Why 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 $\displaystyle {f}'\left( c \right)\left( {b-a} \right)=f\left( b \right)-f\left( a \right)$ .

The phrase “there exists” also means “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 shorthand for these phrases is an upper-case E written backwards: $\displaystyle \exists$ .

…then there is a number c in the open interval (a, b) such that…
…then there is at least one number c in the open interval (a, b) such that…
…then $\displaystyle \exists$ c in the open interval (a, b) such that…

Textbooks, after presenting an existence theorem, usually follow-up with some exercises asking you to actually find the value that exists: “Find the value of c guaranteed by the Mean Value Theorem for the function … on the interval ….” These exercises may help you remember the formula involved.

But, the important thing about most existence theorems is that the number exists, not what the number is.

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.

Another wording of the IVT: 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 Theorem

If f is continuous on the closed interval [a, b], then there exists a number c in [a, b] such that $\displaystyle f\left( c \right)\ge f\left( x \right)$ for all x in the interval.

Another wording: 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 $\displaystyle f\left( c \right)\le f\left( x \right)$ for all x in the interval. Or: 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 $\displaystyle {f}'\left( x \right)$ 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 $\displaystyle {f}'\left( c \right)=0$ .

Mean Value Theorem – 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.

Cogito, ergo sum

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