Đang học Bunyakovsky đúng hong :D
1)
\(S=\sqrt{a^2+4ab+b^2}+\sqrt{b^2+4bc+c^2}+\sqrt{c^2+4ac+a^2}\)
\(S^2=\left(\sqrt{a^2+4ab+b^2}+\sqrt{b^2+4bc+c^2}+\sqrt{c^2+4ac+a^2}\right)^2\)
\(\le\left(1^2+1^2+1^2\right)\left(a^2+4ab+b^2+b^2+4bc+c^2+c^2+4ac+a^2\right)\)
\(=3.2\left(a^2+b^2+c^2+2ab+2bc+2ac\right)=6.\left(a+b+c\right)^2=6.6^2=216\)
\(\Leftrightarrow S\le6\sqrt{6}."="\Leftrightarrow a=b=c=2\)
2) \(M^2=\left(\sqrt{x+1}+\sqrt{y+1}\right)^2\le\left(1^2+1^2\right)\left(x+1+y+1\right)=2.8=16\)
\(M\le4."="\Leftrightarrow x=y=3\)
3)
\(S=ab+2\left(a+b\right)\le\dfrac{\left(a+b\right)^2}{4}+\dfrac{8\left(a+b\right)}{4}\)
\(=\dfrac{\left(a+b\right)^2+8\left(a+b\right)}{4}\)
\(\left(a+b\right)^2\le\left(1^2+1^2\right)\left(a^2+b^2\right)=2\Leftrightarrow a+b\le\sqrt{2}\)
\(\dfrac{\left(a+b\right)^2+8\left(a+b\right)}{4}\le\dfrac{2+8\sqrt{2}}{4}=\dfrac{1+4\sqrt{2}}{2}\)
\(S\le\dfrac{1+4\sqrt{2}}{2}."="\Leftrightarrow a=b=\dfrac{1}{\sqrt{2}}\)
