# Non-explicit use of Weirestrass Extreme Value Theorem

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

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