Minimal Distance between two curves What is the minimal distance between curves?


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*$y = |x| + 1$ 

*$y = \arctan(2x)$


I need to set a point with $\cos(t), \sin(t)$?

One shortcut here is to note that curves 1, 2 (say $f(x)$, $g(x)$) have a co-normal line passing between the closest two points.
Therefore, since $f'(x) = 1$ for all $x>0$ then just find where $g'(x) = 1$ or 
\begin{align}&\frac{2}{4x^2 +1} = 1\\
&2 = 4x^2 + 1\\
&\bf{x = \pm 1/2}\end{align}
