- Give an example of a continuous function defined on which does not
achieve its maximum on
- Give an example of a continuous function defined on which is bounded
but which does not achieve either its maximum or its minimum.
- Give an example of a discontinuous function defined on which is bounded
but does not achieve either its maximum or its minimum.
- Give an example of a continuous function defined on [0,1) ∪ (1,2] which is
positive at 2, negative at 0 but is not equal to zero for any value of x.
- Let f =
x5 + ax4 + bx3 + cx2 + dx + e where a,b,c,d, and e are numbers.
Show there exists real x such that f = 0
- Give an example of a function which is one to one but neither strictly
increasing nor strictly decreasing.
- Show that the function f =
xn − a, where n is a positive integer and a is
a number, is continuous.
- Use the intermediate value theorem on the function f =
x7 − 8 to show
must exist. State and prove a general theorem about
nth roots of positive
- Prove is irrational.
Hint: Suppose =
p∕q where p,q are positive
integers and the fraction is in lowest terms. Then 2q2 = p2 and so p2 is even.
Explain why p = 2r so p must be even. Next argue q must be even.
- Let f =
x − for
x ∈ ℚ, the rational numbers. Show that even though
< 0 and f
> 0, there is no point in ℚ where f = 0
. Does this
contradict the intermediate value theorem? Explain.
- A circular hula hoop lies partly in the shade and partly in the hot sun. Show
there exist two points on the hula hoop which are at opposite sides of the
hoop which have the same temperature. Hint: Imagine this is a circle and
points are located by specifying their angle, θ from a fixed diameter. Then
letting T be the temperature in the hoop,
. You need
to have T =
T for some
θ. Assume T is a continuous function of θ.
- A car starts off on a long trip with a full tank of gas. The driver intends to
drive the car till it runs out of gas. Show that at some time the number of
miles the car has gone exactly equals the number of gallons of gas in the tank.
- Suppose f is a continuous function defined on which maps
. Show there exists x ∈ such that
x = f
. Hint: Consider
≡ x−f and the intermediate value theorem. This is a one dimensional
version of the Brouwer fixed point theorem.
- Let f be a continuous function on such that
. Let n be a
positive integer larger than 2. Show there must exist c ∈ such that
Hint: Consider h
− f. Consider the
k = 1,
,n−1. You want to show that h equals zero
on one of these intervals. If h changes sign between two successive intervals,
then you are done. Assume then, that this does not happen. Say h remains
positive. Argue that f
. Thus f
> f =
It follows that h
< 0 but h