Find the point c

You will write a program that finds the point $c$ of the Mean Value Theorem, and of Rolle's theorem as the special case with a flat chord. It uses a numerical derivative and halves the interval.

The idea. The point $c$ is where the tangent has the same slope as the chord. Let $g(x) = f'(x) - m$, where $m$ is the chord slope. If $g$ changes sign on $[a, b]$, halving the interval finds its zero.

Your tasks.

1. Finish chord_slope(f, a, b). Return the slope of the chord.
2. Finish derivative(f, x). Return the slope taken on both sides, with H = 0.00001.
3. Finish find_c(f, a, b). Halve the interval 50 times for the zero of derivative(f, x) - chord_slope(f, a, b), and return the midpoint.