{"id":11738,"date":"2014-02-03T17:55:31","date_gmt":"2014-02-03T17:55:31","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11738"},"modified":"2021-12-26T01:38:31","modified_gmt":"2021-12-26T01:38:31","slug":"prove-que-7","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11738","title":{"rendered":"Prove que"},"content":{"rendered":"<p><ul id='GTTabs_ul_11738' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11738' class='GTTabs_curr'><a  id=\"11738_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11738' ><a  id=\"11738_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_11738'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Prove que a fun\u00e7\u00e3o definida por $f\\left( x \\right) = \\frac{1}{x}$ n\u00e3o \u00e9 mon\u00f3tona no seu dom\u00ednio.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11738' onClick='GTTabs_show(1,11738)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11738'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>A\u00a0fun\u00e7\u00e3o \u00e9 estritamente decrescente em ${\\mathbb{R}^ &#8211; }$, quer em ${\\mathbb{R}^ + }$, pois ${x_1} &gt; {x_2} \\Rightarrow f\\left( {{x_1}} \\right) &lt; f\\left( {{x_2}} \\right),\\forall x \\in {\\mathbb{R}^ &#8211; }$ e ${x_1} &gt; {x_2} \\Rightarrow f\\left( {{x_1}} \\right) &lt; f\\left( {{x_2}} \\right),\\forall x \\in {\\mathbb{R}^ + }$, respetivamente.<\/p>\n<p>No entanto, a fun\u00e7\u00e3o n\u00e3o \u00e9 mon\u00f3tona no seu dom\u00ednio, pois, por exemplo, $\\begin{array}{*{20}{c}}<br \/>\n{\\begin{array}{*{20}{c}}<br \/>\n{2 &gt;\u00a0 &#8211; 1}&amp; \\wedge &amp;{f\\left( 2 \\right) &gt; f\\left( { &#8211; 1} \\right)}<br \/>\n\\end{array}}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}<br \/>\n{2 &gt;\u00a0 &#8211; 1}&amp; \\wedge &amp;{\\frac{1}{2} &gt;\u00a0 &#8211; 1}<br \/>\n\\end{array}}<br \/>\n\\end{array}$.<\/p>\n<\/p>\n<div id=\"attachment_11739\" style=\"width: 810px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2pag49-11.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-11739\" data-attachment-id=\"11739\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11739\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2pag49-11.png\" data-orig-size=\"946,529\" 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=\"&lt;p&gt;A fun\u00e7\u00e3o $f\\left( x \\right) = \\frac{1}{x}$ n\u00e3o \u00e9 decrescente&lt;\/p&gt;\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2pag49-11.png\" class=\"  wp-image-11739\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2pag49-11.png\" alt=\"A fun\u00e7\u00e3o $f\\left( x \\right) = \\frac{1}{x}$ n\u00e3o \u00e9 decrescente\" width=\"800\" height=\"447\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2pag49-11.png 946w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2pag49-11-300x167.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2pag49-11-150x83.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2pag49-11-400x223.png 400w\" sizes=\"auto, (max-width: 800px) 100vw, 800px\" \/><\/a><p id=\"caption-attachment-11739\" class=\"wp-caption-text\">A fun\u00e7\u00e3o $f\\left( x \\right) = \\frac{1}{x}$ n\u00e3o \u00e9 decrescente<\/p><\/div>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11738' onClick='GTTabs_show(0,11738)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Prove que a fun\u00e7\u00e3o definida por $f\\left( x \\right) = \\frac{1}{x}$ n\u00e3o \u00e9 mon\u00f3tona no seu dom\u00ednio. Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":19189,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,125],"tags":[422,126,294],"series":[],"class_list":["post-11738","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-monotonia"],"views":2771,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat75.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11738","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=11738"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11738\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11738"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11738"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11738"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11738"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}