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 is continuous on the closed interval [a,b], and k is a number between f(a) and f(b), then there exists a number c between a and b for which f(c) = k.
To explain it further:
1) The theorem assumes that the function f is continuous (meaning it has no jumps or breaks) over the interval [a, b].
2) It also assumes that f(a) and f(b) have values on opposite sides of the number k, which means k is between the two values of f(a) and f(b).
3) The Intermediate Value Theorem then concludes that there must be at least one point c between a and b where the function f takes a value of k.
In simpler terms, the Intermediate Value Theorem guarantees that if we have a continuous function on an interval, and we know the function takes on different signs at the two endpoints, then the function must have crossed zero at some point in between the endpoints.
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