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

Question

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?

Answer 1

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.