Intermediate Value Theorem
if f(x) is continuous on [a,b], then there will be a point x=c that lies in between [a,b] such that f(c) is between f(a) and f(b)
The Intermediate Value Theorem (IVT) is a fundamental theorem in calculus that describes an important property of continuous functions. It states that if a function f(x) is continuous on a closed interval [a, b], and f(a) and f(b) have opposite signs (i.e., one is positive and one is negative), then there exists at least one value c in the interval (a, b) such that f(c) = 0.
In other words, the Intermediate Value Theorem guarantees that continuous functions must take on all values between two points a and b.
For example, consider a continuous function that passes through f(a) = -2 and f(b) = 3 on the interval [a, b] = [0, 5]. The intermediate value theorem tells us that there must exist at least one value c in (0, 5) such that f(c) = 0. This is because the function changes sign between a and b, and since it is continuous, it must pass through 0 at some point.
The Intermediate Value Theorem has many important applications in calculus, such as finding roots of equations, determining the existence of solutions to certain differential equations, and proving the existence of maxima and minima of functions.
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