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Intermediate Value Theorem 

The intermediate value theorem is used to establish that a function passes through a certain yvalue 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 \(f(b)\), 
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 nonstandard 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 \(x^33x+1=0\) has a root in the interval \((0,1)\). 
solution 
Practice 2 

Show that \(f(x)=x^4+x3\) 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 