**Existence Theorems**

An *existence theorem* is a theorem that says, if the hypotheses are met, that something, usually a number, must exist.

For example, the **Mean Value Theorem **is an existence theorem: If a function *f *is defined on the closed interval [*a*,* b*] and differentiable on the open interval (*a*, *b*), then *there exists* a number *c* in the open interval (*a*, *b*) such that .

The phrase “there exists” can also mean “there is” and “there is at least one.” In fact, it is a good idea when seeing an existence theorem to reword it using each of these other phrases. “There is at least one” reminds you that there may be more than one number that satisfies the condition. The mathematical symbol for these phrases is an upper-case E written backwards: .

Textbooks, after presenting an existence theorem, usually follow-up with some exercises asking students to find the value for a given function on a given interval: “Find the value of *c* guaranteed by the Mean Value Theorem for the function … on the interval ….” These exercises may help students remember the formula involved but are not very useful otherwise.

The important thing about most existence theorems is *that* the number exists, not *what* the number is. To illustrate this, let’s consider the **Fundamental Theorem of Calculus.** After partitioning the interval [*a*,* b*] into subintervals at various values, *x _{i}*, we consider the limit of the sum

.

Write out a few terms and you will see that is a telescoping series and its limit is .

The expression resembles the right side of the Mean Value Theorem above. Since all the conditions are met, the MVT tells us that in each subinterval *there exists* a number, call it *c _{i }*, such that

and therefore

No one is concerned what these *c _{i}* are, just that there are such numbers, that

*they exist*. (The second limit above is then defined as the definite integral so – The Fundamental Theorem of Calculus.)

**Other important existence theorems in calculus**

__The Intermediate Value Theorem__.

If *f* is continuous on the interval [*a*, *b*] and *M *is any number between* f(a*) and *f*(*b*), then *there exists* a number *c* in the open interval (*a*, *b*) such that *f*(*c*) = *M.*

If *f* is continuous on an interval and *f* changes sign in the interval, then *there must be at least one* number *c* in the interval such that *f*(*c*) = 0

__Extreme Value Theorems__

If *f* is continuous on the closed interval [*a*, *b*], then *there exists* a number *c* in [*a*, *b*] such that for all *x* in the interval. Every function continuous on a closed interval has (i.e. there exists) a maximum value in the interval.

If *f* is continuous on the closed interval [*a*, *b*], then *there exists* a number *c* in [*a*, *b*] such that for all *x* in the interval. Every function continuous on a closed interval has (i.e. there exists) a minimum value in the interval.

__Critical Points__

If *f* is differentiable on a closed interval and changes sign in the interval, then *there exists* a critical point in the interval.

__Rolle’s theorem__

If a function *f *is defined on the closed interval [*a*,* b*] and differentiable on the open interval (*a*, *b*) and *f*(*a*) = *f*(*b*), then *there must exist* a number *c* in the open interval (*a*, *b*) such that .

__MVT – other forms__

If I drive a car continuously for 150 miles in three hours, then there is a time when my speed was exactly 50 mph.

If a function *f *is defined on the closed interval [*a*,* b*] and differentiable on the open interval (*a*, *b*), then *there is* a point on the graph of *f* where the tangent line is parallel to the segment between the endpoints.

Taylor’s Theorem

If *f* is a function with derivatives through order *n* + 1 on an interval *I* containing *a*, then, for each *x* in *I* , *there exists* a number *c* between *x* and *a* such that

The number is called the remainder. The equation above says that if you can find the correct *c* the function is exactly equal to *T _{n}*(

*x*) +

*R*. Notice the form of the remainder is the same as the other terms, except it is evaluated at the mysterious

*c*. The trouble is we almost never can find the

*c*without knowing the exact value of f(x), but; if we knew that, there would be no need to approximate. However, often without knowing the exact values of

*c*, we can still approximate the value of the remainder and thereby, know how close the polynomial

*T*(

_{n}*x*) approximates the value of

*f*(

*x*) for values in

*x*in the interval,

*i*. See Error Bounds and the Lagrange error bound.

*Cogito, ergo sum *

And finally, we have Descartes’ famous “theorem” *Cogito, ergo sum *(in Latin) or the original French, *Je pense, donc je suis*, translated as “I think, therefore I am” proving his own existence.