I want to verify the following claim that I found in some paper. Suppose f is a smooth real-valued function on the real line satisfying $f'(x)x-f(x)\ge x^2$ for all x. Then there is a constant C, s.t. $f(x)\ge \frac{x^2}{2}-C$ for all $x.$


The connection with the title is that the authors claim that any function satisfying the first inequality is of quadratic growth, i.e. satisfies the second inequality.



on a property of functions of quadratic growth
Note that 
$$\left( \frac{f(x)}{x} \right)' = \frac{xf'(x) - f(x)}{x^2} \geq 1,$$
hence the result.


