{"id":7715,"date":"2020-05-19T20:59:38","date_gmt":"2020-05-19T19:59:38","guid":{"rendered":"http:\/\/mathsguyon.fr\/?page_id=7715"},"modified":"2021-01-13T20:38:14","modified_gmt":"2021-01-13T19:38:14","slug":"fonction-affine-exo-15-corr","status":"publish","type":"page","link":"https:\/\/mathsguyon.fr\/?page_id=7715","title":{"rendered":"fonction affine exo 15 corr"},"content":{"rendered":"<p><strong>Exercice 15 :<\/strong><\/p>\n<p>R\u00e9soudre dans \\(\\mathbb{R}\\) les in\u00e9quations suivantes :<\/p>\n<p>\\((3x-2)^2-(x+1)^2\\geq 0\\)<\/p>\n<p>Pour \u00e9tudier le signe de cette expression, il faut la transformer en produit, pour se ramener \u00e0 un tableau de signes.<\/p>\n<p>Pour factoriser, on pense \u00e0 la 3\u00e8me identit\u00e9 remarquable \\(a^2-b^2=(a-b)(a+b)\\)<\/p>\n<p>Il vient donc :<\/p>\n<p>\\((3x-2)^2-(x+1)^2\\geq 0\\)<\/p>\n<p>\\(\\iff \\left(3x-2-(x+1)\\right)\\left(3x-2+(x+1)\\right)\\geq 0\\)<\/p>\n<p>\\(\\iff \\left(3x-2-x-1\\right)\\left(3x-2+x+1\\right)\\geq 0\\)<\/p>\n<p>\\(\\iff \\left(2x-3\\right)\\left(4x-1\\right)\\geq 0\\)<\/p>\n<p>On reconna\u00eet le produit de deux fonctions affines \\(f\\) et \\(g\\) d\u00e9finies sur \\(\\mathbb{R}\\)<\/p>\n<p>par \\(f(x)=2x-3\\) et \\(g(x)=\\)<\/p>\n<p>Pour \\(f\\) : \\(a=2&gt;0\\) et \\(\\dfrac{-b}{a}= &#8211; \\dfrac{-3}{2}=\\dfrac{3}{2}\\)<\/p>\n<p>Pour \\(g\\) : \\(a=4&gt;0\\) et \\(\\dfrac{-b}{a}= &#8211; \\dfrac{-1}{4}=\\dfrac{1}{4}\\)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7724 aligncenter\" src=\"http:\/\/mathsguyon.fr\/wp-content\/uploads\/2020\/05\/tab.png\" alt=\"\" width=\"493\" height=\"191\" srcset=\"https:\/\/mathsguyon.fr\/wp-content\/uploads\/2020\/05\/tab.png 694w, https:\/\/mathsguyon.fr\/wp-content\/uploads\/2020\/05\/tab-300x116.png 300w\" sizes=\"auto, (max-width: 493px) 100vw, 493px\" \/><\/p>\n<p style=\"text-align: center;\">On trouve alors : \\(S=]-\\infty;\\dfrac{1}{4}]\\cup [\\dfrac{3}{2};+\\infty[\\)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Exercice 15 : R\u00e9soudre dans \\(\\mathbb{R}\\) les in\u00e9quations suivantes : \\((3x-2)^2-(x+1)^2\\geq 0\\) Pour \u00e9tudier le signe de cette expression, il faut la transformer en produit, pour se ramener \u00e0 un tableau de signes. Pour factoriser, on pense \u00e0 la 3\u00e8me &hellip; <a href=\"https:\/\/mathsguyon.fr\/?page_id=7715\">Continuer la lecture <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-7715","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/pages\/7715","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=7715"}],"version-history":[{"count":11,"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/pages\/7715\/revisions"}],"predecessor-version":[{"id":8033,"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/pages\/7715\/revisions\/8033"}],"wp:attachment":[{"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7715"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}