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I'm trying to prove that there exists $x\in (-1,1)$ $$\frac{a}{x^3+2x^2-1}+\frac{b}{x^3+x-2}=0$$

For any $a,b \in \mathbb{R}^+$.

I know I have to use the Intermediate Value Theorem here, but I'm not sure how to formulate a proof. Could I have some help?

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Let $f(x) = a({x^3+x-2})+{b}({x^3+2x^2-1})$. Now check that $f$ is continuous and $f(-1)=-4a$ < $f(1)=2b$. By IVT, $f$ has a solution in $(-1,1)$. Now you can conclude that your equation also has that solution.

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  • $\begingroup$ +1. I think $f(-1)=-4a$. But that does not a make a difference to the proof. $\endgroup$ – Kavi Rama Murthy Apr 30 '20 at 8:00
  • $\begingroup$ Corrected @KaviRamaMurthy. Thanks for noting it. $\endgroup$ – aNumosh Apr 30 '20 at 8:05

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