Simple question, but every time I think I have the answer, I get stuck.
Let $ F(x)=\frac{1}{x^2}*(2*\Phi(x)-1) +(2/3)*\frac{1}{x}*(\phi(x)) +(2/3)*\frac{4}{x^3}*(\phi(x)-\phi(0))$
where $\phi$ is the standard normal PDF and $\Phi$ is the standard normal CDF.
Show that $F(x)>0$ for all $x>0$
So far I have that the limit as x->0 = $\infty$ and that as x->$\infty$, $ F(x)->0$
But, then showing the first derivative is negative leads to the same problems as showing the equation above is positive and I'm left where I started.