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 \(x=c\) 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.
video by Krista King Math 

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

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.
video by Dr Chris Tisdell 

Instructions   Use the intermediate value theorem to solve these problems. Give your answers in exact form.
Problem Statement 

Show that \(x^33x+1=0\) has a root in the interval \((0,1)\).
Solution 

video by PatrickJMT 

close solution

Problem Statement 

Show that \(f(x)=x^4+x3\) has a root in the interval \((1,2)\).
Solution 

video by Krista King Math 

close solution

Problem Statement 

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

video by PatrickJMT 

close solution

Problem Statement 

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

He chose \(x=0\) and \(x=\pi/2\) to show that the function has a positive value and negative value. You can choose other values as long as one results in a positive value and the other is negative.
video by Dr Chris Tisdell 

close solution

Problem Statement 

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

video by Krista King Math 

close solution

Here is a playlist of the videos on this page.
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