1. ## Intermediate Value Theorem

The problem:
If $a$ and $b$ are positive numbers, prove that the equation
$\frac{a}{x^3+2x-1}+\frac{b}{x^3+x-2}=0$
has at least one solution in the interval (-1,1).
I was hoping for some pointers on how to proceed.
Sorry, I made a mistake earlier that $2x$ was supposed to be a $2x^2$.

2. Hi,

say we write f(x) = $\frac{a}{x^3+2x-1}+\frac{b}{x^3+x-2}$.
Then as $x \rightarrow -1, f(x) \rightarrow -\frac{a}{4}-\frac{b}{4}$.
Similarly you can show that as $x$ approaches $1$ the function value is positive.
Since $a$ and $b$ are positive numbers, the function $f$ "crosses the x-axis" in the interval $(-1,1)$, and so a solution exists.

I'm sure you can write this in more mathy terms.

Hope this helps!