The Intermediate Value Theorem: Exploring the Existence of Solutions for Continuous Functions on Closed Intervals

Intermediate Value Theorem

The Intermediate Value Theorem is a fundamental theorem in calculus that guarantees the existence of a solution for a specific type of function

The Intermediate Value Theorem is a fundamental theorem in calculus that guarantees the existence of a solution for a specific type of function. This theorem is particularly useful when trying to prove the existence of roots or solutions to equations.

The statement of the Intermediate Value Theorem is as follows:

Suppose that f(x) is a continuous function on a closed interval [a, b], and let C be any value between f(a) and f(b), where f(a) ≠ f(b). Then, there exists at least one value c in the interval (a, b) such that f(c) = C.

In simpler terms, if we have a function that is continuous on a closed interval and takes on two different values at the endpoints of the interval, then it must also take on every value in between those two endpoints at some point within the interval.

To understand the concept of the Intermediate Value Theorem better, let’s consider an example. Suppose we have a continuous function f(x) = x^2 – 4x + 3 on the interval [0, 5]. We can calculate the values of f(0) and f(5) as follows:

f(0) = (0)^2 – 4(0) + 3 = 3
f(5) = (5)^2 – 4(5) + 3 = 8

According to the Intermediate Value Theorem, since f(0) = 3 and f(5) = 8, any value between 3 and 8 should exist somewhere within the interval (0, 5). Let’s pick a value C = 4. Since C lies between f(0) = 3 and f(5) = 8, there must exist a value c in the interval (0, 5) such that f(c) = 4.

To find the specific value of c, we can use algebra to solve the equation f(x) = 4:

x^2 – 4x + 3 = 4

Rearranging, we get:

x^2 – 4x – 1 = 0

Solving this equation, we find that x ≈ 3.73 or x ≈ 0.27. Both of these values lie within the interval (0, 5), and if we plug them into f(x), we obtain f(3.73) ≈ 4 and f(0.27) ≈ 4.

This example demonstrates that the Intermediate Value Theorem guarantees the existence of a solution, in this case, the value 4, within the specified interval.

In summary, the Intermediate Value Theorem is a powerful tool for demonstrating the existence of solutions for continuous functions on closed intervals. It relies on the principle that if a function takes on different values at the endpoints of an interval, it must also take on every value between those endpoints at some point within the interval.

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