find minimum of a function with abs and squares analytically 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.
 A: The values of $x$ at the minimum in the three cases need to be put back into $f(x)$ in order to decide the minimum. These are
case $x<-c$ : $x=\alpha/(2 \beta)$ and then
$$f(x)=-\alpha c - \frac{\alpha}{4 \beta ^2},$$
case $x=-c$ : 
$$f(x)=\beta c^2,$$
case $x>-c$ : $x=- \alpha/(2 \beta)$ and then
$$f(x)= \alpha c - \frac{\alpha}{4 \beta ^2}.$$
Now the decision as to which if these three is the minimum value of $f(x)$ will depend on the values of $\alpha, \beta, c.$ It looks like, depending on these values, any one of the three possible optima might be the actual minimum. 
A: As in @coffeemath's answer, there are three points where the minimum can occur:
$$\begin{align}
x_1 &= \frac\alpha{2\beta} & \text{for} & x<-c, \\
x_2 &= -c & \text{for} & x=-c, \\
x_3 &= \frac{-\alpha}{2\beta} & \text{for} & x>-c. \\
\end{align}$$
However, $x_1$ is a minimum only if it lies in its domain $x<-c$, that is, if $-c>\frac\alpha{2\beta}$. Similarly, $x_2$ counts only if $-c<-\frac\alpha{2\beta}$. Also, $f$ is convex on $x\le-c$, so if $x_1$ is a minimum, then $x_2=c$ cannot be; the same goes for $x_3$ vs. $x_2$ over $x\ge-c$. So we get three disjoint cases depending on the value of $-c$ with respect to $\pm\frac\alpha{2\beta}$, and we can write out the solution explicitly:
$$x_\text{opt}=\begin{cases}
\frac\alpha{2\beta} & \text{if }{-}c>\frac\alpha{2\beta}, \\
-c & \text{if }{-}\frac\alpha{2\beta}\le -c\le\frac\alpha{2\beta}, \\
-\frac\alpha{2\beta} & \text{if }{-}c<-\frac\alpha{2\beta}. \\
\end{cases}$$
It amuses me to observe that this can actually be written more compactly as
$$x_{\text{opt}} = \operatorname{clamp}\left(-c, \left[-\frac\alpha{2\beta}, \frac\alpha{2\beta}\right]\right)$$
where $\operatorname{clamp}(t,[a,b]) = \min(\max(t,a),b)$ is the point closest to $t$ in the interval $[a,b]$.
