MHB Ann's question at Yahoo Answers (At most two roots)

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The equation x^4 + 4x + c can have at most two real roots based on the analysis of its derivative and the application of Rolle's theorem. If there are three real roots, it leads to a contradiction since the derivative has only one real root at x = -1. When c is positive, Descartes' rule indicates no positive roots and at most two negative roots. If c is negative, there will be one positive and one negative root. The behavior of the function at infinity and its minimum point further supports these findings, confirming the maximum of two real roots.
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Hello Ann,

Suppose there are three real roots $a<b<d$, then all the hypothesis of the Rolle's theorem are satisfied for the function $f(x)=x^4 +4x+c$ on the intervals $[a,b]$ and $[b,d]$ so, there exists $\xi_1\in (a,b)$ and $\xi_2\in (b,d)$ such that $f'(\xi_1)=f'(\xi_2)=0$. But $$f'(x)=4x^3+4=0\Leftrightarrow x^3=-1\Leftrightarrow x=-1\quad\mbox{ (in }\mathbb{R})$$ We get a contradiction: $f'$ has at the 'same tine' only one real root and more than one real roots ($\xi_1\neq \xi_2)$.
 
Fernando Revilla said:
Here is a link to the question:

Show that the equation x^4 +4x+c has at most 2 roots? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
Assume \(c\) is positive Descartes rule of signs tell you this has no positive roots, and changing the signs of the coeficient of the odd power of x shows that it has at most two negative roots.

Now assume \(c\) is negative then Descartes rule of signs shows it has one positive and one negative root.
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Anoher way:

Denoting $g(x)=x^4+4x$, we have $$\lim_{x\to -\infty}g(x)=+\infty,\;\lim_{x\to +\infty}g(x)=+\infty,\;g'(x)=4(x-1)(x^2-x+1)$$ which implies $g$ has an absolute minimum at $(-1,-3)$. The intersection of the grpah of $g(x)$ with the graph of $y=-c$, easily provides the following information: $$\begin{aligned}&c\in (-\infty,0):\mbox{Two real roots (one positive and one negative)}\\&c=0: \mbox{ Two real roots }(x=-\sqrt[3]{4},x=-1)\\&c\in (0,3):\mbox{Two real roots (both negative)}\\&c=3:\mbox{One real roots }(x=-1)\\&c\in (3,+\infty):\mbox{No real roots}\end{aligned}$$ At any rate, and according to the question, it seems that the 'spirit' is to apply the Rolle Theorem (they only ask for the number of roots and not for more information).
 
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