Intermediate Value Theorem

Suppose that f(x) is continuous on [a, b] and allow M be any number among f(a) and f(b).   There then exists a number c such that,

1. a < c < b

2. f (c ) = M

All of the Intermediate Value Theorem is actually saying is that a continuous function will take on all values among f(a) & f(b).  Below is a graph of continuous function which illustrates the Intermediate Value Theorem.

As we can illustrates from this image if we pick up any value, M, that is among the value of f(a) and the value of f(b) and draw line straight out from this point the line will hit the graph in at least at one point.  In other terms somewhere between a & b the function will take on the value of M.  Also, as the figure illustrates the function might take on the value at more than one place.

It's also significant to note that the Intermediate Value Theorem only says that the function will take on the value of M somewhere among a & b.  It doesn't say just what that value will be.  It just says that it exists.

hence, the Intermediate Value Theorem tells us that a function will take the value of M somewhere among a & b but it doesn't tell us where it will take the value nor does it tell us how several times it will take the value. There is significant idea to remember regarding the Intermediate Value Theorem.

A fine use of the Intermediate Value Theorem is to prove the existence of roots of equations as the given example shows.