Saturday, 2 April 2011

Intermediate Value Theorem

 Intermediate Value Theorem

If f is continuous on the closed interval [ a , b] and K is any number between f(a) and f(b), then there is at least one number c in the interval such that f(c) = K.

You can use this theorem to determine if a polynomial has at least one real root.

To begin using this we first need to know if

f(a) < 0 < f(b)    or  f(b) < 0 <  f(a)

If either of those cases are true. We know that there is a c in the closed interval such that f(c) = 0.

As an example:

we set f(x) = x^2 - 2. We will now test the function for some closed interval. Lets say [1 , 2]. Then lets test these numbers 1 and 2.

f(1) = -1 < 0 and f(2) = 2 > 0.

Therefore we know by the theorem that there exists a number c in the interval [1 , 2] such that f(c) = 0.

Since f increases on [1 , 2], there is only one such number. This number we can see is the square root of 2.