Problem: 1.(10 points) Let f be dened on an open interval (a, (1). Consider the two statements (a);E})|f(x+h)-f(x)l=0 (buIHhJxhnw Prove that (:2) always implies (b), and given an example in which (5) holds but (a) does not. 2. Let f (:17) be a polynomial f0?) = [17:37\" +an1$n_1+"'+ 013? + 30, an g O. Show (1') (5 points) if n is odd, then an) has at least one real root, (it) (5 points) if an and :10 have opposite signs, then f(:r:) has at least one positive root, and, in addition, if n is even, n 75 0, then f (at) has a negative root as well. 3. Suppose that the function f : R > R has the property that f(u + v) = u) + f(u) for all um E R. (a) (2 points) Prove that if f(1) = m, then f(x) = mm for all rational 3:. (b) (4 points) Prove that if f : R > R is continuous, then there exists a constant m such that at) = mm for all a: E R. (c) (4 points) Prove that if f is continuous at some 2:0 6 R, then there exists a constant m such that f(:c) = mm for all :I: E R. 4.(10 points) Let f(x) = a1 sin x+ag sin 2m+- - -+an sin use, where (11,112, . . . , an are real numbers and Where n is a positive integer. Given that I f (x)| S |sinx| for all real 55', prove that |a1 + 2112 + - - - + nan| S 1