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17calculus > limits > intermediate value theorem

Intermediate Value Theorem

The intermediate value theorem is used to establish that a function passes through a certain y-value and relies heavily on continuity. First, let's look at the theorem itself.

Intermediate Value Theorem

For a continuous function, \(f(x)\) on an interval \([a,b]\), if \( t \) is between \(f(a)\) and \(f(b)\),
then there exists a value \(c \in [a,b]\) such that \(f(c) = t\).


Notice that the theorem just tells you that the value \(c\) exists but doesn't tell you what it is or how to find it.
To get the idea of this theorem clear in your head, here are some great videos for you to watch. They use graphs to help you understand what the theorem means.

Here is a video that shows, graphically, how the intermediate value theorem works. She uses color in her graph to make it easy to follow.

Krista King Math - Intermediate Value Theorem

Here is a great video that clearly explains the intermediate value theorem more from a mathematical point of view than in the previous video.

PatrickJMT - Intermediate Value Theorem

Application of the Intermediate Value Theorem - - Here is a great video showing a non-standard application of the IVT. To work this problem, he uses the definition of the limit. Don't skip this video. It will help you understand limits, continuity and the IVT.

Dr Chris Tisdell - IVT

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Practice 1

Show that \(x^3-3x+1=0\) has a root in the interval \((0,1)\).

solution

Practice 2

Show that \(f(x)=x^4+x-3\) has a root in the interval \((1,2)\).

solution

Practice 3

Show that \(x^2=\sqrt{x+1}\) has a root in the interval \((1,2)\).

solution

Practice 4

Show that \(e^x=2\cos(x)\) has at least one positive root.

solution

Practice 5

Prove that \(\cos(x)=x^3\) has at least one real root.

solution

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