Intermediate Value Theorem

What is meant by the Intermediate Value Theorem?

The Intermediate Value Theorem states that a continuous function attains every intermediate value in an interval specified by functional values of the function on two input variables in the domain of the function.

What is the necessary condition for the Intermediate Value Theorem to be applicable?

The function should be continuous in the given interval for Intermediate Value Theorem to be applicable.

How is the Intermediate Value Theorem applicable in the field of physics?

Intermediate Value Theorem can be used to check whether a particular value would be attained by a physical quantity expressed in form of a function in a specified interval.

Discuss some cases where Intermediate Value Theorem can not be applied.

Intermediate Value Theorem is applicable to continuous functions but it can not be applied to discontinuous or piecewise functions.

Does the Intermediate Value Theorem apply if functional values at endpoints of the interval are equal?

Yes, the theorem is applicable if functional values at endpoints are equal too.

Does Intermediate Value Theorem give any information about number of roots in the specified interval?

No, the theorem checks the presence or absence of roots in the specified interval but does not gives any information about number of roots.

Intermediate Value Theorem is a theorem in calculus which defines an important property of continuous functions. It is abbreviated as IVT. The theorem is quite intuitive one but provides a significant result for the interpretation of the behaviour of functions. It can be used to know the range of values for a physical quantity such as temperature if an expression in terms of time or other variables is known for it. Other applications of the theorem include solving equations, proving the existence of roots, and analyzing real-world problems where continuity is observed.

In this article, we will learn the statement of the theorem, its proof by two different approaches, its various applications, the converse of the theorem, some numerical problems and related frequently asked questions.