See 3x5 ((BETTER))
The Intermediate Value Theorem is one of the most important theorems in Introductory Calculus, and it forms the basis for proofs of many results in subsequent and advanced Mathematics courses. The history of this theorem begins in the 1500's and is eventually based on the academic work of Mathematicians Bernard Bolzano, Augustin-Louis Cauchy, Joseph-Louis Lagrange, and Simon Stevin. Generally speaking, the Intermediate Value Theorem applies to continuous functions and is used to prove that equations, both algebraic and transcendental , are solvable. Note that this theorem will be used to prove the EXISTENCE of solutions, but will not actually solve the equations. (Newton's Method could be used to determine a good ESTIMATE for these solutions.) The formal statement of this theorem together with an illustration of the theorem follow. All functions are assumed to be real-valued. INTERMEDIATE VALUE THEOREM: Let $f$ be a continuous function on the closed interval $ [a, b] $. Assume that $m$ is a number ($y$-value) between $f(a)$ and $f(b)$. Then there is at least one number $c$ ($x$-value) in the interval $[a, b]$ which satifies$$ f(c)=m $$
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