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Consider the following well known theorems due to Weierstrass. If $f:[a,b]\to\mathbb{R}$ is continuous on its domain, then

  1. $f$ is bounded from above and from below.
  2. $f$ attains its maximum and minimum on the interval, i.e. - There exists $x_{\min},x_{\max}\in[a,b]$ such that for all $x\in [a,b]$ one has that $$ f(x_{\min} ) \leq f(x) \leq f(x_{\max}) \, .$$

My Question: Most of the examples I've seen for using these theorems are very explicit - show that the function $f$ under some conditions has maximum/minmum. I'm looking for an example in which we need to use either of these theorems, but we are not asked to do it explicitly.

In terms of prerequisites I can only rely on these theorems, basic continuity definitions and the mean value theorem.

Thanks

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  • $\begingroup$ Note that $1$ is a consequence of $2.$ So, actually, there are not two theorems. $\endgroup$ – mfl Dec 5 '16 at 14:39
  • $\begingroup$ Of course. We usually prove (1) before (2), and use (1) to prove (2). Either version is good for my purposes. $\endgroup$ – Amir Sagiv Dec 5 '16 at 14:41
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    $\begingroup$ Since there are many theorems of Weierstrass, I'll add that this one is otherwise known as the Extreme Value Theorem. $\endgroup$ – Matthew Leingang Dec 5 '16 at 14:52
  • $\begingroup$ @MatthewLeingang thanks, I've changed the title accordingly. $\endgroup$ – Amir Sagiv Dec 6 '16 at 6:58

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