The idea behind the Intermediate Value Theorem is this:
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When we have
two points connected by a continuous curve:
- one point below the line
- the other point above the line
then there will be
at least one place where the curve crosses the line!
of course we must cross the line to get from A to B!
Here is the Intermediate Value Theorem stated more formally:
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When:
- The curve is the function y = f(x),
- which is continuous on the interval [a, b],
- and w is a number between f(a) and f(b),
Then there must be at least one value
c within
[a, b] such that
f(c) = w
In other words the function
y = f(x) at some point must be
w = f(c)
Notice that:
- w is between f(a) and f(b), which leads to
- c must be between a and b
It also says "at least one value c", which means we could have more.
Here, for example, are 3 points where f(x)=w:
- How Is This Useful?
Whenever we can show that:
- there is a point above some line
- and a point below that line, and
- that the curve is continuous,
we can then safely say "yes, there is a value somewhere in between that is on the line".