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 $f (x)$ $f(x)$ in a given interval $[a, b]$ $[a,b]$ , for some $y$ $y$ between $f (a)$ $f(a)$ and $f (b)$ $f(b)$ , there is a value $c$ $c$ in the interval to which $f (c) = y$ $f(c) = y$ .

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 $f (x)$ $f(x)$ is such that $f (x) = x^{2} - 6 \cdot x + 8$ $f(x) = x^2 - 6*x + 8$ and we want to know if there is a solution between $1$ $1$ and $3$ $3$ (in the $[1, 3]$ $[1,3]$ interval).
$f (1) = 3$ $f(1) = 3$
$f (3) = - 1$ $f(3) = -1$
From the theorem (since all polynomials are continuous), we know that there is a $c$ $c$ in $[1, 3]$ $[1,3]$ such that $f (c) = 0$ $f(c) = 0$ ( $- 1 \leq 0 \leq 3$ $-1 <= 0 <= 3$ )//

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