\n
Solu\u00e7\u00e3o<\/p>\n
\nNote que: <\/p>\n
[tex]\\qquad f(x)=6x+\\dfrac {24}{x^{2}}=3x+3x+\\dfrac{24}{x^{2}}[\/tex].<\/p>\n
Agora, como [tex]x \\gt 0[\/tex] e [tex]3x \\cdot 3x \\cdot \\dfrac{24}{x^{2}}=216[\/tex], podemos utilizar a desigualdade entre a m\u00e9dia aritm\u00e9tica e a m\u00e9dia geom\u00e9trica para obter:<\/p>\n
[tex]\\qquad\\dfrac{3x+3x+\\dfrac{24}{x^{2}}}{3} \\ge \\sqrt[3]{3x \\cdot 3x \\cdot \\dfrac{24}{x^{2}}}[\/tex].<\/p>\n
[tex]\\qquad\\dfrac{3x+3x+\\dfrac{24}{x^{2}}}{3} \\ge \\sqrt[3]{216}[\/tex].<\/p>\n
[tex]\\qquad\\dfrac{3x+3x+\\dfrac{24}{x^{2}}}{3} \\ge 6[\/tex].<\/p>\n
[tex]\\qquad3x+3x+\\dfrac{24}{x^{2}} \\ge 18[\/tex].<\/p>\n
Portanto, o valor m\u00ednimo de [tex]3x+3x+\\dfrac{24}{x^{2}}[\/tex] \u00e9 [tex]18[\/tex] e isso s\u00f3 ocorre quando as parcelas s\u00e3o todas iguais, ou seja: <\/p>\n
[tex]\\qquad3x=3x=\\dfrac{24}{x^{2}}=6[\/tex].<\/p>\n
[tex]\\qquad 3x=6[\/tex].
\n[tex]\\qquad x=2[\/tex].<\/p>\n
De fato, quando [tex]x=2[\/tex]:<\/p>\n
[tex]\\qquad f(2)=6\\cdot 2+ \\dfrac {24}{2^2} =12+6=18.[\/tex]<\/p>\n
\n
Solu\u00e7\u00e3o elaborada pelos Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n <\/p>\n
\nParticiparam da discuss\u00e3o os Clubes:
Koreil Guys<\/strong>, Obm\u00e9picos<\/strong> e Phidias <\/strong>.<\/div>\n","protected":false},"excerpt":{"rendered":"Problema (Indicado a partir do 3\u00ba ano do E. M.) Calcule o valor m\u00ednimo da fun\u00e7\u00e3o [tex]f:(0, \\infty)\\rightarrow\\mathbb{R}[\/tex] definida por [tex]f(x)=6x+\\dfrac {24}{x^{2}}[\/tex]. Extra\u00eddo de PAPMEM. Lembretes Sejam [tex]a_1, a_2, a_3, \\cdots, a_n[\/tex] n\u00fameros reais positivos. (1) Chamamos de M\u00e9dia Aritm\u00e9tica dos n\u00fameros [tex]a_1, a_2, a_3, \\cdots a_n[\/tex] o n\u00famero assim definido: [tex]\\boxed{MA=\\dfrac{a_1+a_2+a_3+\\cdots+a_n}{n}}.[\/tex] (2) Chamamos de … <\/p>\n
Continue lendo<\/a><\/p>\n","protected":false},"author":668,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-223851","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/223851","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/users\/668"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=223851"}],"version-history":[{"count":6,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/223851\/revisions"}],"predecessor-version":[{"id":224499,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/223851\/revisions\/224499"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=223851"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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