Using the Intermediate Value Theorem to Understand the Behavior of Continuous Functions on Closed Intervals

Intermediate Value Theorem

The Intermediate Value Theorem is a concept in calculus that relates to the behavior of continuous functions on closed intervals

The Intermediate Value Theorem is a concept in calculus that relates to the behavior of continuous functions on closed intervals. It states that if a function is continuous on the closed interval [a, b] and it takes on two different values, say y1 and y2, at the endpoints a and b, then it must also take on every value between y1 and y2 at least once within the interval.

Mathematically, the Intermediate Value Theorem can be stated as follows:
If f(x) is a continuous function on the closed interval [a, b] and y is any number between f(a) and f(b), then there exists at least one number c in the open interval (a, b) such that f(c) = y.

To illustrate this theorem, let’s consider an example. Suppose we have a function f(x) = x^2 – 3x + 2 on the closed interval [-1, 3].

First, we evaluate f(-1) and f(3):
f(-1) = (-1)^2 – 3(-1) + 2 = 1 + 3 + 2 = 6
f(3) = (3)^2 – 3(3) + 2 = 9 – 9 + 2 = 2

Since f(-1) = 6 and f(3) = 2, we can see that the function takes on different values at the endpoints of the interval. Now, let’s choose a value between 2 and 6, say y = 4.

According to the Intermediate Value Theorem, there must exist at least one value c in the open interval (-1, 3) such that f(c) = 4. To find such a value of c, we can visually observe the graph of the function.

By plotting the graph of f(x) = x^2 – 3x + 2, we see that there is a point on the curve where the y-coordinate is equal to 4. This point lies between x = -1 and x = 3, which confirms the Intermediate Value Theorem.

In summary, the Intermediate Value Theorem guarantees that if a continuous function takes on different values at the endpoints of a closed interval, it will also take on every value between those endpoint values at least once within the interval.

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