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\(\dfrac{\dfrac{b+c}{a}+\dfrac{a}{a}}{2}>=\sqrt{\dfrac{b+c}{a}\cdot\dfrac{a}{a}}\)
=>\(\dfrac{a+b+c}{2a}>=\sqrt{\dfrac{b+c}{a}}\)
=>\(\sqrt{\dfrac{a}{b+c}}>=\dfrac{2a}{a+b+c}\)
Tương tự, ta có: \(\sqrt{\dfrac{b}{a+c}}>=\dfrac{2b}{a+b+c};\sqrt{\dfrac{c}{a+b}}>=\dfrac{2c}{a+b+}\)
=>A>=2
Lời giải:
Đặt \(\left ( \frac{\sqrt{a^2+b^2}}{c},\frac{\sqrt{b^2+c^2}}{a}, \frac{\sqrt{c^2+a^2}}{b} \right )=(x,y,z)\)
BĐT cần chứng minh tương đương với:
\(x+y+z\geq 2\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\)\((*)\)
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Từ cách đặt $x,y,z$ ta có:
\(\frac{1}{x^2+1}+\frac{1}{y^2+1}+\frac{1}{z^2+1}=1\)
Áp dụng BĐT Bunhiacopxky:
\(\frac{x^2+1}{x^2}+\frac{y^2+1}{y^2}+\frac{z^2+1}{z^2}=\left(\frac{1}{x^2+1}+\frac{1}{y^2+1}+\frac{1}{z^2+1}\right)\left(\frac{x^2+1}{x^2}+\frac{y^2+1}{y^2}+\frac{z^2+1}{z^2}\right)\)
\(\geq \left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)^2\)
\(\Leftrightarrow 3\geq 2\left(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{xz}\right)\)
\(\Leftrightarrow xyz\geq \frac{2}{3}(x+y+z)\)
\(\Rightarrow xyz(x+y+z)\geq \frac{2}{3}(x+y+z)^2\)
Áp dụng BĐT AM_GM ta lại có:
\((x+y+z)^2\geq 3(xy+yz+xz)\). Do đó:
\(xyz(x+y+z)\geq 2(xy+yz+xz)\)
\(\Leftrightarrow x+y+z\geq 2\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\)
Đúng theo \((*)\)
Do đó ta có đpcm
Dấu bằng xảy ra khi \(a=b=c\)
áp dụng bat dang thuc bunhiacóki
ta có \(\dfrac{\sqrt{a^2+b^2}}{c}\ge\dfrac{a+b}{\sqrt{2}c}\)
ttu vt \(\ge\dfrac{1}{\sqrt{2}}\left(\dfrac{a+b}{c}+\dfrac{b+c}{a}+\dfrac{c+a}{b}\right)\)
=\(\dfrac{a}{\sqrt{2}}\left(\dfrac{1}{b}+\dfrac{1}{c}\right)+\dfrac{b}{\sqrt{2}}\left(\dfrac{1}{a}+\dfrac{1}{c}\right)+\dfrac{c}{\sqrt{2}}\left(\dfrac{1}{a}+\dfrac{1}{b}\right)\) (1)
áp dung bdt \(\dfrac{1}{a}+\dfrac{1}{b}\ge\dfrac{4}{a+b}\)
ta có (1) \(\ge\dfrac{a}{\sqrt{2}}.\dfrac{4}{b+c}\)
tiếp tục áp dụng bunhia ta có \(\dfrac{a}{\sqrt{2}}.\dfrac{4}{b+c}\ge\dfrac{a}{\sqrt{2}}.\dfrac{4}{\sqrt{2\left(b^2+c^2\right)}}=\dfrac{2a}{\sqrt{b^2+c^2}}\)
ttuong tu ta có \(vt\ge2\left(\dfrac{a}{\sqrt{b^2+c2}}+\dfrac{b}{\sqrt{a^2+c^2}}+\dfrac{c}{\sqrt{a^2+b^2}}\right)\left(dpcm\right)\)
Bài 2:
\(\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{c+a}}+\sqrt{\dfrac{c}{a+b}}>2\)
Trước hết ta chứng minh \(\sqrt{\dfrac{a}{b+c}}\ge\dfrac{2a}{a+b+c}\)
Áp dụng BĐT AM-GM ta có:
\(\sqrt{a\left(b+c\right)}\le\dfrac{a+b+c}{2}\)\(\Rightarrow1\ge\dfrac{2\sqrt{a\left(b+c\right)}}{a+b+c}\)
\(\Rightarrow\sqrt{\dfrac{a}{b+c}}\ge\dfrac{2a}{a+b+c}\). Ta lại có:
\(\sqrt{\dfrac{a}{b+c}}=\dfrac{\sqrt{a}}{\sqrt{b+c}}=\dfrac{a}{\sqrt{a\left(b+c\right)}}\ge\dfrac{2a}{a+b+c}\)
Thiết lập các BĐT tương tự:
\(\sqrt{\dfrac{b}{c+a}}\ge\dfrac{2b}{a+b+c};\sqrt{\dfrac{c}{a+b}}\ge\dfrac{2c}{a+b+c}\)
Cộng theo vế 3 BĐT trên ta có:
\(VT\ge\dfrac{2a}{a+b+c}+\dfrac{2b}{a+b+c}+\dfrac{2c}{a+b+c}=\dfrac{2\left(a+b+c\right)}{a+b+c}\ge2\)
Dấu "=" không xảy ra nên ta có ĐPCM
Lưu ý: lần sau đăng từng bài 1 thôi nhé !
1) Áp dụng liên tiếp bđt \(\dfrac{1}{x}+\dfrac{1}{y}\ge\dfrac{4}{x+y}\) với a;b là 2 số dương ta có:
\(\dfrac{1}{2a+b+c}=\dfrac{1}{\left(a+b\right)+\left(a+c\right)}\le\dfrac{\dfrac{1}{a+b}+\dfrac{1}{a+c}}{4}\)\(\le\dfrac{\dfrac{2}{a}+\dfrac{1}{b}+\dfrac{1}{c}}{16}\)
TT: \(\dfrac{1}{a+2b+c}\le\dfrac{\dfrac{2}{b}+\dfrac{1}{a}+\dfrac{1}{c}}{16}\)
\(\dfrac{1}{a+b+2c}\le\dfrac{\dfrac{2}{c}+\dfrac{1}{a}+\dfrac{1}{b}}{16}\)
Cộng vế với vế ta được:
\(\dfrac{1}{2a+b+c}+\dfrac{1}{a+2b+c}+\dfrac{1}{a+b+2c}\le\dfrac{1}{16}.\left(\dfrac{4}{a}+\dfrac{4}{b}+\dfrac{4}{c}\right)=1\left(đpcm\right)\)
Áp dụng bđt cosi schwart ta có:
`VT>=(a+b+c)^2/(a+b+c+sqrt{ab}+sqrt{bc}+sqrt{ca})`
Dễ thấy `sqrt{ab}+sqrt{bc}+sqrt{ca}<a+b+c`
`=>VT>=(a+b+c)^2/(2(a+b+c))=(a+b+c)/2=3`
Dấu "=" `<=>a=b=c=1.`
\(\sqrt{a+b}=\sqrt{a+c}+\sqrt{b+c}\)
\(\Leftrightarrow a+b=a+c+b+c+2\sqrt{\left(a+c\right)\left(b+c\right)}\)
\(\Leftrightarrow2c+2\sqrt{\left(a+c\right)\left(b+c\right)}=0\)
\(\Leftrightarrow c+\sqrt{\left(a+c\right)\left(b+c\right)}=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}c< 0\\-c=\sqrt{\left(a+c\right)\left(b+c\right)}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}c< 0\\c^2=\left(a+c\right)\left(b+c\right)\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}c< 0\\ab+bc+ac=0\end{matrix}\right.\)
\(\Leftrightarrow\dfrac{ab+bc+ac}{abc}=0\Leftrightarrow\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=0\left(đúng\right)\)
Từ 1a+1b+1c=0⇒ab+bc+ac=01a+1b+1c=0⇒ab+bc+ac=0
Khi đó:
(√a+c+√b+c)2=a+c+b+c+2√(a+c)(b+c)(a+c+b+c)2=a+c+b+c+2(a+c)(b+c)
=a+b+2c+2√ab+ac+bc+c2=a+b+2c+2√c2=a+b+2c+2ab+ac+bc+c2=a+b+2c+2c2
=a+b+2c+2|c|=a+b+2c+2|c|
Vì a,ba,b dương nên −1c=1a+1b>0⇒c<0⇒2|c|=−2c−1c=1a+1b>0⇒c<0⇒2|c|=−2c
Do đó:
(√a+c+√b+c)2=a+b+2c+2|c|=a+b+2c+(−2c)=a+b(a+c+b+c)2=a+b+2c+2|c|=a+b+2c+(−2c)=a+b
⇒√a+c+√b+c=√a+b
Ta có \(\Sigma\sqrt{\dfrac{a}{b+c}}=\Sigma\dfrac{a}{\sqrt{a\left(b+c\right)}}\)
Theo AM-GM ta có
\(\Sigma\dfrac{a}{\sqrt{a\left(b+c\right)}}\ge\Sigma\dfrac{a}{\dfrac{a+b+c}{2}}=\Sigma\dfrac{2a}{a+b+c}=\dfrac{2\left(a+b+c\right)}{a+b+c}=2\)
dấu bằng xảy ra khi \(\left\{{}\begin{matrix}a=b+c\\b=a+c\\c=a+b\end{matrix}\right.\Rightarrow a+b+c=2\left(a+b+c\right)\Rightarrow1=2\) (vô lí)
nên\(\Sigma\sqrt{\dfrac{a}{b+c}}>2\)
Theo bất đẳng thức cô si, có:
\(\sqrt{1.\dfrac{b+c}{a}}\le\left(1+\dfrac{b+c}{a}\right):2=\dfrac{a+b+c}{2a}\)
\(\Rightarrow\sqrt{\dfrac{a}{b+c}}\ge\dfrac{2a}{a+b+c}~~~~~\left(1\right)\)
Tương tự: \(\sqrt{\dfrac{b}{c+a}}\ge\dfrac{2b}{a+b+c}~~~~~\left(2\right)\)
\(\sqrt{\dfrac{c}{a+b}}\ge\dfrac{2c}{a+b+c}~~~~~\left(3\right)\)
Cộng vế theo vế \(\left(1\right);\left(2\right);\left(3\right)\), ta có:
\(\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{a+c}}+\sqrt{\dfrac{c}{a+b}}\ge\dfrac{2a}{a+b+c}+\dfrac{2b}{a+b+c}+\dfrac{2c}{a+b+c}=2\)
Áp dụng BĐT cosi:
\(\sqrt{\dfrac{b+c}{a}}\le\dfrac{\dfrac{b+c}{a}+1}{2}=\dfrac{\dfrac{a+b+c}{a}}{2}=\dfrac{a+b+c}{2a}\\ \Leftrightarrow\sqrt{\dfrac{a}{b+c}}\ge\dfrac{2a}{a+b+c}\)
Cmtt \(\sqrt{\dfrac{b}{c+a}}\ge\dfrac{2b}{a+b+c};\sqrt{\dfrac{c}{c+a}}\ge\dfrac{2c}{a+b+c}\)
Cộng vế theo vế 3 BĐT trên:
\(\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{a+c}}+\sqrt{\dfrac{c}{a+b}}\ge\dfrac{2\left(a+b+c\right)}{a+b+c}=2\)
Dấu \("="\Leftrightarrow\left\{{}\begin{matrix}a=b+c\\b=c+a\\c=a+b\end{matrix}\right.\Leftrightarrow a+b+c=2\left(a+b+c\right)\)
\(\Leftrightarrow a+b+c=0\) (vô lí vì \(a,b,c>0\))
Do đó dấu "=" ko xảy ra hay \(\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{a+c}}+\sqrt{\dfrac{c}{a+b}}>2\)