Using the Intermediate Value Theorem to Establish the Existence of Solutions in Calculus

Intermediate Value Theorem (IVT)

The Intermediate Value Theorem (IVT) is a fundamental theorem in calculus that relates to the continuity of a function

The Intermediate Value Theorem (IVT) is a fundamental theorem in calculus that relates to the continuity of a function. It states that if a continuous function f(x) is defined on a closed interval [a, b] and takes on two distinct values, say f(a) and f(b), then it must also take on every value between f(a) and f(b) at some point within the interval.

In simpler terms, the Intermediate Value Theorem guarantees that if a continuous function starts at one value and ends at another value, it must pass through every value in between at least once.

To understand this concept better, let’s consider an example. Suppose we have a function f(x) = x^3 – 2x^2 + x + 1 defined on the interval [0, 2]. We want to determine if there exists a value c in the interval [0, 2] such that f(c) = 0.

Step 1: Check Continuity
First, we need to verify if f(x) is continuous on the interval [0, 2]. In this case, f(x) is a polynomial function, and polynomials are continuous for all real values of x. Therefore, f(x) is indeed continuous on [0, 2].

Step 2: Evaluate f(a) and f(b)
We evaluate f(x) at the endpoints of the interval: f(0) = 0^3 – 2(0)^2 + 0 + 1 = 1 and f(2) = 2^3 – 2(2)^2 + 2 + 1 = 1.

Step 3: Apply the Intermediate Value Theorem
Since f(a) = 1 and f(b) = 1 are both positive and distinct, according to the Intermediate Value Theorem, the function f(x) must take on every value between f(a) and f(b), which includes the value 0. Hence, there must exist a value c in the interval [0, 2] where f(c) = 0.

This is how we can use the Intermediate Value Theorem to determine the existence of a root for a continuous function within a given interval. It provides a powerful tool to establish the existence of solutions even when we cannot find them explicitly.

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