The Intermediate Value Theorem: Using Continuity To Find Solutions And Roots.

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 states that:

1) If f(x) is a continuous function on the closed interval [a, b],
2) and k is a number between f(a) and f(b)
3) then there exists a number c between a and b for which f(c) = k.

In other words, if we have a continuous function on a closed interval, and we know that the function takes on two values at the endpoints of the interval, then it must take on every value in between those two values at some point (or points) inside the interval.

For example, if we have a function f(x) that is continuous on the interval [0, 1], and we know that f(0) = 1 and f(1) = 4, then by the Intermediate Value Theorem, there must be some number c between 0 and 1 for which f(c) = 2 or 3 (or any other value between 1 and 4).

This theorem is useful in many applications, such as in finding roots of equations or proving the existence of solutions to certain problems.

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