I have a question if a method is true and the proof of it:
So, let us have a real function f which satisfies that f(a) < 0 and f(b) > 0, f is continuous in [a,b], and f'(x)>0 in [a,b]. Then we know that there is an unique point c from (a,b) such that f(c) = 0. We now want to see how to reach that zero.
So, the method I found on my own(so, I don't know if it exists already) is that we start out with a line between (a, f(a)) and (b, f(b)) and mark intersection with real line Ox with c0. Then, if f(c0) < 0, we put a line between (c0,f(c0)) and (b, f(b)) otherwise we put a line between (c0,f(c0)) and (a, f(a)) and mark intersection(in any case) with c1. Now we look at c1 and co. If f(c0) < 0 and f(c1) > 0 or f(c0) > 0 and f(c1)<0 then we draw a line from (c0,f(c0)) to (c1, f(c1)), if f(c0) < 0 and f(c1) < 0 then we draw a line between (c1,f(c1)) and (b, f(b)) and if f(c1) > 0 and f(c0) > 0, then we draw a line between (c1, f(c1)) and (a, f(a)), and intersection(in any case) mark with c2. We continue this way, obtaining array (cn). Then (cn) converges to the unique solution c* of the equation f(x) = 0 in (a,b).
