I will write my proof in short words: If I write that ths set of maximum points is uncountable so if I have a Polynomial function of n degree so the Polynomial function derivative could have a root with a transcendental number and that is an absurdity. So this is way the set of all maximum point is countable or finite. It is a correct proof ? Thanks
Can I prove that Cardinality of the set of all maximum point (for any function $\mathfrak{f}$) is countable or finite by Reductio ad absurdum
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elementary-set-theory
proof-writing
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0With maximumum points, you mean points where the local maximum is attained? W.r.t. correctness of the proof: why do you consider a polynomial of degree $n$ only if you are interested in the general case? – 2012-08-29
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0What kind of functions are you talking about? Where do polynomials come in? If you mean by *maximum point* the highest value archived by a function, there can be at most one maximum point. If you mean the set of points where a maximum is archived, a constant function on the reals has an uncountable number of such points. – 2012-08-29
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0The set of all maximum points(for any function $\mathfrak{f}$) is either finite or countable. This is what I want to prove. – 2012-08-29
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0Ilya: I limited to myself to a Polynomial – 2012-08-29
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0Transcendency is not relevant. Anyway, $x^2-\pi^2$ has transcendental roots. But it is true that if you have a polynomial $P(x)$ of degree $\ge 1$, then $P'(x)$ will only have finitely many roots. So a polynomial of degree $\ge 1$ has only finitely many local maxima. I don't really hink the problem is about polynomials. – 2012-08-29
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0@Hernan: would you please define a maximum point? – 2012-08-29
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0@Ilya:A point $\mathbf{x}\in \mathbb{R}$ is called a $\textit{maximal point}$ for a function $\mathfrak{f}:\mathbb{R}\rightarrow \mathbb{R}$ if there exists some $\epsilon >0$ such that $\mathfrak{f}(x)>\mathfrak{f}(x+h)$ for any h that $\left | h \right |<\epsilon$ and $h\neq 0$ – 2012-08-29
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0I think that if you take the open environment of any local extremum point you can find an Injective correspondence with any two rational point one from the left and one from the right. I think that is another way to prove but I am not shure. – 2012-08-29