Suppose I would like to maximize $$ (\Phi (m+\frac{1}{2}x)-\Phi (m-\frac{1}{2}x)) \cdot \Phi (g(x)) $$ with respect to $x $. Taking the log and then a derivative, I get the following first order condition: $$ \frac{\frac{1}{2}\phi(m+\frac{1}{2}x)+\frac{1}{2}\phi(m-\frac{1}{2}x)}{\Phi(m+\frac{1}{2}x)-\Phi(m-\frac{1}{2}x)}+\frac{\phi(g(x))}{\Phi(g(x))}\cdot \frac{\partial g(x)}{\partial x} = 0 $$ where $ \phi $ and $ \Phi $ are the standard normal PDF and CDF, respectively, and $ m $ is some constant. Note that $g(x) $ is a continuous and differentiable function, $ g'(x) < 0 $, $ g''(x) > 0 $. I am interested in the value of $ x $ that solves this. Now, this seems impossible to solve analytically but suppose I would like to prove the existence of a unique solution. Any ideas on how to approach this type of a proof?
Edit: I forgot to mention, it is OK to restrict the domain of $ x $ to $ x > 0 $.