Cho \(\hept{\begin{cases}a,b,c>0\\abc=1\end{cases}}\)Tìm max \(A=\frac{1}{a+b+1}+\frac{1}{b+c+1}+\frac{1}{c+a+1}\)
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4 tháng 5 2018
Từ đề bài \(\Rightarrow\frac{1}{1+a}=1-\frac{1}{1+b}+1-\frac{1}{1+c}=\frac{b}{1+b}+\frac{c}{1+c}\ge2\sqrt{\frac{bc}{\left(1+b\right)\left(1+c\right)}}\) (AM-GM)
Tương tự \(\hept{\begin{cases}\frac{1}{1+b}\ge2\sqrt{\frac{ac}{\left(a+1\right)\left(c+1\right)}}\\\frac{1}{1+c}\ge2\sqrt{\frac{ab}{\left(a+1\right)\left(b+1\right)}}\end{cases}}\)
Nhân các vế tương ứng của các bđt vừa cm đc ta có :
\(\frac{1}{\left(1+a\right)\left(1+b\right)\left(1+c\right)}\ge\frac{8abc}{\left(1+a\right)\left(1+b\right)\left(1+c\right)}\)\(\Rightarrow abc\le\frac{1}{8}\)
Dấu "=" xảy ra \(\Leftrightarrow a=b=c=\frac{1}{2}\)
20 tháng 6 2018
\(A=\frac{1}{a^2+b^2+c^2}+\frac{1}{abc}=\frac{1}{a^2+b^2+c^2}+\frac{a+b+c}{abc}=\frac{1}{a^2+b^2+c^2}+\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}\)
\(>=\frac{1}{a^2+b^2+c^2}+\frac{9}{ab+ac+bc}\)(bđt svacxo)\(=\frac{1}{a^2+b^2+c^2}+\frac{1}{ab+ac+bc}+\frac{1}{ab+ac+bc}+\frac{7}{ab+ac+bc}\)
\(>=\frac{9}{a^2+b^2+c^2+ab+ac+bc+ac+ac+bc}+\frac{7}{ab+ac+bc}\)(bđt svacxo)
\(=\frac{9}{a^2+b^2+c^2+2ab+2ac+2bc}+\frac{7}{ab+ac+bc}=\frac{9}{\left(a+b+c\right)^2}+\frac{7}{ab+ac+bc}\)
\(=\frac{9}{1}+\frac{7}{ab+ac+bc}=9+\frac{7}{ab+ac+bc}\)
\(\left(a+b+c\right)^2=a^2+b^2+c^2+2ab+2ac+2bc>=ab+ac+bc+2ab+2ac+2bc\)
\(=3ab+3ac+3bc=3\left(ab+ac+bc\right)\Rightarrow\frac{1}{3}\left(a+b+c\right)^2=\frac{1}{3}\cdot1=\frac{1}{3}>=ab+ac+bc\Rightarrow ab+ac+bc< =\frac{1}{3}\)
\(\Rightarrow9+\frac{7}{ab+ac+bc}>=9+\frac{7}{\frac{1}{3}}=9+7\cdot3=9+21=30\)
\(\Rightarrow A>=30\)dấu = xảy ra khi \(a=b=c=\frac{1}{3}\)
vậy min A là 30 khi \(a=b=c=\frac{1}{3}\)
9 tháng 9 2017
1,
\(A=1+a+\frac{1}{b}+\frac{a}{b}+1+b+\frac{1}{a}+\frac{b}{a}\)
\(\ge1+1+2\sqrt{\frac{a}{b}.\frac{b}{a}}+a+b+\frac{a+b}{ab}=4+a+b+\frac{4\left(a+b\right)}{\left(a+b\right)^2}=4+a+b+\frac{4}{a+b}\)
lại có \(\left(1+1\right)\left(a^2+b^2\right)\ge\left(a+b\right)^2\Rightarrow a+b\le\sqrt{2}\)
\(4+a+b+\frac{4}{a+b}=4+\left(a+b+\frac{2}{a+b}\right)+\frac{2}{a+b}\ge4+2\sqrt{2}+\sqrt{2}=4+3\sqrt{2}\)
\(\Rightarrow A\ge4+3\sqrt{2}\)
9 tháng 9 2017
câu 2
ta có:\(\left(2b^2+a^2\right)\left(2+1\right)\ge\left(2b+a\right)^2\Rightarrow3c\ge a+2b\)
\(\frac{1}{a}+\frac{2}{b}=\frac{1}{a}+\frac{4}{2b}\ge\frac{9}{a+2b}\ge\frac{9}{3c}=\frac{3}{c}\left(Q.E.D\right)\)
29 tháng 6 2017
Ta có:
\(\left(a+b+c+d\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)\ge\left(a+b+c+d\right).\frac{16}{\left(a+b+c+d\right)}=16\)
\(\Rightarrow\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\ge4\)
Dấu = xảy ra khi \(a=b=c=d=1\)
18 tháng 11 2019
Bài 2:
\(\frac{1}{\sqrt[3]{81}}\cdot P=\frac{1}{\sqrt[3]{9\cdot9\cdot\left(a+2b\right)}}+\frac{1}{\sqrt[3]{9\cdot9\cdot\left(b+2c\right)}}+\frac{1}{\sqrt[3]{9\cdot9\cdot\left(c+2a\right)}}\)
\(\ge\frac{3}{a+2b+9+9}+\frac{3}{b+2c+9+9}+\frac{3}{c+2a+9+9}\ge3\left(\frac{9}{3a+3b+3c+54}\right)=\frac{1}{3}\)
\(\Rightarrow P\ge\sqrt[3]{3}\)
Dấu bằng xẩy ra khi a=b=c=3
18 tháng 11 2019
Bài 1:
\(ab+bc+ca=5abc\Rightarrow\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=5\)
Theo bđt côsi-shaw ta luôn có: \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{t}+\frac{1}{k}\ge\frac{25}{x+y+z+t+k}\)(x=y=z=t=k>0 ) (*)
\(\Leftrightarrow\left(x+y+z+t+k\right)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{t}+\frac{1}{k}\right)\ge25\)
Áp dụng bđt AM-GM ta có:
\(\hept{\begin{cases}x+y+z+t+k\ge5\sqrt[5]{xyztk}\\\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{t}+\frac{1}{k}\ge5\sqrt[5]{\frac{1}{xyztk}}\end{cases}}\)
\(\Rightarrow\left(x+y+z+t+k\right)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{t}+\frac{1}{k}\right)\ge25\)
\(\Rightarrow\)(*) luôn đúng
Từ (*) \(\Rightarrow\frac{1}{25}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{t}+\frac{1}{k}\right)\le\frac{1}{x+y+z+t+k}\)
Ta có: \(P=\frac{1}{2a+2b+c}+\frac{1}{a+2b+2c}+\frac{1}{2a+b+2c}\)
Mà \(\frac{1}{2a+2b+c}=\frac{1}{a+a+b+b+c}\le\frac{1}{25}\left(\frac{1}{a}+\frac{1}{a}+\frac{1}{b}+\frac{1}{b}+\frac{1}{c}\right)\)
\(\frac{1}{a+2b+2c}=\frac{1}{a+b+b+c+c}\le\frac{1}{25}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{b}+\frac{1}{c}+\frac{1}{c}\right)\)
\(\frac{1}{2a+b+2c}=\frac{1}{a+a+b+c+c}\le\frac{1}{25}\left(\frac{1}{a}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{c}\right)\)
\(\Rightarrow P\le\frac{1}{25}\left[5.\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\right]=1\)
\(\Rightarrow P\le1\left(đpcm\right)\)Dấu"="xảy ra khi a=b=c\(=\frac{3}{5}\)
17 tháng 6 2019
Áp dụng bđt Cô-si có'
\(\frac{1}{x}+\frac{1}{y}\ge\frac{2}{\sqrt{xy}}\ge\frac{2}{\frac{x+y}{2}}=\frac{4}{x+y}\)
\(\Rightarrow\frac{1}{x}+\frac{1}{y}\ge\frac{4}{x+y}\)
\(\Rightarrow\frac{1}{x+y}\le\frac{1}{4}\left(\frac{1}{x}+\frac{1}{y}\right)\)(1)
Áp dụng bđt trên ta được
\(\frac{1}{2a+b+c}=\frac{1}{\left(a+b\right)+\left(a+c\right)}\le\frac{1}{4}\left(\frac{1}{a+b}+\frac{1}{a+c}\right)\)
\(\Rightarrow\left(\frac{1}{2a+b+c}\right)^2\le\frac{1}{16}\left(\frac{1}{a+b}+\frac{1}{a+c}\right)^2\)
Chứng minh tương tự rồi cộng các vế lại cho nhau ta được
\(A\le\frac{1}{16}\left(\frac{1}{a+b}+\frac{1}{a+c}\right)^2+\frac{1}{16}\left(\frac{1}{a+c}+\frac{1}{b+c}\right)^2+\frac{1}{16}\left(\frac{1}{a+b}+\frac{1}{b+c}\right)^2\)
\(\Rightarrow16A\le\left(\frac{1}{a+b}+\frac{1}{a+c}\right)^2+\left(\frac{1}{a+c}+\frac{1}{b+c}\right)^2+\left(\frac{1}{a+b}+\frac{1}{b+c}\right)^2\)
\(=\frac{2}{\left(a+b\right)^2}+\frac{2}{\left(b+c\right)^2}+\frac{2}{\left(c+a\right)^2}+\frac{2}{\left(a+b\right)\left(a+c\right)}+\frac{2}{\left(b+c\right)\left(a+b\right)}+\frac{2}{\left(a+c\right)\left(b+c\right)}\)
Đặt \(\left(\frac{1}{a+b};\frac{1}{b+c};\frac{1}{c+a}\right)\rightarrow\left(x;y;z\right)\)
Khi đó \(16A\le2x^2+2y^2+2z^2+2xy+2yz+2zx\)
Ta có bđt phụ sau : \(xy+yz+zx\le x^2+y^2+z^2\)(tự chứng minh) (2)
Áp dụng ta được
\(16A\le4x^2+4y^2+4z^2=\frac{4}{\left(a+b\right)^2}+\frac{4}{\left(b+c\right)^2}+\frac{4}{\left(c+a\right)^2}\)
\(\Rightarrow4A\le\frac{1}{\left(a+b\right)^2}+\frac{1}{\left(b+c\right)^2}+\frac{1}{\left(c+a\right)^2}\)
Từ (1) \(\Rightarrow\frac{1}{\left(x+y\right)^2}\le\frac{1}{16}\left(\frac{1}{x}+\frac{1}{y}\right)^2\)(Bình phương 2 vế lên)
Áp dụng bđt này ta được
\(4A\le\frac{1}{16}\left(\frac{1}{a}+\frac{1}{b}\right)^2+\frac{1}{16}\left(\frac{1}{b}+\frac{1}{c}\right)^2+\frac{1}{16}\left(\frac{1}{c}+\frac{1}{a}\right)^2\)
\(\Rightarrow64A\le\frac{1}{a^2}+\frac{2}{ab}+\frac{1}{b^2}+\frac{1}{b^2}+\frac{2}{bc}+\frac{1}{c^2}+\frac{1}{c^2}+\frac{2}{ac}+\frac{1}{a^2}\)
\(\Rightarrow64A\le\frac{2}{a^2}+\frac{2}{b^2}+\frac{2}{c^2}+\frac{2}{ab}+\frac{2}{bc}+\frac{2}{ca}\)
\(\Rightarrow32A\le\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\)
Áp dụng bđt (2) ta được \(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\le\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\)
\(\Rightarrow32A\le\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=3+3=6\)
\(\Rightarrow A\le\frac{6}{32}=\frac{3}{16}\)
Dấu "=" xảy ra tại a=b=c = 1
T
17 tháng 6 2019
#)Em thấy có link này có cách giải ngắn gọn hơn nek :
https://h.vn/hoi-dap/tim-kiem?q=cho+c%C3%A1c+s%E1%BB%91+th%E1%BB%B1c+d%C6%B0%C6%A1ng+a,b,c+thay+%C4%91%E1%BB%95i+lu%C3%B4n+th%E1%BB%8Fa+m%C3%A3n+1/a2+++1/b2+++1/c2+=3.T%C3%ACm+Max+P+=+1/(2a+b+c)2++1(2b+a+c)2++1/(2c+a+b)2&id=394201
Ai cần link này ib e nhé ! e gửi cho chị #Diệp Song Thiên đã ^^
YY
4 tháng 7 2018
Đặt x = 1/a ; y = 1/b, z = 1/c với x,y,z > 0
đk <=> 1/x + 1/y + 1/z = 1/(xyz)
<=> xy + yz + zx = 1
A = √[yz/(1+x²)] + √[zx/(1+y²)] + √[xy/(1+z²)]
Ta có:
1 + x² = x² + xy + yz + zx = (x+z)(x+y)
=> √[yz/(1+x²)] = √[y/(x+y)] . √[z/(x+z)]
≤ 1/2 . [y/(x+y) + z/(x+z)] (1)
(áp dụng bđt Cosi: √m .√n ≤ 1/2 . (m+n))
Tương tự:
√[xz/(1+y²)] = √[x/(x+y)] . √[z/(y+z)] ≤ 1/2 . [x/(x+y) + z/(y+z)] (2)
√[xy/(1+z²)] = √[y/(z+y)] . √[x/(x+z)] ≤ 1/2 . [y/(z+y) + x/(x+z)] (3)
Cộng vế của (1),(2) và (3) lại ta được:
A ≤ 1/2 . 3 = 3/2
Vậy Max A = 3/2 xảy ra <=> x = y = z = 1/√3 <=> a = b = c = √3
Ta có \(x^3+y^3=\left(x+y\right)\left(x^2-xy+y^2\right)\ge xy\left(x+y\right)\)
Áp dụng ta có
\(a+b\ge\sqrt[3]{ab}\left(\sqrt[3]{a}+\sqrt[3]{b}\right)\)
=> \(a+b+1\ge\sqrt[3]{ab}\left(\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}\right)\)
Khi đó
\(A\le\frac{\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}}{\sqrt[3]{abc}\left(\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}\right)}=1\)
MaxA=1
Dấu bằng xảy ra khi a=b=c=1