Cho a,b,c>0 t/m a+b+c=2
Tim GTLN của \(Q=\dfrac{ab}{\sqrt{ab+2c}}+\dfrac{bc}{\sqrt{bc+2a}}+\dfrac{ca}{\sqrt{ca+2b}}\)
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AH
Akai Haruma
Giáo viên
7 tháng 1 2019
Lời giải:
Áp dụng BĐT AM-GM:
\(P=\frac{\sqrt{ab}}{(a+c)+(b+c)}+\frac{\sqrt{bc}}{(b+a)+(c+a)}+\frac{\sqrt{ca}}{(c+b)+(a+b)}\)
\(\leq \underbrace{\frac{\sqrt{ab}}{2\sqrt{(a+c)(b+c)}}+\frac{\sqrt{bc}}{2\sqrt{(b+a)(c+a)}}+\frac{\sqrt{ca}}{2\sqrt{(c+b)(a+b)}}}_{M}(*)\)
Xét:
\(M=\frac{1}{2}\frac{\sqrt{ab(a+b)}+\sqrt{bc(b+c)}+\sqrt{ca(c+a)}}{\sqrt{(a+b)(b+c)(c+a)}}(1)\)
Theo BĐT Bunhiacopxky và AM-GM:
\((\sqrt{ab(a+b)}+\sqrt{bc(b+c)}+\sqrt{ca(c+a)})^2\leq (ab+bc+ac)(a+b+b+c+c+a)\)
\(=2(ab+bc+ac)(a+b+c)=2[(a+b)(b+c)(c+a)+abc]\)
\(\leq 2[(a+b)(b+c)(c+a)+\frac{(a+b)(b+c)(c+a)}{8}]=\frac{9}{4}(a+b)(b+c)(c+a)\)
\(\Rightarrow \sqrt{ab(a+b)}+\sqrt{bc(b+c)}+\sqrt{ca(c+a)}\leq \frac{3}{2}\sqrt{(a+b)(b+c)(c+a)}(2)\)
Từ \((1);(2)\Rightarrow M\leq \frac{1}{2}.\frac{3}{2}=\frac{3}{4}(**)\)
Từ \((*); (**)\Rightarrow P\leq M\leq \frac{3}{4}\)
Vậy \(P_{\max}=\frac{3}{4}\Leftrightarrow a=b=c\)
30 tháng 12 2021
\(\dfrac{ab}{\sqrt{ab+2c}}=\dfrac{ab}{\sqrt{ab+\left(a+b+c\right)c}}=\dfrac{ab}{\sqrt{\left(a+c\right)\left(b+c\right)}}=ab\cdot\sqrt{\dfrac{1}{a+b}\cdot\dfrac{1}{b+c}}\le ab\cdot\dfrac{1}{2}\left(\dfrac{1}{a+b}+\dfrac{1}{b+c}\right)=\dfrac{1}{2}\left(\dfrac{ab}{a+b}+\dfrac{ab}{b+c}\right)\)
CMTT: \(\dfrac{bc}{\sqrt{bc+2a}}\le\dfrac{1}{2}\left(\dfrac{bc}{a+b}+\dfrac{bc}{a+c}\right);\dfrac{ac}{\sqrt{ac+2b}}\le\dfrac{1}{2}\left(\dfrac{ac}{b+c}+\dfrac{ac}{b+a}\right)\)
\(\Leftrightarrow P\le\dfrac{1}{2}\left(\dfrac{ab}{c+a}+\dfrac{ab}{c+b}+\dfrac{bc}{b+a}+\dfrac{bc}{c+a}+\dfrac{ac}{b+c}+\dfrac{ac}{b+c}\right)\\ \Leftrightarrow P\le\dfrac{1}{2}\left[\dfrac{b\left(a+c\right)}{a+c}+\dfrac{a\left(b+c\right)}{b+c}+\dfrac{c\left(a+b\right)}{a+b}\right]=\dfrac{1}{2}\left(a+b+c\right)=1\)
Dấu \("="\Leftrightarrow a=b=c=\dfrac{2}{3}\)
NV
Nguyễn Việt Lâm
Giáo viên
3 tháng 8 2021
\(\dfrac{\sqrt{ab+2c^2}}{\sqrt{1+ab-c^2}}=\dfrac{\sqrt{ab+2c^2}}{\sqrt{a^2+b^2+ab}}=\dfrac{ab+2c^2}{\sqrt{\left(a^2+b^2+ab\right)\left(ab+2c^2\right)}}\ge\dfrac{2\left(ab+2c^2\right)}{a^2+b^2+2ab+2c^2}\)
\(\ge\dfrac{2\left(ab+2c^2\right)}{a^2+b^2+a^2+b^2+2c^2}=\dfrac{ab+2c^2}{a^2+b^2+c^2}=ab+2c^2\)
Tương tự và cộng lại:
\(VT\ge ab+bc+ca+2\left(a^2+b^2+c^2\right)=2+ab+bc+ca\)
12 tháng 1 2021
Đặt \(\left(\dfrac{1}{a},\dfrac{1}{b},\dfrac{1}{c}\right)=\left(x,y,z\right)\) với x, y, z > 0 thì ta có \(x+y+z=1\).
Đặt biểu thức ở VT là A. Ta có:
\(A=\sqrt{\dfrac{b^2+2a^2}{a^2b^2}}+\sqrt{\dfrac{c^2+2b^2}{b^2c^2}}+\sqrt{\dfrac{a^2+2c^2}{c^2a^2}}=\sqrt{x^2+2y^2}+\sqrt{y^2+2z^2}+\sqrt{z^2+2x^2}\).
Ta có bất đẳng thức \(\sqrt{a_1^2+a_2^2}+\sqrt{a_3^2+a_4^2}\ge\sqrt{\left(a_1+a_3\right)^2+\left(a_2+a_4\right)^2}\).
Đây là bđt Mincopxki cho hai bộ số thực và dễ dàng cm bằng biến đổi tương đương.
Do đó \(A\ge\sqrt{\left(x+y\right)^2+\left(\sqrt{2}y+\sqrt{2}z\right)^2}+\sqrt{z^2+2x^2}\ge\sqrt{\left(x+y+z\right)^2+\left(\sqrt{2}y+\sqrt{2}z+\sqrt{2}x\right)^2}=\sqrt{1+2}=\sqrt{3}=VP\).
Đẳng thức xảy ra khi a = b = c = 3.
Vậy...
NV
Nguyễn Việt Lâm
Giáo viên
12 tháng 1 2021
Tương tự: \(GT\Rightarrow\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=1\)
\(VT=\dfrac{\sqrt{a^2+a^2+b^2}}{ab}+\dfrac{\sqrt{b^2+b^2+c^2}}{bc}+\dfrac{\sqrt{c^2+a^2+a^2}}{ca}\)
\(VT\ge\dfrac{\sqrt{\dfrac{1}{3}\left(a+a+b\right)^2}}{ab}+\dfrac{\sqrt{\dfrac{1}{3}\left(b+b+c\right)^2}}{bc}+\dfrac{\sqrt{\dfrac{1}{3}\left(c+c+a\right)^2}}{ca}\)
\(VT\ge\sqrt{3}\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)=\sqrt{3}\)
Dấu "=" xảy ra khi \(a=b=c=3\)
16 tháng 5 2017
Lợi dụng Cauchy-Schwarz' inequality ta có:
\(\dfrac{ab}{\sqrt{ab+2c}}=\dfrac{ab}{\sqrt{ab+\left(a+b+c\right)c}}=\dfrac{ab}{\sqrt{ab+ac+bc+c^2}}\)
\(=\dfrac{ab}{\sqrt{\left(a+c\right)\left(b+c\right)}}\le\dfrac{1}{2}\left(\dfrac{ab}{a+c}+\dfrac{ab}{b+c}\right)\)
Tương tự ta cũng có:
\(\dfrac{bc}{\sqrt{bc+2a}}\le\dfrac{1}{2}\left(\dfrac{bc}{a+b}+\dfrac{bc}{a+c}\right);\dfrac{ca}{\sqrt{ca+2b}}\le\dfrac{1}{2}\left(\dfrac{ca}{a+b}+\dfrac{ca}{b+c}\right)\)
Cộng theo vế 3 BĐT trên ta có:
\(P\le\dfrac{1}{2}\left(\dfrac{ab+bc}{a+c}+\dfrac{bc+ca}{a+b}+\dfrac{ab+ca}{b+c}\right)\)
\(=\dfrac{1}{2}\left(\dfrac{b\left(a+c\right)}{a+c}+\dfrac{c\left(a+b\right)}{a+b}+\dfrac{a\left(b+c\right)}{b+c}\right)\)
\(=\dfrac{1}{2}\left(a+b+c\right)=\dfrac{1}{2}\cdot2=1\)
Dấu "=" xảy ra khi \(a=b=c=\dfrac{2}{3}\)
20 tháng 5 2017
Ta có P=\(\dfrac{ab}{\sqrt{ab+\left(a+b+c\right)c}}+\dfrac{bc}{\sqrt{bc+\left(a+b+c\right)a}}+\dfrac{ac}{\sqrt{ac+\left(a+b+c\right)b}}\)
=\(\dfrac{ab}{\sqrt{ab+ac+bc+c^2}}+\dfrac{bc}{\sqrt{bc+ac+ab+a^2}}+\dfrac{ac}{\sqrt{ac+ab+bc+b^2}}\)
=\(\dfrac{ab}{\sqrt{a\left(b+c\right)+c\left(b+c\right)}}+\dfrac{bc}{\sqrt{b\left(a+c\right)+a\left(a+c\right)}}+\dfrac{ac}{\sqrt{c\left(a+b\right)+b\left(a+b\right)}}\)
=\(\dfrac{ab}{\sqrt{\left(a+c\right)\left(b+c\right)}}+\dfrac{bc}{\sqrt{\left(b+a\right)\left(c+a\right)}}+\dfrac{ac}{\sqrt{\left(a+b\right)\left(c+b\right)}}\)
áp dụng bđt Cói ta có:
\(\sqrt{\left(a+c\right)\left(b+c\right)}\)\(\le\)\(\dfrac{2+c}{2}=1+\dfrac{c}{2}\)
\(\sqrt{\left(b+á\right)\left(c+a\right)}\)
30 tháng 8 2021
\(VT=\sqrt{\dfrac{a^2b^2}{c\left(a+b+c\right)+ab}}+\sqrt{\dfrac{b^2c^2}{a\left(a+b+c\right)+bc}}+\sqrt{\dfrac{a^2c^2}{b\left(a+b+c\right)+ac}}\\ VT=\sqrt{\dfrac{a^2b^2}{ac+ab+bc+c^2}}+\sqrt{\dfrac{b^2c^2}{a^2+ac+ab+bc}}+\sqrt{\dfrac{a^2c^2}{ab+bc+b^2+ac}}\\ VT=\sqrt{\dfrac{a^2b^2}{\left(c+a\right)\left(b+c\right)}}+\sqrt{\dfrac{a^2c^2}{\left(b+c\right)\left(a+b\right)}}+\sqrt{\dfrac{b^2c^2}{\left(a+b\right)\left(a+c\right)}}\)
Áp dụng BĐT Cauchy-Schwarz:
\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{\dfrac{b^2c^2}{\left(a+b\right)\left(a+c\right)}}\le\dfrac{\dfrac{bc}{a+b}+\dfrac{bc}{a+c}}{2}\\\sqrt{\dfrac{a^2c^2}{\left(a+b\right)\left(b+c\right)}}\le\dfrac{\dfrac{ca}{a+b}+\dfrac{ca}{b+c}}{2}\\\sqrt{\dfrac{a^2b^2}{\left(b+c\right)\left(a+c\right)}}\le\dfrac{\dfrac{ab}{b+c}+\dfrac{ab}{a+c}}{2}\end{matrix}\right.\)
\(\Rightarrow VT\le\dfrac{\left(\dfrac{bc}{a+b}+\dfrac{ca}{a+b}\right)+\left(\dfrac{ca}{b+c}+\dfrac{ab}{b+c}\right)+\left(\dfrac{bc}{a+c}+\dfrac{ab}{a+c}\right)}{2}\\ \Rightarrow VT\le\dfrac{a+b+c}{2}=\dfrac{2}{2}=1\)
Dấu \("="\Leftrightarrow a=b=c=\dfrac{2}{3}\)
TL
1 tháng 11 2018
Ta có:
1+a2 = ab+bc+ca+a2 = a(a+b)+c(a+b)=(a+b)(a+c)
Tương tự: 1+b2 = (b+c)(b+a)
1+c2 = (c+a)(c+b)
\(\Rightarrow\) P = \(2a\sqrt{\dfrac{1}{\left(a+b\right)\left(a+c\right)}}+2b\sqrt{\dfrac{1}{\left(b+c\right)\left(b+a\right)}}+2c\sqrt{\dfrac{1}{\left(c+a\right)\left(c+b\right)}}\)
Áp dụng BĐT Cô-si ta có:
P\(\le\)\(a\left(\dfrac{1}{a+b}+\dfrac{1}{a+c}\right)+b\left(\dfrac{1}{4\left(b+c\right)}+\dfrac{1}{b+a}\right)+c\left(\dfrac{1}{4\left(c+b\right)}+\dfrac{1}{c+a}\right)\)\(\le\)\(\dfrac{a}{a+b}+\dfrac{a}{a+c}+\dfrac{b}{4\left(b+c\right)}+\dfrac{b}{b+a}+\dfrac{c}{4\left(c+b\right)}+\dfrac{c}{c+a}\)
= \(\dfrac{1}{4}+2=\dfrac{9}{4}\)
\(\Rightarrow\)Pmin = \(\dfrac{9}{4}\)
Dấu "=" xảy ra\(\Leftrightarrow\) b=c=\(\dfrac{a}{7}\)=\(\dfrac{\sqrt{15}}{15}\) \(\Rightarrow\) a = \(\dfrac{7\sqrt{15}}{15}\)
18 tháng 9 2018
\(\sqrt{\dfrac{a+b}{c+ab}}+\sqrt{\dfrac{b+c}{a+bc}}+\sqrt{\dfrac{c+a}{b+ac}}\)
N
30 tháng 9 2017
Bài này có xuất hiện rồi ,you vào mục tìm kiếm là thấy liền.
Lời giải vắn tắt:
\(A=\sum\sqrt{\dfrac{ab+2c^2}{1+ab-c^2}}=\sum\dfrac{ab+2c^2}{\sqrt{\left(ab+2c^2\right)\left(1+ab-c^2\right)}}\ge\sum\dfrac{2\left(ab+2c^2\right)}{1+2ab+c^2}=\sum\dfrac{2\left(ab+2c^2\right)}{\left(a+b\right)^2+2c^2}\ge\sum\dfrac{2\left(ab+2c^2\right)}{2\left(a^2+b^2\right)+2c^2}=\sum\left(ab+2c^2\right)=ab+bc+ca+2\)
( thay \(a^2+b^2+c^2=1\))
Ta có: $\sqrt[]{ab+2c}=\sqrt[]{ab+(a+b+c)c}=\sqrt[]{ab+ac+bc+c^2}=\sqrt[]{(c+a)(c+b)}$ (do $a+b+c=2$)
Nên $\dfrac{ab}{\sqrt[]{ab+2c}}=\dfrac{ab}{\sqrt[]{(c+a).(c+b)}}=ab.\sqrt[]{\dfrac{1}{a+c}.\dfrac{1}{b+c}}$
Áp dụng bất đẳng thức Cauchy cho $\dfrac{1}{a+c};\dfrac{1}{b+c}>0$ có:
$\sqrt[]{\dfrac{1}{a+c}.\dfrac{1}{b+c}} \leq \dfrac{1}{2}.(\dfrac{1}{a+c}+\dfrac{1}{b+c})$
Nên $\dfrac{ab}{\sqrt[]{ab+2c}} \leq \dfrac{1}{2}.ab.(\dfrac{1}{a+c}+\dfrac{1}{b+c})= \dfrac{1}{2}.(\dfrac{ab}{a+c}+\dfrac{ab}{b+c})$
Tương tự ta có: $\dfrac{bc}{\sqrt[]{bc+2a}} \leq \dfrac{1}{2}.(\dfrac{bc}{a+b}+\dfrac{bc}{a+c})$
$\dfrac{ca}{\sqrt[]{ca+2b}} \leq \dfrac{1}{2}.(\dfrac{ca}{b+a}+\dfrac{ca}{b+c})$
Nên $Q \leq \dfrac{1}{2}.(\dfrac{ab}{a+c}+\dfrac{ab}{b+c})+\dfrac{1}{2}.(\dfrac{bc}{a+b}+\dfrac{bc}{a+c})+ \dfrac{1}{2}.(\dfrac{ca}{b+a}+\dfrac{ca}{b+c})=\dfrac{1}{2}(\dfrac{ab}{a+c}+\dfrac{ab}{b+c}+\dfrac{bc}{a+b}+\dfrac{bc}{a+c}+\dfrac{ca}{b+a}+\dfrac{ca}{b+c})=\dfrac{1}{2}.(\dfrac{b(a+c)}{a+c}+\dfrac{a(b+c)}{b+c}+\dfrac{c(a+b)}{a+b}=\dfrac{1}{2}.(a+b+c)=1$ (do $a+b+c=2$)
Dấu $=$ xảy ra khi $a=b=c=\dfrac{2}{3}$