What is Intermediate Value Theorem?
Intermediate Value Theorem also called IVT is a theorem in calculus about values that continuous functions attain between a defined interval. It provides a formal statement for the intuitive understanding of continuous functions. For instance, if we drive from one city to another where the starting point is at sea level and the ending point is at some higher elevation. The theorem supports the fact that at some point in the journey, we must pass through every intermediate elevation. The formal statement of the theorem is stated as follows.
Statement of the Intermediate Value Theorem
The Intermediate Value Theorem states that,
For a continuous function f defined on the closed interval [a,b], where f(a) ≠ f(b) and d is real number between f(a) and f(b), then there exists at least one real number c between in the interval (a,b) such that f(c) = d.
The theorem is also stated for a specific case when f(a) and f(b) have opposite signs, then there exists at least one real number c between a and b such that f(c) = 0. This statement helps to get inference about roots of the function.
Intermediate Value Theorem
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.
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