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soopo
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Homework Statement
Prove [tex] 1 - x \leq e^{-x}[/tex] for [tex]0 \leq x \leq 1[/tex] by calculus.
The Attempt at a Solution
Sketching shows that the statement seems to be true.
Let's assume that
[tex]f(x) = x + e^{-x} -1[/tex], and
[tex]f(x) \geq 0[/tex].
If the function is continuous and increasing, then the function gets all the
values between the interval [0, [tex]e^{-1}[/tex]], by Boltzman's Min-Max theorem.
Thus, the initial assumptions are true.
Please, point out any mistakes.