More MVT and IVT
1.
Name:
 2 if  3  x  0
 x 2 if 0  x  1

Consider the function f  x   
on the interval [ 3, 3] .
2
if
x

1

2  x if 1  x  3
a.
Do the hypotheses of the IVT hold for this function on
[ 3, 3] ? Why or why not?
b.
Do the conclusions of the IVT hold for this function on
[ 3, 3] ? Why or why not?
c.
Do the hypotheses of the MVT hold for this function on [ 3, 3] ? Why or why not?
d.
Do the conclusions of the MVT hold for this function on [ 3, 3] ? Why or why not?
f (b)  f (a )
If yes, find the value of c for which f (c) 
.
ba
e.
Do the hypotheses of the IVT hold for this function on [0, 3] ? Why or why not?
f.
Do the conclusions of the IVT hold for this function on [0, 3] ? Why or why not?
g.
 3
Do the hypotheses of the MVT hold for this function on  0,  ? Why or why not?
 2
h.
 3
Do the conclusions of the MVT hold for this function on  0,  ? Why or why not?
 2
f (b)  f (a )
If yes, find the value of c for which f (c) 
.
ba
More MVT and IVT
Name:
2. Suppose that y  f ( x) is differentiable on the interval [ 3, 3] with f (0)  2 and 1  f ( x)  2 for
all x  [3,3] .
a. Find the largest possible value for f (3) .Explain/Show work.
b. Find the largest possible value for f (3) . Explain/Show work.
 
3. Consider f ( x)  x sin( x) and g ( x)  ( x  5) cos( x) . Prove that there exists a c  0,  such that
 2
 
f (c)  g (c) . That is, show that the graphs of f and g intersect on 0,  .
 2
More MVT and IVT
Name:
4. For each of the following conjectures, decide if it is true or false. For the true statements, give
some coherent justification for the statement using complete sentences. For the false statement,
draw or otherwise describe a function f which provides a counterexample, and explain why it
does. Always assume that [a, b] implies that a  b and (a, b] [c, d ) implies that a  b  c  d .
a. Let a function f on [a, b] be given which is differentiable on (a, b) and continuous on [a,b]. If
f ( s)  f (r )
.
c  (a, b) , then there are r , s  [a, b] such that r < c < s and f (c) 
sr
b. Let f be a function with domian [a, b] , then f has a maximum and a minimum value on [a,b].
c. Let f be a continuous function with domain (a, b) , then f has a maximum and a minimum
value on (a,b).
d. Let f be a function with domian [a, b] , then f has a maximum and a minimum value on [a,b].
e. Suppose that f is differentiable on (a, b) . Then for r , s  (a, b) and a y between f (r ) and f ( s)
there is some c  (r , s ) such that f (c)  y .