tag:blogger.com,1999:blog-2400513859305780710.post7277037145128809938..comments2023-04-04T07:53:53.789-07:00Comments on Mathematical Food For Thought: Up, Up, And Away! Topic: Algebra/Calculus. Level: AIME.Jeffrey Wanghttp://www.blogger.com/profile/11114458640271201663noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-2400513859305780710.post-19866856599649759202006-05-23T09:36:05.000-07:002006-05-23T09:36:05.000-07:00Let F(x) = \int_{0}^{x} f(t) dt. F(x) is a functi...Let F(x) = \int_{0}^{x} f(t) dt. F(x) is a function [0, 1] --> [0, 1), and moreover F(0) = 0.<br><br>2x - F(x) - 1 takes on the value -1 at 0 and a positive value at 1, so it has at least one root. Its derivative, 2 - f(x), is always positive, so it is monotonically increasing; hence, it has exactly one root. <br><br>(It seems a little too easy...)QCnoreply@blogger.comtag:blogger.com,1999:blog-2400513859305780710.post-2258136251707395822006-05-23T10:08:39.000-07:002006-05-23T10:08:39.000-07:00Yeah, it did seem a little easy, but oh well.Yeah, it did seem a little easy, but oh well.paladin8noreply@blogger.com