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Áp dụng BĐT Bunyakovsky dạng phân thức ta có:
\(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=\frac{a^2}{ab+ac}+\frac{b^2}{bc+ba}+\frac{c^2}{ca+cb}\)
\(\ge\frac{\left(a+b+c\right)^2}{ab+bc+bc+ca+ca+ab}=\frac{\left(a+b+c\right)^2}{2\left(ab+bc+ca\right)}\)
Mà \(ab+bc+ca\le\frac{\left(a+b+c\right)^2}{3}\)
\(\Rightarrow\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge\frac{\left(a+b+c\right)^2}{\frac{2\left(a+b+c\right)^2}{3}}=\frac{3}{2}\)
Dấu "=" xảy ra khi: a = b = c
Hy vọng a;b;c dương
Khi đó: \(\frac{a^2}{b^2}+1\ge\frac{2a}{b}\) ; \(\frac{b^2}{c^2}+1\ge\frac{2b}{c}\) ; \(\frac{c^2}{a^2}+1\ge\frac{2c}{a}\)
\(\Rightarrow\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}+3\ge2\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\)
\(\Leftrightarrow\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}-3\right)\)
\(\Rightarrow\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+3\sqrt[3]{\frac{abc}{abc}}-3\)
\(\Rightarrow\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\)
Dấu "=" xảy ra khi \(a=b=c\)
Áp dụng bất đẳng thức \(a^2+b^2\ge2ab\)
ta có\(\frac{a^2}{b^2}+\frac{b^2}{c^2}\ge2\frac{ab}{bc}=2\frac{a}{c}\)
tương tự:\(\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge2\frac{b}{a}\)
\(\frac{a^2}{b^2}+\frac{c^2}{a^2}\ge\frac{c}{b}\)
Cộng 3 về bất đẳng thức trên lại với nhau ta đươc:\(2\left(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\right)\ge2\left(\frac{c}{b}+\frac{b}{a}+\frac{a}{c}\right)\)
\(\Leftrightarrow\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{c}{b}+\frac{b}{a}+\frac{a}{c}\)
Dấu "=" xảy ra khi \(\frac{a}{b}=\frac{b}{c}=\frac{c}{a}\)
Áp dụng BĐT Cô - si cho các số dương ta có :
+ ) \(\frac{a^2}{b^2}+\frac{b^2}{c^2}\ge2\sqrt{\frac{a^2}{b^2}.\frac{b^2}{c^2}}=\frac{2a}{c}\left(1\right)\)
Cmt ta có : \(\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{2b}{a}\left(2\right)\)
+ ) \(\frac{a^2}{b^2}+\frac{c^2}{a^2}\ge\frac{2c}{b}\left(3\right)\)
Cộng vế với vế của các BĐT \(\left(1\right),\left(2\right),\left(3\right)\) ta được :
\(2\left(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\right)\ge2\left(\frac{c}{b}+\frac{b}{a}+\frac{a}{c}\right)\)
\(\Leftrightarrow\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{c}{b}+\frac{b}{a}+\frac{a}{c}\left(đpcm\right)\)
Chúc bạn học tốt !!!
Áp dụng bất đẳng thức AM-GM:
\(\frac{a^2}{b^2}+\frac{b^2}{c^2}\ge2\sqrt{\frac{a^2b^2}{b^2c^2}}=2\sqrt{\frac{a^2}{c^2}}=2\left|\frac{a}{c}\right|\ge\frac{2a}{c}\)
Chứng minh tương tự: \(\hept{\begin{cases}\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{2b}{a}\\\frac{a^2}{b^2}+\frac{c^2}{a^2}\ge\frac{2c}{b}\end{cases}}\)
Cộng theo vế: \(2\left(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\right)\ge2\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\)
\(\Rightarrow\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\)
Dấu "=" khi \(a=b=c\)
Ta có: \(\left(\frac{a}{b}-\frac{b}{c}-\frac{c}{a}\right)^2\ge0\)
<=>\(\frac{a^2}{b^2}-\frac{2a}{c}+\frac{b^2}{c}+\frac{c^2}{a^2}-\frac{2c}{b}-\frac{2b}{a}\ge0\)
<=>\(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{2c}{b}+\frac{2b}{a}+\frac{2a}{c}\)
cùng đường nếu a,b,c > hoặc = 0 thì dễ
Cauchy-Schwarz trực tiếp là được, hoặc chúng ta có thể dùng Cô-si:
\(\frac{a^2}{b+c}+\frac{b+c}{4}\ge2\sqrt{\frac{a^2\left(b+c\right)}{4\left(b+c\right)}}=a\)
Tương tự: \(\frac{b^2}{c+a}+\frac{c+a}{4}\ge b\); \(\frac{c^2}{a+b}+\frac{a+b}{4}\ge c\)
Cộng vế với vế:
\(\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}+\frac{a+b+c}{2}\ge a+b+c\)
\(\Leftrightarrow\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\ge\frac{a+b+c}{2}\)
Dấu "=" xảy ra khi \(a=b=c\)
Đặt \(b+c=x;a+c=y;a+b=z\)
Áp dụng bđt Bunhiacopxki ta có :
\(\left(\frac{a^2}{x}+\frac{b^2}{y}+\frac{c^2}{z}\right)\left(\sqrt{x}^2+\sqrt{y}^2+\sqrt{z}^2\right)\ge\left(\frac{a}{\sqrt{x}}.\sqrt{x}+\frac{b}{\sqrt{y}}.\sqrt{y}+\frac{c}{\sqrt{z}}.\sqrt{z}\right)^2\)
\(\Leftrightarrow\left(\frac{a^2}{x}+\frac{b^2}{y}+\frac{c^2}{z}\right)\left(x+y+z\right)\ge\left(a+b+c\right)^2\)
\(\Leftrightarrow\frac{a^2}{x}+\frac{b^2}{y}+\frac{c^2}{z}\ge\frac{\left(a+b+c\right)^2}{x+y+z}\)
\(\Leftrightarrow\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b}\ge\frac{\left(a+b+c\right)^2}{\left(b+c\right)+\left(a+c\right)+\left(a+b\right)}\)
\(\Leftrightarrow\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b}\ge\frac{\left(a+b+c\right)^2}{2\left(a+b+c\right)}=\frac{a+b+c}{2}\) (đpcm)
Dấu "=" xay ra \(\Leftrightarrow a=b=c\)
Áp dụng S-vác-sơ, ta có
\(\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b}\ge\frac{\left(a+b+c\right)^2}{b+c+a+c+a+b}\)
\(=\frac{\left(a+b+c\right)^2}{2\left(a+b+c\right)}=\frac{a+b+c}{2}\)