{"id":1370,"date":"2017-10-31T12:00:40","date_gmt":"2017-10-31T16:00:40","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1370"},"modified":"2017-11-05T23:48:26","modified_gmt":"2017-11-06T04:48:26","slug":"hw-7-challenge-3","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/10\/31\/hw-7-challenge-3\/","title":{"rendered":"HW 7: Challenge 3"},"content":{"rendered":"

Problem:<\/strong> Let \"f be a differentiable function on an interval \"A\". Show that if\u00a0\"f'(x) on\u00a0\"A, then\u00a0\"f is one-to-one. Provide an example to show that the converse of the statement need not be true.<\/p>\n

Proof:\u00a0<\/strong> Suppose\u00a0\"f is a differentiable function on some interval \"A. Let \"f(x) for some arbitrary\u00a0\"x,. By Theorem 5.2.3, \"f\" is also continuous on \"A\" and so by the Mean Value Theorem, \"\exists such that \"f'(c). By some algebra, \"f(x)-f(y). Since\u00a0\"f(x),\u00a0\"f'(c)(x-y). So either\u00a0\"f'(c) or \"(x-y)=0\". However, by assumption, \"f'(x) so \"(x-y)=0\", so\u00a0\"x=y\" and thus\u00a0\"f is one-to-one.<\/p>\n

 <\/p>\n

Corrected Proof: (Proof by Contradicition)\u00a0<\/span><\/strong>Suppose \"f is a differentiable function on some interval \"A with the condition that \"f'(c). Let \"f(x) for some arbitrary \"x,. By Theorem 5.2.3, \"f\" is also continuous on \"A\" and so by the Mean Value Theorem, \"\exists such that (WLOG) for \"y>x\", \"f'(c). By some algebra, \"f(x)-f(y). Since \"f(x), \"f'(c)(x-y). Since \"y>x\" by contruction of the mean value thereorem, \"f'(c)=0\" However, this is a contradiction to the assumption that \"f'(c), thus \"x=y\" and thus \"f is one-to-one.\u00a0<\/span><\/p>\n

Example:\u00a0<\/strong>Let \"f(x) on the interval \"(-3,3)\". We see that \"f'(x). If \"f(x), then \"x^5 and so \"x=y\" and we have that this is a 1-1 function. However at \"x=0\" on \"(-3,3),.<\/p>\n","protected":false},"excerpt":{"rendered":"

Problem: Let be a differentiable function on an interval . Show that if\u00a0 on\u00a0, then\u00a0 is one-to-one. Provide an example to show that the converse of the statement need not be true. Proof:\u00a0 Suppose\u00a0 is a differentiable function on some interval . Let for some arbitrary\u00a0. By Theorem 5.2.3, is also continuous on and so […]<\/p>\n","protected":false},"author":3525,"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-1370","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-m6","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1370","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\/3525"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=1370"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1370\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1370"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1370"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1370"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}