A calculus problem by Sourabh Jangid

Calculus Level 3

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 \(f(n) = e^n\).

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