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In an exercise I have to make I am asked to show that a function $f$ has a global maximum and minimum, given that:

$f:\mathbb{R}\to\mathbb{R}$ is continuous and

$$\lim_{x\to\infty}f(x)=0.$$

However, in my mind, the function $f(x)=e^{-x^2}$ satisfies those conditions and has an global maximum, but no minimum at all.

Am I misinterpreting the given information?

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  • $\begingroup$ You are right, another counter-example is $\frac{1}{x^2+1}$. $\endgroup$ – Yanko Dec 7 '18 at 14:57
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Yes you are correct, in that case by the EVT we can only claim that the function has (at least) a maximum or a minimum.

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  • $\begingroup$ Okay thank you! I was beginning to doubt myself haha $\endgroup$ – Renze Dec 7 '18 at 14:59
  • $\begingroup$ @Renze Maybe the question contain just a typo for "and/or". You are welcome! Bye $\endgroup$ – gimusi Dec 7 '18 at 15:05

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