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Is it true or false: $$1+e^{-\alpha(1-x)/\epsilon}\leq 1?$$ when $0\leq x\leq1$, $\alpha>0$ and $\epsilon$ is a small parameter. If so how do I show this.

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No, $e^x$ is always positive. – njguliyev Aug 27 '13 at 13:56
Try $x=1$ and see that it immediately breaks. Also try $x=0$ it is still bigger than 1. – Ali Caglayan Aug 27 '13 at 13:57

False since $$\forall x\in\mathbb R,\quad e^x>0$$

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