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\(M=\left(a-\frac{6}{a+1}\right)+\left(2b-\frac{3}{b+1}\right)+\left(3c-\frac{2}{c+1}\right)\)
\(M=\left(a+2b+3c\right)-6\left(\frac{1}{a+1}+\frac{1}{2b+2}+\frac{1}{3c+3}\right)\)
\(M\le6-\frac{6.\left(1+1+1\right)^2}{a+1+2b+2+3c+3}\)
\(M\le6-\frac{6.9}{6+6}=6-\frac{9}{2}=\frac{3}{2}\)
Đẳng thức xảy ra khi \(a=3;b=1;c=\frac{1}{3}\)
ta có:\(\frac{a}{a+1}=1-\frac{1}{a+1};\frac{2b}{2+b}=2-\frac{4}{2+b};\frac{3c}{3+c}=3-\frac{9}{3+c}\)
\(\Rightarrow\frac{a}{1+a}+\frac{2b}{2+b}+\frac{3c}{3+c}\le\left(1+2+3\right)-\left(\frac{1}{a+1}+\frac{4}{b+2}+\frac{9}{c+3}\right)\)
\(\le6-\frac{\left(1+2+3\right)^2}{a+b+c+1+2+3}=6-\frac{36}{7}=\frac{6}{7}\left(Q.E.D\right)\)
a/ \(\frac{2}{a}.\frac{4\left|a\right|}{3}=\frac{-8a}{3a}=-\frac{8}{3}\)
b/ \(\frac{3}{a-1}\sqrt{\frac{4\left(a-1\right)^2}{25}}=\frac{3}{\left(a-1\right)}.\frac{2\left|a-1\right|}{5}=\frac{6\left(a-1\right)}{5\left(a-1\right)}=\frac{6}{5}\)
c/ \(\frac{3\sqrt{9a^2b^4}}{\sqrt{a^2b^2}}=\frac{9.\left|a\right|.b^2}{\left|a\right|\left|b\right|}=9\left|b\right|\)
d/ \(\left(1+\frac{\sqrt{a}\left(\sqrt{a}+1\right)}{\sqrt{a}+1}\right)\left(1-\frac{\sqrt{a}\left(\sqrt{a}-1\right)}{\sqrt{a}-1}\right)=\left(1+\sqrt{a}\right)\left(1-\sqrt{a}\right)=1-a\)
a/ \(=\frac{2}{a}.\frac{4\left|a\right|}{3}=\frac{2}{a}.\frac{-4a}{3}=\frac{-8}{3}\)
b/ \(=\frac{3}{a-1}.\frac{\left|2a-2\right|}{5}=\frac{3}{a-1}.\frac{2\left(a-1\right)}{5}=\frac{6}{5}\)
c/ \(=\sqrt{\frac{162a^2b^4}{2a^2b^2}}=\sqrt{81b^2}=9\left|b\right|\)
d/ \(=\left(1+\frac{\sqrt{a}\left(\sqrt{a}+1\right)}{\sqrt{a}+1}\right)\left(1-\frac{\sqrt{a}\left(\sqrt{a}-1\right)}{\sqrt{a}-1}\right)\)
\(=\left(1+\sqrt{a}\right)\left(1-\sqrt{a}\right)=1-a\)
\(S=a+b+c+\frac{3}{a}+\frac{9}{2b}+\frac{4}{c}\)
\(=\left(\frac{3a}{4}+\frac{3}{a}\right)+\left(\frac{b}{2}+\frac{9}{2b}\right)+\left(\frac{c}{4}+\frac{4}{c}\right)+\frac{1}{4}\left(a+2b+3c\right)\)
\(\ge2\sqrt{\frac{3a}{4}.\frac{3}{a}}+2\sqrt{\frac{b}{2}.\frac{9}{2b}}+2\sqrt{\frac{c}{4}.\frac{4}{c}}+\frac{1}{4}.20\)
\(\Rightarrow S\ge13\)
Đẳng thức xảy ra khi a = 2, b = 3, c = 4
Vậy minS = 13 tại (a,b,c) = (2,3,4)
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