MEAN VALUE THEOREM LAB

 

1.   a)   What are the conditions a function must satisfy in order to guarantee the conclusion of the Mean Value Theorem applies?

 

      b)  Give an example of a function which is continuous on an interval I = [a,b] but for which the conclusion of the Mean Value Theorem does not hold.

 

      c)   Give an example of a function that is both

            i) defined on some closed interval I = [a,b] and

            ii) continuous and differentiable on the open interval (a,b)

                        and for which the conclusion of the Mean Value Theorem does not hold.

 

d)  Give an example of a function that does not satisfy the hypotheses for the Mean Value Theorem, but for which the conclusion still holds.

 

 

2.   Let  f(x) = sin(2x).  Find all real numbers c in [0,2¹] such that

 

                                                           

 

3.   Let  f  be a quadratic function, and let a and b be any real numbers.  Show that the value, c, that satisfies the Mean Value Theorem is the midpoint of a and b.

 

4.   Let f be a function that is differentiable on .  Show that if f has two zeroes, then  must have one zero.

 

5.   Let the function f be continuous on [a,b] and differentiable on (a,b) with  for all x in (a,b).  Show that f is decreasing on (a,b).  That is, show that if .

 

 

6.  

a)   Show that if  f  is continuous on [a,b] and on (a,b) then f is a constant function (i.e., there is a number k such that f(x) = k for all x in [a,b]).

 

b)  Suppose  f and  g are continuous on [a,b]  with   for all x in (a,b).  Show that there is a number k such that f(x) = g(x) + k for all x in [a,b] .

 

7.   Suppose that f is differentiable on   Show that   for some  in (0,1).