High school senior and was helping my friend with Rolle's theorem and came across this neat point.
Given $f(x)$ is a continuous and differentiable function, and $f(-x)=f(x)$. Additionally, $f(x)=0$ must occur at least twice. Finding the mean of the $x$ values of any two consecutive roots and evaluating it in the function will give you exactly where $f'(x)=0$. I’m not sure, but I think if you take the arithmetic mean of the $x$ values of any two roots for an even function, plugging in this value will go a slope of zero?
Any feedback would be awesome!
The function $$f(x)=x^4-5x^2+4=(x-1)(x+1)(x-2)(x+2)$$ is even. The zeros $x=1$ and $x=2$ are "consecutive" and have an arithmetic mean of $\bar x={3\over 2}$.
However, $f'\left({3\over 2}\right)=-{3\over 2}\neq 0.$