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<p align="justify"><font size="4" face="times new roman,times,serif"><em>El gráfico de la función <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> que satisface <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»§apos;§apos;«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§nbsp;«/mo»«mi»#cond«/mi»«mo»§nbsp;«/mo»«mn»0«/mn»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> , si y sólo si </em></font></p> <p align="center"><font size="4" face="times new roman,times,serif"><em><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»§#8712;«/mo»«mi»#i«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> es:</em></font></p> <p align="justify"><font size="4" face="times new roman,times,serif"><em></em></font></p>
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<p align="justify"><font size="4" face="times new roman,times,serif"><em>Solución:</em></font></p> <p align="justify"><font size="4" face="times new roman,times,serif"><em>Para resolver, debemos recordar el criterio de la segunda derivada que postula lo siguiente:</em></font></p> <p align="justify"><font size="4" face="times new roman,times,serif"><em>Sea <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»f«/mi»«/math»</span> una función dos veces derivable en <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mo»]«/mo»«mi»a«/mi»«mo»,«/mo»«mi»b«/mi»«mo»[«/mo»«/math»</span>:</em></font></p> <p align="justify"><font face="times new roman,times,serif"><font size="4"><em>- <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»f«/mi»«mo»§apos;§apos;«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§gt;«/mo»«mn»0«/mn»«/math»</span></span></span></span></span></span> </span></span></span></span></span> en <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mo»]«/mo»«mi»a«/mi»«mo»,«/mo»«mi»b«/mi»«mo»[«/mo»«/math»</span> si y sólo si <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»f«/mi»«/math»</span> es cóncava hacia arriba en <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mo»]«/mo»«mi»a«/mi»«mo»,«/mo»«mi»b«/mi»«mo»[«/mo»«/math»</span></em></font></font></p> <p align="justify"><font face="times new roman,times,serif"><font size="4"><em>- <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»f«/mi»«mo»§apos;§apos;«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§lt;«/mo»«mn»0«/mn»«/math»</span></span></span></span></span></span></span></span></span></span></span> en <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mo»]«/mo»«mi»a«/mi»«mo»,«/mo»«mi»b«/mi»«mo»[«/mo»«/math»</span> si y sólo si <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»f«/mi»«/math»</span> es cóncava hacia abajo en <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mo»]«/mo»«mi»a«/mi»«mo»,«/mo»«mi»b«/mi»«mo»[«/mo»«/math»</span></em></font></font></p> <p align="justify"><font size="4" face="times new roman,times,serif"><em>En nuestro caso, <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»f«/mi»«mo»§apos;§apos;«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§nbsp;«/mo»«mi»#cond«/mi»«mo»§nbsp;«/mo»«mn»0«/mn»«/math»</span></span></span></span></span></span></span></span></span></span></span> si y sólo si <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»x«/mi»«mo»§#8712;«/mo»«mi»#i«/mi»«/math»</span>.</span></span></span></span></span></span></span></span></span></span><br /></em></font></p> <p align="justify"><font size="4" face="times new roman,times,serif"><em>Por lo tanto, <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»f«/mi»«/math»</span> es cóncava hacia #co en #i.<br /></em></font></p> <p align="justify"><font size="4" face="times new roman,times,serif"><em>Lo que se refleja en el siguiente gráfico:</em></font></p> <p align="center"><font size="4" face="times new roman,times,serif"><em>#sol</em></font></p> <p></p>
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Choice 1
Answer
#sol
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<p align="justify"><font size="4" face="times new roman,times,serif"><em>¡Excelente!</em></font></p>
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Choice 2
Answer
#sol2
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Grade
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100%
90%
83.33333%
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75%
70%
66.66667%
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<p align="justify"><font size="4" face="times new roman,times,serif"><em>Identifica bien los intervalos de concavidad según el criterio de la segunda derivada.<font size="4" face="times new roman,times,serif"><em> ¡Tú puedes!</em></font></em></font></p>
Format
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Choice 3
Answer
#sol3
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Grade
None
100%
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83.33333%
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75%
70%
66.66667%
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<p align="justify"><font size="4" face="times new roman,times,serif"><em>Identifica bien los intervalos de concavidad según el criterio de la segunda derivada. <font size="4" face="times new roman,times,serif"><em>¡Tú puedes!</em></font></em></font></p>
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Moodle auto-format
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Choice 4
Answer
#sol4
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<p align="justify"><font size="4" face="times new roman,times,serif"><em>Identifica bien los intervalos de concavidad según el criterio de la segunda derivada. <font size="4" face="times new roman,times,serif"><em>¡Tú puedes!</em></font></em></font></p>
Format
HTML format
Moodle auto-format
Plain text format
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Choice 5
Answer
Grade
None
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83.33333%
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70%
66.66667%
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Choice 6
Answer
Grade
None
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70%
66.66667%
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Choice 7
Answer
Grade
None
100%
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83.33333%
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70%
66.66667%
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33.33333%
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Feedback
Combined feedback
For any correct response
Format
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Markdown format
For any partially correct response
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Options
Show the number of correct responses once the question has finished
For any incorrect response
Format
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Settings for multiple tries
Penalty for each incorrect try
100%
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33.33333%
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Wednesday, 15 February 2017, 3:41 PM
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