Math 5307 Additional Problems Chapter 5 Fall 2003

Dr. Cordero

- 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.

- Prove that |sin x-sin y| |x-y| for every x, y in R.
- Suppose with . Prove that there exists a such that f(x)>f(a) for every x with a < x < a + .
- 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. - 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].
- (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. - Give an example of a function that is differentiable and 1-1, but for some x in R.
- If is differentiable at c, a < c < b and , prove there exists x, c < x < b, such that f(x) > f(c) .
- 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 . - 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.