This simple but slowly convergent method for finding a solution of f(x) = 0 with continuous f is based on the intermediate value theorem, which states that if a continuous function f has opposite signs at some x = α and x = b (> α), that is, either f(α) < 0, f(b) > 0 or f(α) > 0, f(b) < 0, then f.

must be 0 somewhere on [α, b]. The solution is found by repeated bisection of the interval and in each iteration picking that half which also satisfies that sign condition. 

(a) Write an algorithm for the method.

(b) Solve x = cos x by Newton’s method and by bisection. Compare.

(c) Solve e-x = ln x and e^x + x⁴ + x = 2 by bisection.