An * existence proof* shows that an object
exists. In some cases, this means displaying the object, or giving a
method for finding it.

* Example.* Show that there is a real number x
such that but .

There are many possibilities; for example,

* Example.* Show that there is a real number x
such that .

Rewrite the inequality as

The graph of looks like this:

The graph lies below the x-axis between and . So, for example, meets the conditions:

In some cases, you can know that an object exists without having any way of finding it (or finding it exactly). By analogy:

(a) If you throw your keys into a corn field, you *know* your
keys are in the field --- but you may have trouble finding them!

(b) You know that Calvin Butterball has a birthday, even though you
don't know *what day* it is.

You've seen results of this kind in calculus. One such result is:

* Theorem.* (* The Intermediate
Value Theorem:*) Let f be a continuous function on the interval
. Suppose that c is a number between and . Then for some x in the interval .

The Intermediate Value Theorem does not tell you how to *find*
an x such that --- it simply *guarantees*
that such an x exists.

* Example.* Show that there is a real number x
such that .

The assertion means that the graphs of and intersect:

It looks like they do. Note, however, that a picture is not a proof.

Let . Then

Since is positive and is negative, and since f is continuous for all x, the Intermediate Value Theorem implies that there is an x between 0 and for which . Then , so .

Notice that the Intermediate Value Theorem doesn't tell you what x is, or how to find it. (It's approximately 0.73909.)

* Example.* Suppose f is a continuous function
satisfying

Prove that there is a number c such that and

The function is continuous.

Since 10 is between 26 and 4, there is a number c such that and

To say that there is an x satisfying a certain property does not mean
that there is *only one* x satisfying the property. If that is
what is meant, it has to be stated explicitly. Hence, there might be
*many* values which satisfy the conclusion of the Intermediate
Value Theorem.

Here's another existence theorem from calculus:

* Theorem.* (* Mean Value
Theorem*) Suppose f is function which is continuous on the closed
interval and differentiable on the open
interval . Then there is a number c such that and

* Example.* Find a number c which satisfies the
conclusion of the Mean Value Theorem when it is applied to on the interval .

Note that

Now , so setting , I find that . Both of these values satisfy the conclusion of the Mean Value Theorem.

* Example.* Suppose f is a differentiable
function satisfying

Prove that .

Applying the Mean Value Theorem to f on the interval gives a number c such that and

Then

* Rolle's theorem* is special case of the Mean
Value Theorem: With the assumptions of the theorem, if , then there is a number c such that and

That is, c is a * critical point* of f.

* Example.* Let . Prove
that there is a number c between 1 and such that .

f is differentiable. Moreover,

By Rolle's theorem, there is a number c between 0 and such that .

In the last example, I *found* numbers satisfying the
conclusion of the theorem --- but again, there is no guarantee that I
can find such numbers explicitly. The theorem just says that at least
one such number *exists*.

Copyright 2019 by Bruce Ikenaga