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\(GT\Rightarrow\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}=6\)
Ta có:
\(2\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)\ge2\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\right)\)
\(\frac{1}{a^2}+1+\frac{1}{b^2}+1+\frac{1}{c^2}+1\ge2\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)
Cộng vế với vế:
\(3\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)+3\ge2\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\right)=12\)
\(\Rightarrow\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\ge3\)
cho a, b, c > 0 thỏa mãn a+b+c=3. Cmr:
\(\frac{a+1}{b^2+1}+\frac{b+1}{c^2+1}+\frac{c+1}{a^2+1}\ge3\)
\(\frac{a+1}{b^2+1}=a+1-\frac{b^2\left(a+1\right)}{b^2+1}\ge a+1-\frac{b^2\left(a+1\right)}{2b}=a+1-\frac{b\left(a+1\right)}{2}\)
Tương tự: \(\frac{b+1}{c^2+1}\ge b+1-\frac{c\left(b+1\right)}{2}\) ; \(\frac{c+1}{a^2+1}\ge c+1-\frac{a\left(c+1\right)}{2}\)
Cộng vế với vế:
\(VT\ge6-\frac{1}{2}\left(ab+bc+ca+a+b+c\right)\)
\(VT\ge\frac{9}{2}-\frac{1}{2}\left(ab+bc+ca\right)\ge\frac{9}{2}-\frac{1}{6}\left(a+b+c\right)^2=3\)
Dấu "=" xảy ra khi \(a=b=c=1\)
\(\frac{1}{2-a}+\frac{1}{2-b}+\frac{1}{2-c}\ge3\)
\(\Leftrightarrow\frac{\left(2-b\right)\left(2-c\right)+\left(2-c\right)\left(2-a\right)+\left(2-a\right)\left(2-b\right)}{\left(2-a\right)\left(2-b\right)\left(2-c\right)}\ge3\)\(\Leftrightarrow\frac{4-2b-2c+bc+4-2c-2a+ca+4-2a-2b+ab}{\left(4-2a-2b+ab\right)\left(2-c\right)}\ge3\)\(\Leftrightarrow\frac{12-4\left(a+b+c\right)+\left(ab+bc+ca\right)}{8-4\left(a+b+c\right)+2\left(ab+bc+ca\right)-abc}\ge3\)
\(\Leftrightarrow12-4\left(a+b+c\right)+\left(ab+bc+ca\right)\ge\) \(24-12\left(a+b+c\right)+6\left(ab+bc+ca\right)-3abc\)
\(\Leftrightarrow8\left(a+b+c\right)+3abc\ge12+5\left(ab+bc+ca\right)\)
Đặt \(a+b+c=p;ab+bc+ca=q;abc=r\)thì giả thiết trở thành \(p^2-2q=3\)hay \(4q-p^2=2q-3\)
và ta cần chứng minh \(8p+3r\ge12+5q\)
Theo Schur, ta có: \(r\ge\frac{p\left(4q-p^2\right)}{9}\)hay \(3r\ge\frac{p\left(4q-p^2\right)}{3}=\frac{p\left(2q-3\right)}{3}\)(*)
Có \(p^2-2q=3\Rightarrow q=\frac{p^2-3}{2}\)(**)
Sử dụng hai điều kiện (*) và (**) ta đưa điều phải chứng minh về dạng \(8p+\frac{p\left(p^2-6\right)}{3}\ge12+\frac{5\left(p^2-3\right)}{2}\)
\(\Leftrightarrow\left(2p-3\right)\left(p-3\right)^2\ge0\)*đúng*
Đẳng thức xảy ra khi a = b = c = 1
\(a-b+b+\frac{1}{b\left(a-b\right)}\ge3\sqrt[3]{\frac{\left(a-b\right)b.1}{b\left(a-b\right)}}=3\)
Dấu "=" xảy ra khi \(\left\{{}\begin{matrix}a=2\\b=1\end{matrix}\right.\)
\(VT=a-b+\frac{4}{\left(a-b\right)\left(b+1\right)^2}+\frac{b+1}{2}+\frac{b+1}{2}-1\)
\(VT\ge4\sqrt[4]{\frac{4\left(a-b\right)\left(b+1\right)^2}{4\left(a-b\right)\left(b+1\right)^2}}-1=3\)
Dấu "=" xảy ra khi \(\left\{{}\begin{matrix}b=1\\a=2\end{matrix}\right.\)
\(\frac{a-b}{2}+\frac{a-b}{2}+\frac{1}{b\left(a-b\right)^2}+b\ge4\sqrt[4]{\frac{b\left(a-b\right)^2}{4b\left(a-b\right)^2}}=\frac{4}{\sqrt{2}}=2\sqrt{2}\)
Dấu "=" xảy ra khi \(\left\{{}\begin{matrix}a=\frac{3\sqrt{2}}{2}\\b=\frac{\sqrt{2}}{2}\end{matrix}\right.\)
\(A=\frac{a^2}{b}+\frac{b^2}{a}+\frac{8}{a^2+b^2+6}=\frac{a^3+b^3}{ab}+\frac{8}{a^2+b^2+6}=a^3+b^3+\frac{8}{a^2+b^2+6}\)
\(A=\left(a+b\right)\left(a^2+b^2-ab\right)+\frac{8}{a^2+b^2+6}\ge2\sqrt{ab}\left(a^2+b^2-1\right)+\frac{8}{a^2+b^2+6}\)
\(A\ge2\left(a^2+b^2-1\right)+\frac{8}{a^2+b^2+6}=2a^2+2b^2-2+\frac{8}{a^2+b^2+6}\)
\(A\ge\frac{a^2+b^2+6}{8}+\frac{8}{a^2+b^2+6}+\frac{15}{8}\left(a^2+b^2\right)-\frac{11}{4}\)
\(A\ge2\sqrt{\frac{\left(a^2+b^2+6\right).8}{8\left(a^2+b^2+6\right)}}+\frac{15}{8}.2ab-\frac{11}{4}=3\)
Dấu "=" xảy ra khi \(a=b=1\)
Ta có:
\(\frac{1}{a}+\frac{1}{b}+\frac{2}{a+b}=\frac{a+b}{ab}+\frac{2}{a+b}\) . Do giả thiết cho \(ab=1\)
\(\Rightarrow\frac{a +b}{ab}+\frac{2}{a+b}=a+b+\frac{2}{a+b}=\frac{a+b}{2}+\frac{a+b}{2}+\frac{2}{a+b}\)
Áp dụng Bất đẳng thức Cô-si: \(\frac{x+y}{2}\ge\sqrt{xy}\)
Ta có: \(\frac{a+b}{2}\ge\sqrt{ab}=1\)
Ta sẽ chứng minh BĐT phụ sau: với z >0 thì
\(z+\frac{1}{z}\ge2\Leftrightarrow\frac{z^2+1-2z}{z}\ge0\Leftrightarrow\frac{\left(z-1\right)^2}{z}\ge0\)
Áp dụng BĐT trên => \(\frac{a+b}{2}+\frac{2}{a+b}\ge2\) (khi a+b>0)Vậy \(a+b+\frac{2}{a+b}\ge3\Leftrightarrow\frac{1}{a}+\frac{1}{b}+\frac{2}{a+b}\ge3\)