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 if a function f(x) is continuous on a closed interval [a, b], and k is any number between f(a) and f(b), then there exists a number c between a and b such that f(c) = k.
In other words:
1) The function f(x) must be continuous on the closed interval [a, b].
2) The number k must be between f(a) and f(b).
3) There must exist a number c between a and b, inclusively, for which f(c) is equal to k.
This theorem has many applications and is commonly used to prove the existence of roots or solutions to equations. For example, if we want to prove that a polynomial equation has a real root between two given values, we can use the Intermediate Value Theorem to show that the polynomial must cross the x-axis at some point between those values, and hence must have a real root.
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