I have to prove to following statement and I am having a really hard time here.
There it is:
Prove that the following polynomial has a minimum in R
I tried to make the following proof, but I am stuck:
The polynomial can be shown like this:
So we can now write the polynomial like
Since we can say that (from limit arithmetic).
We know that it is always the case that , so the sign of
is determined by g(x). For certain values, so we can say that there is some such that and .
p(x) is continouos everywhere, certainly in [a,b], so from the Extreme Value Theorem, the function has a minimum in [a,b].
There are two problems in my proof:
I can't find any value in which g(x) is negative, only if there is some a which is negative. What if a is always positive?
If the first problem is solved, I managed to prove the statement for some [a,b]. Is it just enough? It seems to me that it isn't.