Hi I am dealing with an optimization problem with quartic function:
$$x = \arg\min\limits_{x\in \mathbb{R}+} x^4 + (\frac{\alpha}{2} - 2y) x^2 - d\alpha x +y^2 + \frac{\alpha d^2}{2}$$
where $\alpha, y, d \in \mathbb{R}+$ are all positive constants.
So the natural idea is to find the stationary point of this quartic function. The derivative of this function is a cubic equation, involving three roots. But how to find out which root is right one? Or is there any other better solution? Thanks in advance!