A continuous function g : [a, b] + [a, b] must have at least one fixed point, that is, at least one solution x e [a, b] for the equation x = g(x). This is an immediate consequence of the intermediate value theorem (see Lecture 6, page 3). If in addition, we would like to prove that the fixed point is unique, that is, there is exactly one solution for x = g(x), then one useful result is the so-called contraction mapping theorem that says: if there exists 1 € (0, 1) such that g(x) – g(y)<\x – y for all x, y € [a, b], then the fixed point is unique. Moreover, the fixed point recursion Xk+1 = g(xx) starting from any initial guess xo € [a, b] will converge to this unique fixed point as k → 0. (a) Kepler's equation in celestial mechanics is of the form: x = m + esinx, where e, m are known g(2) parameters, and one would like to solve for the fixed point x, which physically represents "eccen- tric anomaly" - an angular variable that defines the position of a planet moving along an elliptic orbit. The known parameters satisfy 0