Constant function

Let f:[0, 1] \rightarrow \mathbb{R} be a continous function such that \bigintsss_0^1 f(x) \, {\rm d}x=1 and

(1)   \begin{equation*} \int_0^1 \left(1-f(x) \right)e^{-f(x)}\, \mathrm{d}x \leq 0 \end{equation*}

Prove that f(x)=1 forall x\in \mathbb{R}.

Solution

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