halfspaces question How do I find the supporting halfspace inequality to epigraph of 
$$f(x) = \frac{x^2}{|x|+1}$$
at point $(1,0.5)$

For $x>0$, we have
$$
f(x)=\frac{x^2}{x+1}\quad\Rightarrow\quad f'(x)=\frac{x^2+2x}{(x+1)^2}.
$$
Hence $f'(1)=\frac{3}{4}$ and an equation of the tangent to the graph of $f$ at $(1,f(1))$ is 
$$y=f'(1)(x-1)+f(1)=\frac{3}{4}(x-1)+\frac{1}{2}=\frac{3}{4}x-\frac{1}{4}.$$ 
An inequality defining the halfspace above this tangent is therefore
$$
y\geq  \frac{3}{4}x-\frac{1}{4}.
$$
See here for a picture of the graph and the tangent.
