I am trying to prove the monotonicity of the following function. $$f(x) = x + \frac{\Phi'\left(x\right)}{\Phi\left(x\right)}$$ for $x\in \mathbb{R}$ where $\Phi\left(x\right)$ is normal cdf and $\Phi'\left(x\right)$ is the derivative. It seems that it is monotonically increasing in $x$, and its slope is less than 1 for all $x$.
I also wonder if $f(x) > 0$ for all $x$, with $\lim_{x\rightarrow -\infty} f(x) = 0$. Thank you in advance.
