# Relation between CDF and PDF

I want to prove there is no random variable X that satisfies the following relation: $$f_X(x) = \bar{F}_X(x)\bar{F}_X(-x)F_X(x)$$ where $$f_X$$, $$\bar{F}_X$$ and $$F_X$$ are the PDF, Complementary CDF and CDF of random variable X, respectively. Any idea is really appreciated.

Note: Pointmass at $$+\infty$$ is indeed a solution, I want to show there is no solution with full support.

Note: The followings has a unique solution which is the logistic distribution. $$f_X(x) = \bar{F}_X(x)\bar{F}_X(-x)$$

If the support is bounded from below, there is $$x_0$$ such that $$F_X(x_0)=0$$. But you have also $$F_X'(x)=f_X(x)\leq F_X(x)$$, which then implies $$F_X\equiv 0$$, contradiction.