3. In this problem we study another approach to show that certain equations have a unique solution on an interval [a, b]. The goal is to first write the equation in

the form x = f(x) for some function f defined on [a, b]. (a) Suppose that f is continuous on [a, b] and that a < f(x) < b for all x = [a, b]. Prove that there exists c = (a, b) such that c= f(c). Hint: Consider the function g(x)= x - f(x). (b) Suppose, in addition, that f is differentiable on (a, b) and that f'(x) < 1 for all r = (a,b). Prove that there is a unique point c = (a, b) such that c = f(c). Hint: Use MVT./n= V √2r +1 has a unique solution on (c) Using parts (a) and (b), show that the equation tan(3x) (0, 1). Hint: Formulate this equation as r = f(x) for some function f(x). Then use the results from part (a) and part (b).