I've tried to show that the following equation has a solution:
\begin{equation*} g(x)=\left[1-\left(2\int _{\mu}^{x}f(y)dy\right)^2\right]-8xf(x)\int _{\mu}^{x}f(y)dy=0, \end{equation*} where $f(x)$ is the probability density of the normal distribution and $\mu >0$ is the mean of the distribution.
Here is what I've already done (thought):
(i) $g(\mu)=1$, such that we must have $x^{\ast}>\mu$.
(ii) I've tried to show that $G(x)<0$ for some $x$, such that I could apply Intermediate value theorem, but I've failed.
(iii) Numerical simulations show that there exists an unique solution and it is higher than the mean.
Could someone give me a clue? Thank you!