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 theorem of calculus that states the following:
1) Suppose that f(x) is a continuous function on the closed interval [a,b], meaning that f(x) has no sudden jumps or breaks on that interval.
2) Let k be any number between f(a) and f(b), meaning that k lies strictly between the minimum and maximum values of f(x) on the interval [a,b].
3) Then there exists a number c between a and b for which f(c) = k, meaning that f(x) takes on every value between f(a) and f(b) on that interval.
In other words, if a function is continuous on an interval and takes on different values at the endpoints of that interval, then it must also take on every intermediate value between those endpoints at some point within the interval. This is a powerful tool for proving the existence of solutions to many types of mathematical problems, including finding roots and zeroes of functions, determining if functions have local maximum and minimum values, and solving equations that cannot be solved algebraically.
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