+2c+a}+\frac{1}{3c+2a+b}\right)\) (đpcm)
Dấu bằng xảy ra khi $a=b=c$
Sử dụng AM-GM:
\(\Sigma\frac{\sqrt{ab}}{a+b+2c}=\Sigma\frac{\sqrt{ab}}{a+c+b+c}\le\frac{1}{2}\Sigma\frac{\sqrt{ab}}{\sqrt{\left(a+c\right)\left(b+c\right)}}\le\frac{1}{4}\Sigma\left(\frac{a}{a+c}+\frac{b}{b+c}\right)=\frac{3}{4}\)
Đẳng thức xảy ra tại a=b=c
Cauchy-Schwarz dạng ENgel:
\(P=\dfrac{a}{2a+b}+\dfrac{b}{2b+c}+\dfrac{c}{2c+a}\)
\(=\dfrac{1}{2}\cdot3-\left(\dfrac{b}{4a+2b}+\dfrac{c}{4b+2c}+\dfrac{a}{4c+2a}\right)\)
\(=\dfrac{3}{2}-\left(\dfrac{b^2}{4ab+2b^2}+\dfrac{c^2}{4bc+2c^2}+\dfrac{a^2}{4ac+2a^2}\right)\)
\(\le\dfrac{3}{2}-\dfrac{\left(a+b+c\right)^2}{2\left(a^2+b^2+c^2+2ab+2bc+2ca\right)}\)
\(=\dfrac{3}{2}-\dfrac{\left(a+b+c\right)^2}{2\left(a+b+c\right)^2}=1\)
\("="\Leftrightarrow a=b=c=\dfrac{1}{3}\)
Có a,b,c dương ko nhỉ ?