Suppose that $f$ is a continuous function and that $f(-1)=f(1)=0$. Show that there is $c\in(-1,1)$ such that $f(c)=\frac{c}{1-c^2}$
I am not sure if this is a new question as I set it this morning, after solved a similar question. I wanted to prove it using the same idea (Intermediate Value Theorem) but it was not that nice...
Can anyone help me on this?
By the way, the 'similar question' I mentioned above is as follow:
Suppose that $f$ is a continuous function and that $f(0)=1$ and $f(1)=2$. Show that there is $c\in(0,1)$ such that $f(c)=\frac{1}{c}$ The hint given was to let $g(x)=xf(x)$ and use the Intermediate Value Theorem.