TITLE: Minimum of $\left| \sin x- 1\right| + \left|\sin x- 2\right| + \left| \sin x -3\right| + \left| \sin x+1\right|$
QUESTION [7 upvotes]: I would like to know the minimum value of $$\left| \sin x- 1\right| + \left|\sin x- 2\right| + \left| \sin x -3\right| + \left| \sin x+1\right|$$ 
for $x \in \mathbb{R}$.

REPLY [12 votes]: The number $\sin x$ is trying to decide where to live on the number line. 
Her friends live at $-1$, $1$, $2$, and $3$.
She wants the sum of her distances to her friends to be as small as possible.
Where should $\sin x$ live? Unfortunately, she has to live between $-1$ and $1$. She starts at $-1$ and begins to travel to the right. For every tiny step she takes, her distance from $-1$ increases by $1$ step, and the sum of her distances from $1$, $2$, and $3$ decreases by $3$ steps, for a net decrease of $2$ steps. 
So she should go to the right as far as her nature allows. I hope that $1$ doesn't mind when $\sin x$ moves in.