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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](http://math.stackexchange.com/a/165217/6179)?2012-07-02
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    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.2012-07-02
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    @did: yes!$ $ $ $ $ $ I don't know why it is natural for you to know when to use it.2012-07-02
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    @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.2012-07-02
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    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.2012-07-02

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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.2012-07-01
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    Thanks! I understand the inequality. My question is more of looking for results similar to that.2012-07-01
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    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 $a \le x \le b$.2012-07-01
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    @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 $c=0$?2012-07-02
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    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*?2012-07-02
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    @t.b.: I happened to google out [this page](http://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtII.aspx), which explains their usage in USA.2012-07-02
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    @Tim: Thanks. There's also [this thread here](http://math.stackexchange.com/q/3399/5363).2012-07-02
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    @t.b. : Whether the function goes up or down has nothing to do with whether the concavity is upward or downward. The parabola $y=x^2$ goes downward and then goes upward, but it's concave upward everywhere. I think there is something to be said for "convex" and "concave", but yuor comment is a mistake, IMO.2012-07-02
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    I have taught calculus for many years and the common parlance there is "concave up" or "concave down". It's an ingrained habit.2012-07-02