I have been trying to characterize the roots of the equation
$$ f(x) = x \left(1-x^2 \right) + \left(1+x^2 \right) \frac{\sum w_i y_i }{n} - x \frac{\sum w_i^2 + \sum y_i^2}{n} = 0 $$
where the $w$s and $y$s are real numbers and $x$ has to lie in the interval $[-1, 1]$. Now since the function is continuous and changes signs between $-1$ and $1$, there is at least one root and there can be as many as three. My book, nevertheless, claims that the equation can have at most two such solutions. If anyone sees how this is true, I would appreciate some help understanding it. Thank you.