Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
làm rõ \(\sum_{cyc}\frac{a}{a+b}-\frac{3}{2}=\sum_{cyc}\left(\frac{a}{a+b}-\frac{1}{2}\right)=\sum_{cyc}\frac{a-b}{2(a+b)}\)
\(=\sum_{cyc}\frac{(a-b)(c^2+ab+ac+bc)}{2\prod\limits_{cyc}(a+b)}=\sum_{cyc}\frac{c^2a-c^2b}{2\prod\limits_{cyc}(a+b)}\)
\(=\sum_{cyc}\frac{a^2b-a^2c}{2\prod\limits_{cyc}(a+b)}=\frac{(a-b)(a-c)(b-c)}{2\prod\limits_{cyc}(a+b)}\geq0\) (đúng)
ok thỏa thuận rồi tui làm nửa sau thui nhé :D
Đặt \(a^2=x;b^2=y;c^2=z\) thì ta có:
\(VT=\sqrt{\dfrac{x}{x+y}}+\sqrt{\dfrac{y}{y+z}}+\sqrt{\dfrac{z}{x+z}}\)
Lại có: \(\sqrt{\dfrac{x}{x+y}}=\sqrt{\dfrac{x}{\left(x+y\right)\left(x+z\right)}\cdot\sqrt{x+z}}\)
Tương tự cộng theo vế rồi áp dụng BĐT C-S ta có:
\(VT^2\le2\left(x+y+z\right)\left[\dfrac{x}{\left(x+y\right)\left(x+z\right)}+\dfrac{y}{\left(y+z\right)\left(y+x\right)}+\dfrac{z}{\left(z+x\right)\left(z+y\right)}\right]\)
\(\Leftrightarrow VT^2\le\dfrac{4\left(x+y+z\right)\left(xy+yz+xz\right)}{\left(x+y\right)\left(y+z\right)\left(x+z\right)}\)
Vì \(VP^2=\dfrac{9}{2}\) nên cần cm \(VT\le \frac{9}{2}\)
\(\Leftrightarrow9\left(x+y\right)\left(y+z\right)\left(x+z\right)\ge8\left(x+y+z\right)\left(xy+yz+xz\right)\)
Can you continue
áp dụng bdt côsi \(\dfrac{a^2}{b^3}+\dfrac{1}{a}+\dfrac{1}{a}\ge\dfrac{3}{b}\)
tuông tu \(\dfrac{b^2}{c^3}+\dfrac{1}{b}+\dfrac{1}{b}\ge\dfrac{3}{c}\)
\(\dfrac{c^2}{a^3}+\dfrac{1}{c}+\dfrac{1}{c}\ge\dfrac{3}{a}\)
suy ra vt +\(2\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\ge3\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\)
suy ra dpcm
dau = xay ra khi a=b=c
\(\dfrac{a^3}{b^3}+\dfrac{a^3}{b^3}+1+\dfrac{b^3}{c^3}+\dfrac{b^3}{c^3}+1+\dfrac{c^3}{a^3}+\dfrac{c^3}{a^3}+1\ge3\left(\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\right)\)
\(\Leftrightarrow2\left(\dfrac{a^3}{b^3}+\dfrac{b^3}{c^3}+\dfrac{c^3}{a^3}\right)\ge3\left(\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\right)-3\)
\(\ge2\left(\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\right)+3-3=2\left(\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\right)\)
\(\Leftrightarrow\dfrac{a^3}{b^3}+\dfrac{b^3}{c^3}+\dfrac{c^3}{a^3}\ge\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\)
giả sử \(a>b>c>0\) thì ta có :
\(\dfrac{a^2}{b^2}\left(\dfrac{a}{b}-1\right)+\dfrac{b^2}{c^2}\left(\dfrac{b}{c}-1\right)+\dfrac{c^2}{a^2}\left(\dfrac{c}{a}-1\right)\ge2\dfrac{a}{b}+\dfrac{c^2}{a^2}\left(\dfrac{c}{a}-1\right)\)
\(=\dfrac{2a}{b}+\dfrac{c^3}{a^3}-\dfrac{c^2}{a^2}\ge0\)
làm tương tự cho trường hợp \(c>b>a>0\) ; \(b>a>c\) và \(b>c>a\)
\(\Rightarrow\left(đpcm\right)\)
có: \(\dfrac{1}{c}=\dfrac{2}{b}-\dfrac{1}{a}=\dfrac{2a-b}{ab}\Rightarrow2a-b=\dfrac{ab}{c}\)
tương tự ta cũng có \(2c-b=\dfrac{bc}{a}\)
\(VT=\dfrac{c\left(a+b\right)}{ab}+\dfrac{a\left(c+b\right)}{bc}=\dfrac{c}{a}+\dfrac{c}{b}+\dfrac{a}{b}+\dfrac{a}{c}=\left(\dfrac{c}{a}+\dfrac{a}{c}\right)+\dfrac{a+c}{b}\)
Áp dụng BĐt AM-GM:\(\dfrac{c}{a}+\dfrac{a}{c}\ge2\)
và \(\dfrac{2}{b}=\dfrac{1}{a}+\dfrac{1}{c}\ge\dfrac{4}{a+c}\Leftrightarrow a+c\ge2b\)
do đó \(VT\ge2+2=4\)
Dấu = xảy ra khi a=b=c
Áp dụng bất đẳng thức \(AM-GM\) cho 2 số dương ta có:
\(VT=\dfrac{a^3+b^3+c^3}{2abc}+\dfrac{a^2+b^2}{c^2+ab}+\dfrac{b^2+c^2}{a^2+bc}+\dfrac{a^2+c^2}{b^2+ac}\ge\dfrac{3abc}{2abc}+\dfrac{2ab}{c^2+ab}+\dfrac{2bc}{a^2+bc}+\dfrac{2ac}{b^2+ac}=\dfrac{3}{2}+2\left(\dfrac{ab}{c^2+ab}+\dfrac{bc}{a^2+bc}+\dfrac{ac}{b^2+ac}\right)\)
Áp dụng bất đẳng thức \(Cauchy-Schwarz\) \(\dfrac{ab}{c^2+ab}+\dfrac{bc}{a^2+bc}+\dfrac{ac}{b^2+ac}=\dfrac{a^2b^2}{c^2ab+a^2b^2}+\dfrac{b^2c^2}{a^2bc+b^2c^2}+\dfrac{a^2c^2}{b^2ac+a^2c^2}\ge\dfrac{\left(ab+bc+ac\right)^2}{c^2ab+a^2b^2+a^2bc+b^2c^2+b^2ac+a^2c^2}\)
Đặt: \(\left\{{}\begin{matrix}ab=x\\bc=y\\ac=z\end{matrix}\right.\) ta được: \(\dfrac{ab}{c^2+ab}+\dfrac{bc}{a^2+bc}+\dfrac{ac}{b^2+ac}\ge\dfrac{\left(x+y+z\right)^2}{x^2+y^2+z^2+xy+xz+xy}\ge\dfrac{3\left(xy+yz+xz\right)}{2\left(xy+yz+xz\right)}=\dfrac{3}{2}\)
Nên: \(\dfrac{3}{2}+2\left(\dfrac{ab}{c^2+ab}+\dfrac{bc}{a^2+bc}+\dfrac{ac}{b^2+ac}\right)\ge\dfrac{3}{2}+2.\dfrac{3}{2}=\dfrac{9}{2}\)
Mà: \(VT\ge\dfrac{3}{2}+2\left(\dfrac{ab}{c^2+ab}+\dfrac{bc}{a^2+bc}+\dfrac{ac}{b^2+ac}\right)\Leftrightarrow VT\ge\dfrac{3}{2}\left(đpcm\right)\)
Lời giải:
Áp dụng BĐT AM-GM ta có: \(\frac{a^3+b^3+c^3}{2abc}\geq \frac{3\sqrt[3]{a^3b^3c^3}}{2abc}=\frac{3abc}{2abc}=\frac{3}{2}\) (1)
Áp dụng BĐT Cauchy-Schwarz:
\(\frac{a^2+b^2}{c^2+ab}+\frac{b^2+c^2}{a^2+bc}+\frac{a^2+c^2}{b^2+ac}\geq \frac{(\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2})^2}{a^2+b^2+c^2+ab+bc+ac}\) (2)
Có:
\((\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2})^2=2(a^2+b^2+c^2)+2\sqrt{(a^2+b^2)(b^2+c^2)}+2\sqrt{(b^2+c^2)(c^2+a^2)}+\sqrt{(a^2+b^2)(c^2+a^2)}\)
Áp dụng BĐT Bunhiacopxky:
\(\sqrt{(a^2+b^2)(b^2+c^2)}\geq \sqrt{(ac+b^2)^2}=ac+b^2\)
\(\sqrt{(b^2+c^2)(c^2+a^2)}\geq \sqrt{(ba+c^2)^2}=ba+c^2\)
\(\sqrt{(a^2+b^2)(c^2+a^2)}\geq \sqrt{(a^2+bc)^2}=a^2+bc\)
\(\Rightarrow (\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2})^2\geq 2(a^2+b^2+c^2)+2(a^2+b^2+c^2+ab+bc+ac)\)
\(\geq a^2+b^2+c^2+ab+bc+ac+2(a^2+b^2+c^2+ab+bc+ac)\) (AM-GM)
Hay \((\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2})^2\geq 3(a^2+b^2+c^2+ab+bc+ac)\) (3)
Từ \((2); (3)\Rightarrow \frac{a^2+b^2}{c^2+ab}+\frac{b^2+c^2}{a^2+bc}+\frac{a^2+c^2}{b^2+ac}\geq 3\) (4)
Từ \((1); (4)\Rightarrow \frac{a^3+b^3+c^3}{2abc}+\frac{a^2+b^2}{c^2+ab}+\frac{b^2+c^2}{a^2+bc}+\frac{c^2+a^2}{b^2+ac}\geq \frac{9}{2}\)
Ta có đpcm.
Dấu bằng xảy ra khi $a=b=c$
\(\dfrac{a^5}{b^3}+\dfrac{a^5}{b^3}+\dfrac{a^5}{b^3}+\dfrac{a^5}{b^3}+b^2\ge5\sqrt[5]{\dfrac{a^{20}b^2}{b^{12}}}=5.\dfrac{a^4}{b^2}\)
\(\Rightarrow4.\dfrac{a^5}{b^3}+b^2\ge5.\dfrac{a^4}{b^2}\)
Tương tự: \(4.\dfrac{b^5}{c^3}+c^2\ge5\dfrac{b^4}{c^2};4\dfrac{c^5}{a^3}+a^2\ge5.\dfrac{c^4}{a^2}\)
\(\Rightarrow4\left(\dfrac{a^5}{b^3}+\dfrac{b^5}{c^3}+\dfrac{c^5}{a^3}\right)+a^2+b^2+c^2\ge5\left(\dfrac{c^4}{a^2}+\dfrac{a^4}{b^2}+\dfrac{b^4}{c^2}\right)\)
Lại có: \(\dfrac{a^5}{b^3}+\dfrac{a^5}{b^3}+b^2+b^2+b^2\ge5a^2\)
\(\Rightarrow2.\dfrac{a^5}{b^3}+3b^2\ge5a^2\), tương tự: \(2.\dfrac{b^5}{c^3}+3c^2\ge5b^2;2\dfrac{c^5}{a^3}+3a^2\ge5c^2\)
\(\Rightarrow\dfrac{a^5}{b^3}+\dfrac{b^5}{c^3}+\dfrac{c^5}{a^3}\ge a^2+b^2+c^2\)
\(\Rightarrow\dfrac{a^5}{b^3}+\dfrac{b^5}{c^3}+\dfrac{c^5}{a^3}+4.\left(\dfrac{a^5}{b^3}+\dfrac{b^5}{c^3}+\dfrac{c^5}{a^3}\right)\ge4.\left(\dfrac{a^5}{b^3}+\dfrac{b^5}{c^3}+\dfrac{c^5}{a^3}\right)+a^2+b^2+c^2\ge5.\left(\dfrac{c^4}{a^2}+\dfrac{a^4}{b^2}+\dfrac{b^4}{c^2}\right)\)
\(\Rightarrow dpcm\)
giả sử \(a>b>c>0\) thì ta có :
\(\dfrac{a^4}{b^2}\left(\dfrac{a}{b}-1\right)+\dfrac{b^4}{c^2}\left(\dfrac{b}{c}-1\right)+\dfrac{c^4}{a^2}\left(\dfrac{c}{a}-1\right)\ge\dfrac{2a^2b}{c}+\dfrac{c^5}{a^3}-\dfrac{c^4}{a^2}\)
\(\ge\dfrac{2c^4b}{a}-\dfrac{c^4}{a^2}=\dfrac{c^4}{a}\left(2b-\dfrac{1}{a}\right)>0\)
làm tương tự cho trường hợp \(c>b>a>0\) ; \(b>a>c\) và \(b>c>a\)
\(\Rightarrow\left(đpcm\right)\)
mấy câu cậu câu đăng khác bn làm tương tự nha . nếu bn lm không được thì có j mk lm luôn cho còn h mk bạn rồi :(
Áp dụng BĐT Cauchy cho 2 số dương:
\(\left\{{}\begin{matrix}\dfrac{a^2}{b}+b\ge2\sqrt{\dfrac{a^2}{b}.b}=2a\\\dfrac{b^2}{c}+c\ge2\sqrt{\dfrac{b^2}{c}.c}=2b\\\dfrac{c^2}{a}+a\ge2\sqrt{\dfrac{c^2}{a}.a}=2c\end{matrix}\right.\)
\(\Rightarrow\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{a}+a+b+c\ge2a+2b+2c\)
\(\Rightarrow\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{a}\ge a+b+c\left(đpcm\right)\)
Dấu "=" xay ra \(\Leftrightarrow a=b=c\)
Áp dụng BĐT cosi cho 3 số a,b,c dương:
\(\dfrac{a^2}{b}+b\ge2\sqrt{\dfrac{a^2b}{b}}=2a\\ \dfrac{b^2}{c}+c\ge2\sqrt{\dfrac{b^2c}{c}}=2b\\ \dfrac{c^2}{a}+a\ge2\sqrt{\dfrac{c^2a}{a}}=2c\)
Cộng vế theo vế 3 BĐT trên
\(\Leftrightarrow\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{a}+a+b+c\ge2\left(a+b+c\right)\\ \Leftrightarrow\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{a}\ge a+b+c\)
Dấu \("="\Leftrightarrow a=b=c\)