## 17Calculus - 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 $$x=c$$ 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.

### Krista King Math - Intermediate Value Theorem [4min-5secs]

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.

### PatrickJMT - Intermediate Value Theorem [7min-53secs]

video by PatrickJMT

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 [5min-58secs]

video by Dr Chris Tisdell

### Practice

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

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

Problem Statement

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

Solution

### 415 video

video by PatrickJMT

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

Problem Statement

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

Solution

### 422 video

video by Krista King Math

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

Problem Statement

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

Solution

### 416 video

video by PatrickJMT

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

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.

### 414 video

video by Dr Chris Tisdell

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

Problem Statement

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

Solution

### 421 video

video by Krista King Math

### intermediate value theorem 17calculus youtube playlist

You CAN Ace Calculus

 limits continuity

WikiBooks: Calculus/Continuity

### Calculus Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem. The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free.
 Intermediate Value Theorem Practice

How to Ace the Rest of Calculus: The Streetwise Guide, Including MultiVariable Calculus Save 20% on Under Armour Plus Free Shipping Over \$49! Prime Student 6-month Trial When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.