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I often see the following inequality is used over and over again $$ 1−x⩽e^{−x} $$ for $x \in \mathbb{R}$, for proving or deriving various statements.

As a layman, I haven't seen this inequality appearing in any class I have taken in my life. So it seems quite unnatural to me, and seems just a special result for linear function and exponential function. It is not yet part of my instinct to use it for solving problems. So I want to fill up this indescribable gap within my knowledge.

I wonder if there are other similar results for possibly other commonly seen functions (elementary functions?).

Is there some source listing such results?

Thanks and regards!

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You mean... like there? – Did Jul 2 '12 at 0:01
It is a serious mistake to expect everything to be covered in some class you take. This inequality is something you would expect to be true if you apply what you learned in those classes to the situation that the inequality is about. – Michael Hardy Jul 2 '12 at 0:01
@did: yes!$ $ $ $ $ $ I don't know why it is natural for you to know when to use it. – Tim Jul 2 '12 at 0:08
@MichaelHardy: Whenever something doesn't seem natural to me, in the sense that I don't know how and when to apply it, I always think there is some link missing within my knowledge base. Sometimes I blame it on my lacking formal math training/education. But I know It is not necessarily acquirable from school education, and I didn't enjoy my past school education actually. – Tim Jul 2 '12 at 0:20
As I said: It is reasonable to expect you to understand things that have never been explicitly covered in courses, since many things are covered only implicitly. ncmathsadist's answer pretty much covers the reasons why one would expect this inequality to hold. – Michael Hardy Jul 2 '12 at 1:21
up vote 3 down vote accepted

This inequality occurs for two reasons. The line $y = 1-x$ is tangent to the curve $y=e^{-x}$ at $(0,1)$ and $x\mapsto e^{-x}$ is concave up everywhere. Hence the line lies below the curve.

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Try cooking up some other examples this way. – ncmathsadist Jul 1 '12 at 23:53
Thanks! I understand the inequality. My question is more of looking for results similar to that. – Tim Jul 1 '12 at 23:53
Well, take any function $f$ that is concave up on an interval $(a,b)$ and differentiable at $c$ with $a \le c \le b$. Then $f(x) \ge f(c) + f'(c)(x-c)$ for $a \le x \le b$. – Robert Israel Jul 1 '12 at 23:56
@RobertIsrael: That really brings things to a higher level! Thanks! What are some commonly seen examples of $f$ and $c$ in such inequalities, besides $f(x) = e^{-x}$ for $c=0$? – Tim Jul 2 '12 at 0:23
Concave up... (doesn't $e^{-x}$ go down as $x$ increases?) I suspect many non-Americans like me simply don't understand what is meant by that term because they started reading math books in English only during undergrad studies (or later). What's wrong with convex? – t.b. Jul 2 '12 at 0:28

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