{"id":11734,"date":"2014-02-03T00:27:26","date_gmt":"2014-02-03T00:27:26","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11734"},"modified":"2022-01-22T01:23:50","modified_gmt":"2022-01-22T01:23:50","slug":"um-ponto-b","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11734","title":{"rendered":"Um ponto $B$"},"content":{"rendered":"<p><ul id='GTTabs_ul_11734' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11734' class='GTTabs_curr'><a  id=\"11734_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11734' ><a  id=\"11734_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_11734'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Seja $B$ o ponto de coordenadas $\\left( {1,2} \\right)$.<br \/>\nA cada ponto $C\\left( {x,0} \\right)$ do eixo $Ox$, com $x &gt; 1$, fa\u00e7a corresponder um ponto $D\\left( {0,y} \\right)$ do eixo $Oy$, de modo que $B$, $C$ e $D$ sejam colineares.<\/p>\n<ol>\n<li>Exprima $y$ em fun\u00e7\u00e3o de $x$.<\/li>\n<li>Mostre que a \u00e1rea $A\\left( x \\right)$ do tri\u00e2ngulo $\\left[ {ODC} \\right]$ \u00e9 dada por: \\[A\\left( x \\right) = \\frac{{{x^2}}}{{x &#8211; 1}},\\,x &gt; 1\\]<\/li>\n<li>Represente o gr\u00e1fico de $A$ e indique o maior intervalo onde $A$ \u00e9 crescente e o maior intervalo onde \u00e9 decrescente.<\/li>\n<li>Determine para que valores de $x$ se verifica $A\\left( x \\right) &gt; 8$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11734' onClick='GTTabs_show(1,11734)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11734'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p style=\"text-align: center;\" data-tadv-p=\"keep\"><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":546,\r\n\"height\":523,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 59 || 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 || 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 || 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 6\",\r\n\"showToolBarHelp\":false,\r\n\"showResetIcon\":true,\r\n\"enableLabelDrags\":false,\r\n\"enableShiftDragZoom\":false,\r\n\"enableRightClick\":false,\r\n\"errorDialogsActive\":false,\r\n\"useBrowserForJS\":false,\r\n\"preventFocus\":false,\r\n\"showFullscreenButton\":true,\r\n\"language\":\"pt\",\r\n\/\/ use this instead of ggbBase64 to load a material from GeoGebraTube\r\n\/\/ 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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 1,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet')};\r\n<\/script><\/p>\n<ol>\n<li>Seja $B&#8217;$ a proje\u00e7\u00e3o ortogonal de $B$ sobre o eixo $Ox$.\n<p style=\"text-align: left;\">Como os tri\u00e2ngulos $\\left[ {COD} \\right]$ e $\\left[ {CB&#8217;B} \\right]$ s\u00e3o semelhantes, para $x &gt; 1$ vem: \\[\\frac{{\\overline {DO} }}{{\\overline {BB&#8217;} }} = \\frac{{\\overline {OC} }}{{\\overline {B&#8217;C} }} \\Leftrightarrow \\frac{y}{2} = \\frac{x}{{x &#8211; 1}} \\Leftrightarrow \\boxed{y = \\frac{{2x}}{{x &#8211; 1}}}\\]<\/p>\n<\/li>\n<li>Para $x &gt; 1$, a \u00e1rea do tri\u00e2ngulo $\\left[ {ODC} \\right]$ \u00e9 dada por: \\[\\begin{array}{*{20}{l}}<br \/>\n{A\\left( x \\right)}&amp; = &amp;{\\frac{{\\overline {OC}\u00a0 \\times \\overline {OD} }}{2}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{x \\times \\frac{{2x}}{{x &#8211; 1}}}}{2}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{{x^2}}}{{x &#8211; 1}}}<br \/>\n\\end{array}\\]<\/p>\n<p><span style=\"color: #000080;\">Na anima\u00e7\u00e3o acima, arraste o ponto $C$<\/span>.<\/p>\n<\/li>\n<li>Apresenta-se, de seguida, uma representa\u00e7\u00e3o gr\u00e1fica da fun\u00e7\u00e3o e a indica\u00e7\u00e3o da janela de visualiza\u00e7\u00e3o.<br \/>\n<table border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/Janela.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11735\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11735\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/Janela.jpg\" data-orig-size=\"264,136\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Janela de visualiza\u00e7\u00e3o\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/Janela.jpg\" class=\"aligncenter size-full wp-image-11735\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/Janela.jpg\" alt=\"Janela de visualiza\u00e7\u00e3o\" width=\"264\" height=\"136\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/Janela.jpg 264w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/Janela-150x77.jpg 150w\" sizes=\"auto, (max-width: 264px) 100vw, 264px\" \/><\/a><\/td>\n<td><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/Grafico.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11736\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11736\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/Grafico.jpg\" data-orig-size=\"264,136\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Gr\u00e1fico\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/Grafico.jpg\" class=\"aligncenter size-full wp-image-11736\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/Grafico.jpg\" alt=\"Gr\u00e1fico\" width=\"264\" height=\"136\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/Grafico.jpg 264w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/Grafico-150x77.jpg 150w\" sizes=\"auto, (max-width: 264px) 100vw, 264px\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>De acordo com a indica\u00e7\u00e3o fornecida pela ferramenta adequada da calculadora, a fun\u00e7\u00e3o admite $4$ como m\u00ednimo absoluto, para $x = 2$.<\/p>\n<p>Assim sendo, a fun\u00e7\u00e3o \u00e9 decrescente no intervalo $\\left] {1,2} \\right[$ e crescente no intervalo $\\left] {2, + \\infty } \\right[$.<\/p>\n<\/li>\n<li>Tem-se sucessivamente:<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{A\\left( x \\right) &gt; 8}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{\\frac{{{x^2}}}{{x &#8211; 1}} &gt; 8}&amp; \\wedge &amp;{x &gt; 1}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{{x^2} &gt; 8\\left( {x &#8211; 1} \\right)}&amp; \\wedge &amp;{x &gt; 1}&amp;{({\\text{pois }}x &#8211; 1 &gt; 0)}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{{x^2} &#8211; 8x + 8 &gt; 0}&amp; \\wedge &amp;{x &gt; 1}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{x \\in \\left] { &#8211; \\infty ,\\frac{{8 &#8211; \\sqrt {64 &#8211; 32} }}{2}} \\right[ \\cup \\left] {\\frac{{8 &#8211; \\sqrt {64 &#8211; 32} }}{2}, + \\infty } \\right[}&amp; \\wedge &amp;{x &gt; 1}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{x \\in \\left] { &#8211; \\infty ,4 &#8211; 2\\sqrt 2 } \\right[ \\cup \\left] {4 + 2\\sqrt 2 , + \\infty } \\right[}&amp; \\wedge &amp;{x &gt; 1}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x \\in \\left] {1,4 &#8211; 2\\sqrt 2 } \\right[ \\cup \\left] {4 + 2\\sqrt 2 , + \\infty } \\right[}<br \/>\n\\end{array}\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11734' onClick='GTTabs_show(0,11734)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Seja $B$ o ponto de coordenadas $\\left( {1,2} \\right)$. A cada ponto $C\\left( {x,0} \\right)$ do eixo $Ox$, com $x &gt; 1$, fa\u00e7a corresponder um ponto $D\\left( {0,y} \\right)$ do eixo&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20868,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,125],"tags":[422,126,270],"series":[],"class_list":["post-11734","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-funcoes-racionais","tag-11-o-ano","tag-funcao-racional","tag-inequacao"],"views":4212,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11V2Pag49-9_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11734","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=11734"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11734\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20868"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11734"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11734"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11734"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11734"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}