Gabarito -T3 -7h - P3

Prévia do material em texto

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 a)
 
 
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 O \303\272nico ponto cr\303\255tico de f no interior de R \303\251 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 0.2 => Se calculou as derivadas parciais e percebeu que fx(x,y)=0 e fy(x,y)=0
 0.3 => Pelo ponto cr\303\255tico encontrado corretamente
 
 
 
 b)
 
 
 Usando o m\303\251todo dos Multiplicadores de Lagrange:
 
 
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
 PCQvSSJ4RzYiJCIrSkd6Im8iISIqL0kieUdGJUYm
 
 
 Por simetria \303\251 poss\303\255vel achar os demais pontos:
 
 
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 JCIrSkd6Im8iISIq
 
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 JCIrSkd6Im8iISIq
 
 JCErSkd6Im8iISIq
 
 
 QyY+SSN4NEc2IiQhK0pHeiJvIiEiKiIiIj5JI3k0R0YlJCIrSkd6Im8iRihGKQ==
 
 JCErSkd6Im8iISIq
 
 JCIrSkd6Im8iISIq
 
 
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 JCErSkd6Im8iISIq
 
 JCErSkd6Im8iISIq
 
 
 0.2 => Se escreveu o m\303\251todo de Lagrangre corretamente 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
 0.8 => Pelos pontos encontrados corretamente (0.2 para cada ponto)
 
 
 
 c)
 
 
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 QyQtSSJmRzYiNiRJI3gzR0YlSSN5M0dGJSIiIg==
 
 JCIrXFlkNWYhIzY=
 
 
 QyQtSSJmRzYiNiRJI3g0R0YlSSN5NEdGJSIiIg==
 
 JCIrXFlkNWYhIzY=
 
 
 QyQtSSJmRzYiNiRJI3g1R0YlSSN5NUdGJSIiIg==
 
 JCIrcUcpPXAiISIp
 
 
 
 
 
 O valor m\303\241ximo de f em R \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW5HRiQ2JFEsMTYuOTE4ODI4NzBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSJ+RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZHRi8= e o valor m\303\255nimo de f em R \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW5HRiQ2JFEuMC4wNTkxMDU3NDY0OUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIi5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkdGLw==
 
 0.2 => Pelo c\303\241lculos das imagens dos 5 pontos
 0.3 => Pelos valores de m\303\241ximo e m\303\255nimo global corretos
 
 
 
 JSFH