maybe someone here can help me. I want to find the analytical minimum '$x_\mathrm{opt} = \arg\min f(x)$' of the following function: $ f(x) = \alpha |c + x| + \beta x^2 $ where $x$ is a real number ($x$ is_element_of $\mathbb{R}$), $c$ is a real constant ($c$ is_element_of $\mathbb{R}$), $\alpha$ and $\beta$ are positive real constants ($\alpha$ is_element_of $\mathbb{R}^+$, $\beta$ is_element_of $\mathbb{R}^+$) and $|\cdot|$ is the absolute value function.
Looks simple, but the absolute value function makes it tricky (at least for me...). As already mentioned, i want to find the solution to this minimization problem analytically, not numerically.