Intermediate Value Theorem
The Intermediate Value Theorem (IVT) is a fundamental theorem in calculus that states that if a continuous function f(x) is defined on a closed interval [a, b] and takes on two different values, say f(a) and f(b), then it must take on every value between f(a) and f(b) at some point in the interval [a, b]
The Intermediate Value Theorem (IVT) is a fundamental theorem in calculus that states that if a continuous function f(x) is defined on a closed interval [a, b] and takes on two different values, say f(a) and f(b), then it must take on every value between f(a) and f(b) at some point in the interval [a, b].
In other words, if you can draw a continuous curve without lifting your pencil that starts at (a, f(a)) and ends at (b, f(b)), then the curve must cross every horizontal line between f(a) and f(b) at least once within the interval [a, b].
This theorem is particularly useful in situations where we want to demonstrate the existence of a root or a solution to an equation. By applying the IVT, we can show that there exists at least one value c in the interval (a, b) such that f(c) = 0. This is because if f(a) and f(b) have opposite signs (meaning one is positive and the other is negative), then the IVT ensures that the function must pass through the x-axis at some point.
The Intermediate Value Theorem is a powerful tool in calculus and is often used in conjunction with other techniques to prove important results such as the existence of solutions to equations, the existence of fixed points for functions, and to prove the intermediate value property for continuous functions.
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