Problems

The quadratic trinomials $f(x)$ and $g(x)$ are such that $f^{\prime }(x)g^{\prime }(x)\ge |f(x)|+|g(x)|$ for all real $x$. Prove that the product $f(x)g(x)$ is equal to the square of some trinomial.

Prove that $|x|\ge x$. It may be helpful to compare each of $|3|$, $|-4.3|$ and $|0|$ with $3$, $-4.3$ and $0$ respectively.