INTERMEDIATE VALUE THEOREM Meaning and
Definition
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The intermediate value theorem is a fundamental concept in calculus that describes the behavior of continuous functions on a closed interval. It states that if a function is continuous on a closed interval [a, b] and takes on two distinct values, then it must also take on every value in between those two values.
More specifically, the intermediate value theorem asserts that for every value y between the function values f(a) and f(b), there exists at least one value c in the interval [a, b] such that f(c) = y. This theorem implies that if a continuous function starts at one value and ends at another, it must pass through every value in between.
The intermediate value theorem is a fundamental tool for proving the existence of solutions to equations, as it guarantees the existence of at least one value for which the equation is satisfied. It is particularly useful in real-world applications where finding an exact solution may be difficult, as it allows for the estimation of solutions within a given range.
In summary, the intermediate value theorem states that if a function is continuous on a closed interval and takes on two distinct values, it must also take on every value in between. It is a powerful concept in calculus that enables the estimation and determination of solutions to equations.
