[MATH 1A03] - Final Exam Guide - Ultimate 91 pages long Study Guide!
28 Nov 2016
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Let f be a function that satisfies the following three hypotheses: 3) f is continuous on the closed interval [a,b]. f is differentiable on the open interval (a,b). f(a) = f(b). Then there is a number c in (a,b) such that f"( c) =0. Proof of theorem can be found in the book, page 287. Example: show that the function f(x) = x^3 + x - 1 has exactly one root. Last class, we used intermediate value theorem to show that the function f(x) has at least one root. If f"(x)>0 on an interval, then f is increasing on that interval. If f"(x)<0 on an interval, then f is decreasing on that interval. We proved this last class using mean value theorem. Suppose that c is a critical point at which f"{c}=0, that f"{c} exists in a neighborhood of c , and that f"{c} exists.
