Edit: I have two equations. \begin{equation} f'(x_1)-g'(x_1)=0 \end{equation} \begin{equation} k f'(x_2)-g'(x_2)=0 \end{equation} I know additionally that $f,g$ are twice continuously differentiable functions such that: $f'<0,g'<0,f''<0,g''>0$. Do I know something (even adding additional assumptions) about the following relationship: \begin{equation} x_1 \gtreqless \frac{1}{k} x_2 \end{equation}
For instance, if $f',g'$ were linear ($f'''=g'''=0$), then $x_1=\frac{1}{k}x_2$.
I appreciate any help and apologize in advance if this may be somehow meaningless.