The problems are due on December 4. This is an individual assignment.
Problem 1.
Prove: If a function f is differentiable on an interval [a,b]
and f ' is bounded on [a,b], then f satisfies a
Lipschitz-condition on [a,b], i.e. there is a constant K so that
for all 
.
Problem 2.
Let f and g be differentiable on 
and suppose
f(0)=g(0)=0 and 
for all 
. Show that
for all .
Problem 3.
Let 
be continuous on [a,b] and differentiable on (a,b).
Show that if 
exists, then f is
differentiable at a and 
.
Hint: Use The Mean Value Theorem and the definition of
f'(a).
Problem 4. p. 134, # 27.
Problem 5. p. 135, # 39.
Helmut Knaust