Suppose that f(x) and g(x) are positive functions such that f(x) has a local maximum and g(x) has a local minimum at x=c. Prove that
this means if and that
at x = c
I don't know if I am close.
or something along the lines if f(x) is divided by g(x) and is equal to h(x) when f(x) and g(x) have a local minimum and local maximum respectively that would make h(x) also have a local maximum at x = c because if
Then you would not know that there are continuous functions that have no derivatives.
Therefore, you cannot prove this using derivatives.
To say that there is an open neighborhood, , of means where .
In that open interval is an absolute maximum for and is an absolute minimum for .
Okay, I think I have some understanding. I understand the part of absolute maximum and absolute minimum. I still don't understand how this proves is equal to .
I will mark it down as a question I need to return to once I get a a stronger knowledge of the topic.