Solved Examples on Intermediate Value Theorem
Example 1: Check whether the function defined as f(x) = x3 – 8 has a root in the interval [0,4].
Solution:
Here, we have,
f(0) = 0 – 8 = -8, and
f(4) = 43 – 8 = 64 – 8 = 56.
As the function yields values with opposite signs at the endpoints of the given interval, by intermediate value theorem, it implies that the function has at least one root in the interval.
Example 2: Show that the function defined as f(x) = ex – 3x has a root in the interval [0,1].
Solution:
To show whether the function has a root in the given interval, we check the value of function at the endpoints of the interval,
We have, f(0) = e0 – 3(0) = 1 – 0 = 1, and
f(1) = e – 3 = -0.28 (approx.)
Thus, we see that function has opposite signed values at endpoints of the interval, so it has at least one root in the interval.
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.