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Áp dụng bất đẳng thức cộng mẫu số
\(\Rightarrow\dfrac{a^2}{a+\sqrt[3]{bc}}+\dfrac{b^2}{b+\sqrt[3]{ca}}+\dfrac{c^2}{c+\sqrt[3]{ab}}\ge\dfrac{\left(a+b+c\right)^2}{a+b+c+\sqrt[3]{bc}+\sqrt[3]{ca}+\sqrt[3]{ab}}\)
\(\Rightarrow\dfrac{a^2}{a+\sqrt[3]{bc}}+\dfrac{b^2}{b+\sqrt[3]{ca}}+\dfrac{c^2}{c+\sqrt[3]{ab}}\ge\dfrac{9}{3+\sqrt[3]{bc}+\sqrt[3]{ca}+\sqrt[3]{ab}}\)
Chứng minh rằng \(\dfrac{9}{3+\sqrt[3]{bc}+\sqrt[3]{ca}+\sqrt[3]{ab}}\ge\dfrac{3}{2}\)
\(\Leftrightarrow18\ge3\left(3+\sqrt[3]{bc}+\sqrt[3]{ca}+\sqrt[3]{ab}\right)\)
\(\Leftrightarrow18\ge9+3\sqrt[3]{bc}+3\sqrt[3]{ca}+3\sqrt[3]{ab}\)
\(\Leftrightarrow9\ge3\sqrt[3]{ab}+3\sqrt[3]{bc}+3\sqrt[3]{ca}\)
Áp dụng bất đẳng thức Cauchy cho 3 bộ số thực không âm
\(\Rightarrow\left\{{}\begin{matrix}a+b+1\ge3\sqrt[3]{ab}\\b+c+1\ge3\sqrt[3]{bc}\\c+a+1\ge3\sqrt[3]{ca}\end{matrix}\right.\)
\(\Rightarrow2\left(a+b+c\right)+3\ge3\sqrt[3]{ab}+3\sqrt[3]{bc}+3\sqrt[3]{ca}\)
\(\Rightarrow9\ge3\sqrt[3]{ab}+3\sqrt[3]{bc}+3\sqrt[3]{ca}\) ( đpcm )
Vì \(\dfrac{9}{3+\sqrt[3]{bc}+\sqrt[3]{ca}+\sqrt[3]{ab}}\ge\dfrac{3}{2}\)
Mà \(\dfrac{a^2}{a+\sqrt[3]{bc}}+\dfrac{b^2}{b+\sqrt[3]{ca}}+\dfrac{c^2}{c+\sqrt[3]{ab}}\ge\dfrac{9}{3+\sqrt[3]{bc}+\sqrt[3]{ca}+\sqrt[3]{ab}}\)
\(\Rightarrow\dfrac{a^2}{a+\sqrt[3]{bc}}+\dfrac{b^2}{b+\sqrt[3]{ca}}+\dfrac{c^2}{c+\sqrt[3]{ab}}\ge\dfrac{3}{2}\)( đpcm )
Áp dụng BĐT AM-GM và Cauchy-Schwarz ta có:
\(\sum\frac{a^2}{a+\sqrt[3]{bc}}\geq\sum\frac{a^2}{a+\frac{b+c+1}{3}}=\sum\frac{9a^2}{3(3a+b+c)+a+b+c}\)
\(=\sum\frac{9a^2}{10a+4b+4c}\geq\frac{9(a+b+c)^2}{(10a+4b+4c)}=\frac{9(a+b+c)^2}{18(a+b+c)}=\frac{3}{2}\)
Ta có\(\sqrt{2}\) A=\(\sum\sqrt{\dfrac{2ab}{a^2+b^2}}=\sum\dfrac{\sqrt{2ab\left(a^2+b^2\right)}}{a^2+b^2}\ge\sum\dfrac{2ab}{a^2+b^2}\)
=> \(\sqrt{2}A+3=\sum\dfrac{\left(a+b\right)^2}{a^2+b^2}\ge\dfrac{\left(2a+2b+2c\right)^2}{2\left(a^2+b^2+c^2\right)}=\dfrac{2\left(a^2+b^2+c^2+2ab+2bc+2ca\right)}{a^2+b^2+c^2}=\dfrac{4\left(a^2+b^2+c^2\right)}{a^2+b^2+c^2}=4\Rightarrow\sqrt{2}A+3\ge4\)
=> \(A\ge\dfrac{1}{\sqrt{2}}\)
dấu = xảy ra <=> 2 số =1, 1 số =0
a) CM:\(\sqrt{\left(n+1\right)^2}+\sqrt{n^2}=\left(n+1\right)^2-n^2\)
\(\Leftrightarrow n+1+n=\left(n+1-n\right)\left(n+1+n\right)\)
\(\Leftrightarrow2n+1=1\left(2n+1\right)\)
\(\Leftrightarrow2n+1=2n+1\)
\(\Rightarrow\sqrt{\left(n+1\right)^2}+\sqrt{n^2}=\left(n+1\right)^2-n^2\)
Câu b) ý 2:
Áp dụng BĐT cô si ta có :
\(\dfrac{a}{b}+\dfrac{b}{c}\ge2\sqrt{\dfrac{a}{c}}\\ \dfrac{b}{c}+\dfrac{c}{a}\ge2\sqrt{\dfrac{b}{a}}\\ \dfrac{c}{a}+\dfrac{a}{b}\ge2\sqrt{\dfrac{c}{b}}\\ \Leftrightarrow2\left(\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\right)\ge2\left(\sqrt{\dfrac{a}{c}}+\sqrt{\dfrac{b}{a}}+\sqrt{\dfrac{c}{b}}\right)\\ \Rightarrowđpcm\)
\(\sqrt{\dfrac{a+b}{c+ab}}+\sqrt{\dfrac{b+c}{a+bc}}+\sqrt{\dfrac{c+a}{b+ac}}\)
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 \(a^2+b^2+c^2=1\))
Lời giải:
\(a+b+c=abc\Rightarrow a(a+b+c)=a^2bc\)
\(\Rightarrow a(a+b+c)+bc=bc(a^2+1)\)
\(\Leftrightarrow (a+b)(a+c)=bc(a^2+1)\Rightarrow a^2+1=\frac{(a+b)(a+c)}{bc}\)
\(\Rightarrow \frac{1}{\sqrt{a^2+1}}=\sqrt{\frac{bc}{(a+b)(a+c)}}\)
Hoàn toàn tương tự với các phân thức còn lại
\(\Rightarrow \text{VT}=\frac{1}{\sqrt{a^2+1}}+\frac{1}{\sqrt{b^2+1}}+\frac{1}{\sqrt{c^2+1}}=\sqrt{\frac{bc}{(a+b)(a+c)}}+\sqrt{\frac{ac}{(b+a)(b+c)}}+\sqrt{\frac{ab}{(c+a)(c+b)}}\)
Áp dụng BĐT Cauchy:
\(\sqrt{\frac{bc}{(a+b)(a+c)}}+\sqrt{\frac{ac}{(b+a)(b+c)}}+\sqrt{\frac{ab}{(c+a)(c+b)}}\leq \frac{1}{2}\left(\frac{b}{a+b}+\frac{c}{a+c}\right)+\frac{1}{2}\left(\frac{a}{b+a}+\frac{c}{b+c}\right)+\frac{1}{2}\left(\frac{a}{c+a}+\frac{b}{c+b}\right)\)
\(=\frac{1}{2}\left(\frac{b+a}{b+a}+\frac{c+b}{c+b}+\frac{c+a}{c+a}\right)=\frac{3}{2}\)
\(\Rightarrow \text{VT}\leq \frac{3}{2}\) (đpcm)
Dấu "=" xảy ra khi $a=b=c=\sqrt{3}$
Lời giải:
Ta có: \(\text{VT}=\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}=\frac{a^2}{b}-a+b+\frac{b^2}{c}-b+c+\frac{c^2}{a}-c+a\)
\(=\frac{a^2-ab+b^2}{b}+\frac{b^2-bc+c^2}{c}+\frac{c^2-ca+a^2}{a}\)
Áp dụng BĐT AM-GM:
\(\frac{a^2-ab+b^2}{b}+b\geq 2\sqrt{a^2-ab+b^2}\)
\(\frac{b^2-bc+c^2}{c}+c\geq 2\sqrt{b^2-bc+c^2}\)
\(\frac{c^2-ca+a^2}{a}+a\geq 2\sqrt{c^2-ca+a^2}\)
Cộng theo vế:
\(\Rightarrow \text{VT}+(a+b+c)\geq 2(\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}+\sqrt{c^2-ca+a^2})(1)\)
Lại có:
\(\sqrt{a^2-ab+b^2}=\sqrt{\frac{3}{4}(a-b)^2+\frac{1}{4}(a+b)^2}\geq \sqrt{\frac{1}{4}(a+b)^2}=\frac{a+b}{2}\)
TT: \(\sqrt{b^2-bc+c^2}\geq \frac{b+c}{2}; \sqrt{c^2-ca+a^2}\geq \frac{c+a}{2}\)
Suy ra: \(\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}+\sqrt{c^2-ca+a^2}\geq a+b+c(2)\)
Từ \((1);(2)\Rightarrow \text{VT}\geq \sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}+\sqrt{c^2-ca+a^2}\) (đpcm)
Dấu bằng xảy ra khi \(a=b=c\)
Hay,hay lắm!