A calculus problem by Sourabh Jangid

Consider a continuous function $f(x)$ in $(0,1 )$ satisfying $\displaystyle \int_0^1 x \sqrt x f(x) \left(1 - \sqrt x \; f(x) \right) \, dx = \dfrac18$ .

Find the number of solutions of $n$ such that $f(n) = e^n$ .