Let [tex]n \ \epsilon \ \mathbb{N}, \ n \geq 2[/tex] and [tex]p, \ q \ \epsilon \ \mathbb{R}[/tex]. Consider [tex]f: \ \mathbb{R} \ \rightarrow \ \mathbb{R}[/tex] defined by [tex]f(x)=x^{n}+px+q.[/tex](adsbygoogle = window.adsbygoogle || []).push({});

Suppose n is odd, prove that f has at least one and at most three real roots.

I thought about the intermediate value theorem for proving that f has one root, but then you need one x where f is negative and another one where it's positive and it's impossible to expres this x in terms of n, p and q.

To prove that f has at most three real roots, I thought about finding the local extrema (where f'(x)=0) and examining each of the possible combinations of positions of them. However, then I'm kinda facing the same problem. I'm sure there has to be some more elegant way.

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# Analysis problem

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