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?

  • You are right, another counter-example is $\frac{1}{x^2+1}$. – Yanko Dec 7 at 14:57
up vote 3 down vote accepted

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.

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

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.