Math 5307 Additional
Problems Chapter 5 Fall 2003
- 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
- Prove that |sin x-sin
y| |x-y| for every x, y in R.
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.
, 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.
is differentiable at c, a < c < b and
, prove there exists x, c < x < b, such that f(x) > f(c)
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
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.