calculus - Proof using the mean value theorem

I need to prove (using the mean value theorem) that for all $x\in(0,\infty)$, the following inequality holds:

$\exp(-x)\geq 1 - x$

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I don't know how the mean value theorem is applicable here, since we don't have a closed interval.

How do I prove the statement?

Answer

Hint: let $f(t)=e^{-t}$, and (for a fixed $x$) try using the mean value theorem on the interval $[0,x]$.