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
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
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
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.