Determining the sign of an expression

Question

Given a random variable $X$ with continuous density $f$, $\int X f(X)\mathrm{d}X = 0$ and $\int X^2 f(X) \mathrm{d}X = 1$, and a function $h = h(X)$ with $0 < h(X)$, show if the sign of the following expression can be determined: \begin{equation} w \equiv \int X h(X)f(X)\mathrm{d}X \end{equation}

Answer 1

Hint: Take $\ X\ $ to be a standard normal variate (density $\ f\left(x\right) = \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} \ $) and consider a function $\ h\left(x\right)\ $ of the form $$ h\left(x\right) = \cases{a& for $\ x<0\ $\\ b & for $\ 0\le x\ $.} $$ What is the sign of $\ \int_\limits{-\infty}^\infty x\, h\left(x\right)f\left(x\right) dx \ $ in this case, when $\ 0<a<b\ $? What about when $\ 0<b<a\ $.