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I have to prove that the equation $$x^5 +3x- 6$$ can't have more than one real root..so the function is continuous, has a derivative (both in $R$) . In $R$ there must be an interval where $f'(c)=0$, and if I prove this,than the equation has at least one real root. So $5x^4+3 =0$ ..this equation is only true for $x=0$. How to prove that this is the only root?

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    x=1.19334 when the equation is true2012-12-26
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    Note that you don't have to show that the equation $x^5+3x-6$ has a root; you only have to show it can't have more than one. So suppose it has more than one. This gives you the situation described in your second sentence (with the clause "and a $c$ in this interval" added). Now derive a contradiction.2012-12-26
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    Really interesting question...2012-12-26

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