Intermediate Value Theorem Calculator
The Intermediate Value Theorem is a popular concept in Calculus, often studied together with the mean value theorem. Here I will state the theorem and help you understand this with the help of the Intermediate Value Theorem calculator.
Table of Contents
Intermediate Value Theorem
Let f(x) be a function which is continuous on $[a, b]$, $N$ be a real number lying between $f(a)$ and $f(b)$, then there is at least one $c$ with $a \leq c \leq b$ such that $N = f(c)$.
The IVT is also known as Bolzano’s theorem and Weierstrass Intermediate Value Theorem by some mathematicians. Bolzano provided the first proof of the theorem, but at his time the nature of real numbers wasn’t well-defined. The detailed proof was later done by Karl Weierstrass.
The Weierstrass proof of the Intermediate Value Theorem can be found here.
See the definitions of
Intermediate Value Theorem Calculator
Click on the button above to launch the Intermediate Value Theorem calculator and grapher. If the button doesn't appear then you are using an incompatible browser or your browser doesn't have proper Javascript support. Try opening this page in Google Chrome or any other modern browser. This works on mobile devices too. If you are looking for more help, prefer using a math assignment solver or a math tutor.
Note: $x_{min}$ and $x_{max}$ are just graph plotting ranges. These don't have anything to do with the Intermediate Value Theorem. Increase or decrease the values to increase the graph size.
Also, see, mean value theorem calculator.
Recommended Reading
Calculus: Early Transcendentals
Author: James Stewart
One of the best Calculus textbooks out there. Covers all important topics.
Calculus: An Intuitive and Physical Approach
Author: Morris Kline
Interesting textbook with a more reader friendly writing and explanations.
Need some more books on calculus? See the list of the best Calculus Text Books.
Frequently Asked Questions
What does the Intermediate Value Theorem state?
If f is continuous on [a, b] and k is any value between f(a) and f(b), then there exists at least one c in (a, b) with f(c) = k. In short: continuous functions don't skip values.
Why does continuity matter?
Because discontinuous functions can jump over values. Step functions are the classic counterexample — they go from one value to another without ever passing through anything in between.
How do I use the IVT to prove a root exists?
Show f(a) and f(b) have opposite signs, then by IVT there's a c in (a, b) where f(c) = 0. This is the basis of the bisection method for numerical root-finding.
Does the IVT tell you where the root is?
No — only that one exists. To locate it, you apply the theorem repeatedly (bisection) or switch to a faster numerical method like Newton-Raphson.
Is the IVT valid for vector-valued functions?
Not directly. The single-variable IVT applies to real-valued functions. There are higher-dimensional analogs (the Borsuk–Ulam theorem, the Brouwer fixed-point theorem) but they're separate results.
Can the IVT find multiple roots?
It guarantees at least one. If you suspect more, subdivide the interval and apply IVT to each piece. The calculator does this automatically when you enable 'find all roots'.
What's the difference between IVT and Mean Value Theorem?
IVT is about values f(c). MVT is about derivatives — it asserts a slope f'(c) equals the average rate of change. They're related but answer different questions.
