prove the existence of a maximum

I apologize if my question is too elementary for this forum, but i can't find a rigorous proof of the following obvious fact: let f be a real function of one real variable, defined and continous on , with . Then attains a maximum. Which is the standard way to prove this fact?

Re: prove the existence of a maximum

Quote:

Originally Posted by

**tenderline** I apologize if my question is too elementary for this forum, but i can't find a rigorous proof of the following obvious fact: let f be a real function of one real variable, defined and continous on

, with

. Then

attains a maximum. Which is the standard way to prove this fact?

For a start: Have a look here: Rolle's theorem - Wikipedia, the free encyclopedia

You'll find additional information here: Mean value theorem - Wikipedia, the free encyclopedia

Re: prove the existence of a maximum

You might also need to know that a continuous function on a compact set has a maximum value. But is not a compact set. However, you know that for some N, and for . So you just need to make sure that is less than the maximum value on [-N,N]. Then whatever happens for x<-N and x>N doesn't affect the maximum.

- Hollywood

Re: prove the existence of a maximum

Re: prove the existence of a maximum

Quote:

Originally Posted by

**Plato** If

then

If

then

So if

then

has a maximum on

.

than you Plato for your solution. I have two questions:

1) i think in the second identity we have to replace with , secondly i think the conclusion is subject to the restriction: maximum on . Is it right what i'm saying?

2) Can we extend this argument to the plane? I mean, let f be a continous (differentiable if necessary) function of two real variables x,y, such that , uniformly with respect to x. Can we conclude that f attains a maximum?

Thanks in advance for any suggestion.