Chapter 5 – Additional Problems

Math 5307
Fall 2003

Dr. Cordero

  1. Find a polynomial  of degree less than or equal to 2 with P(2)=0 such that the function f(x) given below is differentiable at  x=1.
  2. Prove that |sin x-sin y| |x-y| for every x, y in R.
  3. Suppose  with . Prove that there exists a  such that f(x)>f(a) for every x with a < x < a + .
  4. a) Show that if f is differentiable on R and |  for every x in R, then f has at most one fixed point.
    b) Let . Show that f satisfies the hypothesis of part (a), but that f has no fixed point.
  5. Let  ,  be differentiable with f(0)=g(0) and f’(x)>g’(x) for every x in [0,1]. Prove that f(x)>g(x) for every x in (0,1].
  6. (a) Suppose is differentiable and  for some constant M for every x in (a, b). Prove that f is uniformly continuous on (a, b).
    (b) Give an example of a function  that is differentiable and uniformly continuous on (0, 1) but such that  is not bounded.
  7. Give an example of a function  that is differentiable and 1-1, but for some x in R.
  8. If  is differentiable at c, a < c < b and , prove there exists x, c < x < b, such that f(x) > f(c) .
  9. Suppose  is continuous on [0,2] and differentiable on (0,2). Suppose f(0)=0, f(1)=2 and f(2)=2.
    a. Show there exist a c in (1, 2) such that .
    b. Show there exists a d in (0, 1) such that 
    c. Show there exists k in (0, 2) such that .
  10. a) Show that  is one-to-one on  with .
    b) For x in [-1, 1], let Arcsin x denote the inverse function of f. Show that Arcsin x is differentiable on (-1, 1), and find the derivative of Arcsin x.