Use Cauchy's inequality for positive numbes a,b,c.

\(a+b\ge2\sqrt{ab},b+c\ge2\sqrt{bc},c+a\ge2\sqrt{ca}\)

\(\Rightarrow\left(a+b\right)\left(b+c\right)\left(c+a\right)\ge8\sqrt{ab}\cdot\sqrt{bc}\cdot\sqrt{ca}=8abc\)

\(\Rightarrow A=\dfrac{abc}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\le\dfrac{1}{8}\)

\(MaxA=\dfrac{1}{8}\Leftrightarrow a=b=c>0\)

Use Cauchy's inequality for positive numbes a,b,c.

a+b≥2ab−−√,b+c≥2bc−−√,c+a≥2ca−−√

⇒(a+b)(b+c)(c+a)≥8ab−−√⋅bc−−√⋅ca−−√=8abc

⇒A=abc(a+b)(b+c)(c+a)≤18

MaxA=18⇔a=b=c>0