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Test-Lidia
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<i>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">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»h«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mi»#f«/mi»«/math»</span></span></span></span></span></span></span></span> es #cr si:<br><br><br>obs: Concava= concava hacia arriba; Convexa= concava hacia abajo<br></i>
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<div><i>Solución:<br><br></i></div> <div><em></em></div> <div><em>Para resolver esta pregunta, debemos recordar lo siguiente:<br><br></em></div> <div> <table width="100%" cellspacing="1" cellpadding="1" border="1"><tbody> <tr> <td width="100%" valign="top"> <p><i>Sea <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> una función dos veces derivable en su dominio:</i></p> <p><i>- Si <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¨»«mfrac»«mrow»«msup»«mi»d«/mi»«mn»2«/mn»«/msup»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/mrow»«mrow»«mi»d«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mo»§nbsp;«/mo»«mi»#des«/mi»«mo»§nbsp;«/mo»«mn»0«/mn»«/math»</span></span></span></span></span></span> en <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¨»«mo»]«/mo»«mi»a«/mi»«mo»,«/mo»«mi»b«/mi»«mo»[«/mo»«/math»</span></span></span></span></span></span>, <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> es #cr en <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¨»«mo»]«/mo»«mi»a«/mi»«mo»,«/mo»«mi»b«/mi»«mo»[«/mo»«/math»</span></span></span></span></span></span></i></p> </td> </tr></tbody> </table></div> <div style="text-align: left"><br></div> <div style="text-align: left" align="justify"><i>Entonces, la función <span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»h«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mi»#f«/mi»«/math»</span></span></span> es #cr cuando:</i></div> <div style="text-align: left"><em></em></div> <div style="text-align: left"><em></em></div> <div style="text-align: center"><span class="nolink"><span class="nolink"><span class="nolink"></span></span></span><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»d«/mi»«mn»2«/mn»«/msup»«mi»h«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/mrow»«mrow»«mi»d«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mo»§nbsp;«/mo»«mi»#des«/mi»«mo»§nbsp;«/mo»«mn»0«/mn»«/math»</span></span></span> </div> <div style="text-align: left" align="justify"><i><i>Entonces,</i> calculamos la segunda derivada de <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»h«/mi»«/math»</span></span></span></span></i></div> <div style="text-align: left" align="justify"><i><span class="nolink"></span></i></div> <div style="text-align: left" align="justify"><em></em></div> <div style="text-align: left" align="justify"><em></em></div> <div style="text-align: center"><i style="text-align: center; font-family: 'times new roman',times,serif; font-size: large"><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¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mi»h«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mi»#f«/mi»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»h«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mi»#df«/mi»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mrow»«msup»«mi»d«/mi»«mn»2«/mn»«/msup»«mi»h«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/mrow»«mrow»«mi»d«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mo»=«/mo»«mi»#ddf«/mi»«/mtd»«/mtr»«/mtable»«/math»</span></span></span></span></span></span></span></span></i></div> <div style="text-align: left"><i>Como ya sabemos, para que h(x) sea #cr se debe cumplir que:</i></div> <div style="text-align: left"><em></em></div> <div style="text-align: left"><em></em></div> <div style="text-align: left"><em></em></div> <div style="text-align: left"><em></em></div> <div style="text-align: center"><i><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¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mfrac»«mrow»«msup»«mi»d«/mi»«mn»2«/mn»«/msup»«mi»h«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/mrow»«mrow»«mi»d«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mo»§nbsp;«/mo»«mi»#des«/mi»«mo»§nbsp;«/mo»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mrow»«msup»«mi»d«/mi»«mn»2«/mn»«/msup»«mi»h«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/mrow»«mrow»«mi»d«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mo»=«/mo»«mi»#ddf«/mi»«mo»§nbsp;«/mo»«mi»#des«/mi»«mo»§nbsp;«/mo»«mn»0«/mn»«/mtd»«/mtr»«/mtable»«/math»</span></span></span></span></span></span></span></span></i></div> <div style="text-align: left"><i>Luego,</i></div> <div style="text-align: center"><i><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»§nbsp;«/mo»«mi»#des«/mi»«mo»§nbsp;«/mo»«mi»#pinf«/mi»«/math»</span></span></span></span></span></span></span></span><br></i></div> <div style="text-align: left"><i>Es decir,</i></div> <div style="text-align: center"><i style="font-family: 'times new roman',times,serif; font-size: large"><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»«mi»#pert«/mi»«mi»#r1«/mi»«mi»#r2«/mi»«mi»#r3«/mi»«mi»#r4«/mi»«mi»#r5«/mi»«/math»</span></span></span></span></span></span></span></span></i> <br></div> <div style="text-align: left"><i>Observemos la gráfica de <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»h«/mi»«/math»</span></span></span></span></span></span></i></div> <div style="text-align: center"><i>#t1</i></div> <div style="text-align: center"><em></em></div> <div style="text-align: center" align="justify"><i>Si observas el gráfico, está marcada </i><i><i>en rojo</i> la parte donde <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»h«/mi»«/math»</span></span></span></span> es #cr. Fíjate que coincide con el resultado obtenido.</i></div>
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Choice 1
Answer
x#pert#r1#r2#r3#r4#r5
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<div style="text-align: left" align="justify"><font size="4" face="times new roman,times,serif"><i>¡Excelente!</i></font></div>
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Choice 2
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x#pert#r1#in#r3#sup#r5
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<div style="text-align: left"><font style="font-style: italic; font-family: times new roman,times,serif" size="4">Revisa el desarrollo para identificar tus errores. ¡Tú puedes!</font></div> <div style="text-align: center"></div>
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Choice 3
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x#pert#r1#i2#r3#i4#r5
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<div style="text-align: left"><font style="font-style: italic; font-family: times new roman,times,serif" size="4">Revisa el desarrollo para identificar tus errores. ¡Tú puedes!</font></div> <div style="text-align: center"></div>
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Choice 4
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x#pert#re
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<div style="text-align: left"><font style="font-style: italic; font-family: times new roman,times,serif" size="4">Revisa el desarrollo para identificar tus errores. ¡Tú puedes!</font></div> <div style="text-align: center"></div>
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