I am dealing with an optimization problem in which I want to find $x$ that minimizes $y(x)$. Instead of resorting to numerical optimization, I would like to find a closed-form solution. So, after taking the derivative of $y(x)$, I arrived to the following equation, for which I am not able to find a closed-form solution:
where $\alpha$, $\beta$, $\kappa$ and $\gamma$ are positive real-valued constants. Also, the solution for $x$ should be real-valued and greater than $0$. Using mathematica, it seems that the equation can be solved when isolating each one of the two LHS terms, but not when both are considered together...
Any help would be very welcome! Thanks in advance.