\(5^{x+3y}+5^{xy+1}+xy+1+x+3y=\frac{1}{5^{xy+1}}+\frac{1}{5^{x+3y}}\)
\(\Leftrightarrow5^{x+3y}-5^{-x-3y}+x+3y=5^{-xy-1}-5^{-\left(-xy-1\right)}+\left(-xy-1\right)\)
Xét hàm \(f\left(t\right)=5^t-\frac{1}{5^t}+t\Rightarrow f'\left(t\right)=5^t.ln5+\frac{ln5}{5^t}+1>0\)
\(\Rightarrow f\left(t\right)\) đồng biến
\(\Rightarrow x+3y=-xy-1\)
\(\Rightarrow y\left(x+3\right)=-x-1\)
\(\Rightarrow y=\frac{-x-1}{x+3}\)
\(\Rightarrow T=f\left(x\right)=x-\frac{2x+2}{x+3}+1\)
\(f'\left(x\right)=\frac{\left(x+1\right)\left(x+5\right)}{\left(x+3\right)^2}>0;\forall x\ge0\)
\(\Rightarrow f\left(x\right)_{min}=f\left(0\right)=\frac{1}{3}\Rightarrow m=\frac{1}{3}\)
