{"id":1353,"date":"2017-10-30T16:20:28","date_gmt":"2017-10-30T20:20:28","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1353"},"modified":"2017-10-30T16:20:28","modified_gmt":"2017-10-30T20:20:28","slug":"hw7-challenge-5","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/10\/30\/hw7-challenge-5\/","title":{"rendered":"HW7: Challenge 5"},"content":{"rendered":"

Proof by Contradiction<\/p>\n

Suppose \"f\" is differentiable on an interval\u00a0\"A\" with\u00a0\"f'(x), and\u00a0\"f\" has more than 1 fixed point. Let\u00a0\"x,y such that\u00a0\"x,y\" are fixed points where\u00a0\"x. By the definition of fixed point from Weekly HW 6,\u00a0\"f(x)=x\" and\u00a0\"f(y)=y\". By Theorem 5.2.3, since\u00a0\"f\" is differentiable on\u00a0\"A\", then\u00a0\"f\" is continuous on\u00a0\"A\". Thus, we can apply the Mean Value Theorem so that there exists a\u00a0\"c such that\u00a0\"f'(c)=. By algebra,\u00a0\"f'(c)=\frac{y-x}{y-x}=1\". This is a contradiction because\u00a0\"f'(x). Thus,\u00a0\"f\" has at most one fixed point.<\/p>\n","protected":false},"excerpt":{"rendered":"

Proof by Contradiction Suppose is differentiable on an interval\u00a0 with\u00a0, and\u00a0 has more than 1 fixed point. Let\u00a0 such that\u00a0 are fixed points where\u00a0. By the definition of fixed point from Weekly HW 6,\u00a0 and\u00a0. By Theorem 5.2.3, since\u00a0 is differentiable on\u00a0, then\u00a0 is continuous on\u00a0. Thus, we can apply the Mean Value Theorem so […]<\/p>\n","protected":false},"author":3530,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[58827,58826],"tags":[],"class_list":["post-1353","post","type-post","status-publish","format-standard","hentry","category-challenge-solutions","category-peer-grading"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-lP","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1353","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/users\/3530"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=1353"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1353\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1353"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1353"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1353"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}