Say a continuous function f: [a,b] -> [a,b] is required to have 1 or more fixed points, so then it must have 1 or more solutions where x E [a,b] for the equations x = f(x). Thus, a direct issue caused by the intermediate value theorem. Prove that the fixed point is unique, meaning there is only one solution for x = f(x). A helpful outcome is the contraction mapping theorem: if there exists 1 E[0,1) such that |f(x)-f(y) = N/x-y| for all x,y [a,b], then the fixed point is unique. Lastly, fixed point recursion Xnt = g(xn) beginning from any initial guess x, E[a,b] will converge to this unique fixed point as n-> 00. (i) The form x = m + sin x, also referred to as kepler's equation, where, and m are known parameters, we need to solve for the fixed point x, which signifies a variable with angular properties which defines the position of a planet traveling along an elliptic orbit, known as eccentric anomaly. The parameters satisfy o [0,1]. Write a Matlab code using format long to compute the fixed point of kepler's equation with fixed point recursion of 20 iterations. The code should create a plot with Xn against f(xn) for 50 randomly chosen guesses in [0,1]. To create the initial guess use the command rand. Say a continuous function f: [a,b] -> [a,b] is required to have 1 or more fixed points, so then it must have 1 or more solutions where x E [a,b] for the equations x = f(x). Thus, a direct issue caused by the intermediate value theorem. Prove that the fixed point is unique, meaning there is only one solution for x = f(x). A helpful outcome is the contraction mapping theorem: if there exists 1 E[0,1) such that |f(x)-f(y) = N/x-y| for all x,y [a,b], then the fixed point is unique. Lastly, fixed point recursion Xnt = g(xn) beginning from any initial guess x, E[a,b] will converge to this unique fixed point as n-> 00. (i) The form x = m + sin x, also referred to as kepler's equation, where, and m are known parameters, we need to solve for the fixed point x, which signifies a variable with angular properties which defines the position of a planet traveling along an elliptic orbit, known as eccentric anomaly. The parameters satisfy o [0,1]. Write a Matlab code using format long to compute the fixed point of kepler's equation with fixed point recursion of 20 iterations. The code should create a plot with Xn against f(xn) for 50 randomly chosen guesses in [0,1]. To create the initial guess use the command rand