Khai triển Logarit:InS=Σ[b∗In(a+√a2+1)]
Xét hàm số f(x)=In(x+√x2+1),x>0,ta có:
f′(x)=1√x2+1;f″
\Rightarrow f(x)\leq f'(\frac{3}{4})(x-\frac{3}{4})+f(\frac{3}{4})=\frac{4}{5}x+In2-\frac{3}{5}(Use BĐT tiếp tuyến)
\Rightarrow In S \leq \frac{4}{5}(ab+bc+ca)+(In 2-\frac{9}{5})(a+b+c)\le \frac{27}{20}+In 2*\frac{9}{4}-\frac{27}{20}=ln2*\frac{9}{4}
\Rightarrow S\le e^{\frac{9}{4}ln(2)}
Dấu''='' xra\Leftrightarrow a=b=c=\frac{3}{4}