Intermediate Value Theorem: Calculus Concepts and Proving Real Roots in Functions.

Intermediate Value Theorem:1) ___________________2) ___________________3) ___________________ then there exits a number c between a and b for which f(c) = k

1) f(x) is continues on [a,b]2) f(a) ≠ f(b)3) if k is between f(a) and f(b), then there’s

The Intermediate Value Theorem is a fundamental concept in calculus. It states that:

1) If a function f is continuous on the closed interval [a, b], and
2) k is a number between f(a) and f(b), and
3) a < b, then there exists a number c between a and b for which f(c) = k. This means that if we have a function that is continuous over a given interval, and we know that the function takes on two specific values at the endpoints of the interval, then the function must also take on every value in between those two endpoints. For example, let f(x) = x^3 - 2x^2 - 4x + 1. We want to prove that the equation f(x) = 0 has at least one real root between x = 1 and x = 2. We can see that f(1) = -4 and f(2) = 1, so we know that the function takes on values on opposite sides of 0. Therefore, by the Intermediate Value Theorem, there must exist at least one value c between x = 1 and x = 2 for which f(c) = 0, meaning the equation has at least one real root on the interval [1, 2].

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