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 the following:
1) If f(x) is a continuous function on the closed interval [a,b], and if k 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) = k.
2) Another way to state this theorem is to say that if a continuous function f(x) on a closed interval takes on two different values (let’s say a and b), then it must also take on every value in between a and b at some point within that interval.
3) The Intermediate Value Theorem is important in calculus and real analysis because it provides a guarantee that a continuous function will cross a given horizontal line at least once within a specified interval. This theorem is often used to prove the existence of roots of equations, which is a fundamental concept in many fields of mathematics and science.
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