Intermediate Value Theorem

Key Questions

How do I use the intermediate value theorem to determine whether a polynomial function has a solution over a given interval?

To answer this question, we need to know what the intermediate value theorem says.

The theorem basically sates that:
For a given continuous function #f(x)# in a given interval #[a,b]#, for some #y# between #f(a)# and #f(b)#, there is a value #c# in the interval to which #f(c) = y#.

It's application to determining whether there is a solution in an interval is to test it's upper and lower bound.

Let's say that our #f(x)# is such that #f(x) = x^2 - 6*x + 8# and we want to know if there is a solution between #1# and #3# (in the #[1,3]# interval).
#f(1) = 3#
#f(3) = -1#
From the theorem (since all polynomials are continuous), we know that there is a #c# in #[1,3]# such that #f(c) = 0# (#-1 <= 0 <= 3#)//

Hope it helps.

Bernardo Russo G. Ferreira · 1 · May 16 2015
What is a continuous function?

Answer:

There are several definitions of continuous function, so I give you several...

Explanation:

Very roughly speaking, a continuous function is one whose graph can be drawn without lifting your pen from the paper. It has no discontinuities (jumps).

Much more formally:

If #A sube RR# then #f(x):A->RR# is continuous iff

#AA x in A, delta in RR, delta > 0, EE epsilon in RR, epsilon > 0 :#

#AA x_1 in (x - epsilon, x + epsilon) nn A, f(x_1) in (f(x) - delta, f(x) + delta)#

That's rather a mouthful, but basically means that #f(x)# does not suddenly jump in value.

Here's another definition:

If #A# and #B# are any sets with a definition of open subsets, then #f:A->B# is continuous iff the pre-image of any open subset of #B# is an open subset of #A#.

That is if #B_1 sube B# is an open subset of #B# and #A_1 = { a in A : f(a) in B_1 }#, then #A_1# is an open subset of #A#.

George C. · 1 · Jul 9 2015
What does the intermediate value theorem mean?

Answer:

It means that a if a continuous function (on an interval #A#) takes 2 distincts values #f(a)# and #f(b)# (#a,b in A# of course), then it will take all the values between #f(a)# and #f(b)#.

Explanation:

In order to remember or understand it better, please know that the math vocabulary uses a lot of images. For instance, you can perfectly imagine an increasing function! It's the same here, with intermediate you can imagine something between 2 other things if you know what I mean. Don't hesitate to ask any questions if it's not clear!

Topscooter · 1 · Jan 5 2016