## advanced calc prob Darboux's Property

a) Assume that f(x) is continuous on [a,b] but not monotonic. Show that there exists points a<= x1 < x2 <= b such that f(x1) = f(x2)

b)Show that if the function in part a) is differentiable, then there exists a c in (a,b) such that f'(c) = 0

c) Assume that f(x) is differentiable on [a,b] and that f'(a) > 0 while f'(b) < 0. Show that f(x) is not monotonic on [a,b]

d) Assume that f(x) is differentiable on [a,b] and that f'(b) < p < f'(p). Show that there exists some c in (a,b) such that f'(c) = p.

e) Show that there is no function F(x) such that F'(x) = f(x) where
f(x) = 1 if x is in Q
= 0 if x is in R\Q