# Proving that $\exists x\in (-1,1),\frac{a}{x^3+2x^2-1}+\frac{b}{x^3+x-2}=0$

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?

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.
• +1. I think $f(-1)=-4a$. But that does not a make a difference to the proof. – Kavi Rama Murthy Apr 30 '20 at 8:00