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<i><font size="4" face="times new roman,times,serif">Determine el o los intervalos donde 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»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mi»#fs«/mi»«/math»</span></span></span></span></span></span></span></span> es #c y además, señale los puntos de inflexión si es que estos existen.</font></i>
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<div> <div><i style="font-size: large;"><font face="times new roman,times,serif">Solución</font><font face="times new roman,times,serif">:</font></i></div> <div><br /> <div align="justify"><em><font size="4" face="Times New Roman"></font></em></div> <div align="justify"><em><font size="4" face="Times New Roman"></font></em></div> <div align="justify"><em><font size="4" face="Times New Roman"></font></em></div> <div align="justify"></div> <div align="justify"><em><font size="4" face="Times New Roman"></font></em></div><font face="times new roman,times,serif"><i> <div align="justify"><font size="4"></font></div> <div align="justify"><font size="4">Para resolver, debemos recordar lo siguiente:<br /></font></div> <div align="justify"><i style="text-align: start;"> <div align="justify"> <p align="justify"><em><font size="4">Sea <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»f«/mi»«/math»</span> una función 2 veces derivable en su dominio:</font></em></p> <p align="justify"><em><font size="4">- 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"><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"»«mfrac»«mrow»«msup»«mi»d«/mi»«mn»2«/mn»«/msup»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«mrow»«mi»d«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mi»#h«/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> 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>, se tiene que <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»f«/mi»«/math»</span> es #c en </span></span></span></font></em><font face="times new roman,times,serif"><i><em><font size="4"><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></font></em></i></font></p></div></i></div></i></font></div> <div style="text-align: center;"><br /></div> <div align="justify"><font size="4" face="times new roman, times, serif"><i>Por lo tanto, nuestra función <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»h«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mi»#fs«/mi»«/math»</span> será #c si <span style="text-align: center;"></span><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"»«mfrac»«mrow»«msup»«mi»d«/mi»«mn»2«/mn»«/msup»«mi»h«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«mrow»«mi»d«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mi»#h«/mi»«/math»</span>0</span></span></span></span></span></span></span> </i></font></div> <div align="justify"><font size="4" face="times new roman, times, serif"><i>Para obtener</i></font><i style="font-family: 'times new roman',times,serif; font-size: large;"> </i><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"»«mfrac»«mrow»«msup»«mi»d«/mi»«mn»2«/mn»«/msup»«mi»h«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«mrow»«mi»d«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/math»</span> </span></span></span></span></span></span></span></i><i style="font-family: 'times new roman',times,serif; font-size: large;">utiliza</i><i style="font-size: large;"><font face="times new roman, times, serif">mos la regla de la derivada de cuociente (división) de funciones 2 veces:</font></i></div><font size="4" face="times new roman, times, serif"><i> <div><br /></div></i></font> <div style="text-align: center;"><i style="font-family: 'times new roman',times,serif; font-size: large; text-align: start;"> <div style="text-align: center;"><font size="4" face="times new roman, times, serif"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mfenced»«mfrac»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«mrow»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mfrac»«/mfenced»«mo»=«/mo»«mfrac»«mrow»«mfrac»«mrow»«mi»d«/mi»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»·«/mo»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»-«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»·«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«msup»«mfenced»«mrow»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«/math»</span></font></div></i></div> <div><font size="4" face="times new roman, times, serif"><i>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">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mi»#f1s«/mi»«/math»</span></span></span></span></span></span></span></span> y <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»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mi»#f2s«/mi»«/math»</span></span></span></span></span></span></span></span>, reemplazamos en nuestra fórmula:</i></font></div> <div><font size="4" face="times new roman, times, serif"><i><br /></i></font></div> <div style="text-align: center;"><font size="4" face="times new roman, times, serif"><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"»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mfenced»«mi»#fs«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mrow»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mfenced»«mrow»«mi»#f1s«/mi»«/mrow»«/mfenced»«mo»·«/mo»«mfenced»«mrow»«mi»#f2s«/mi»«/mrow»«/mfenced»«mo»-«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mfenced»«mrow»«mi»#f2s«/mi»«/mrow»«/mfenced»«mo»·«/mo»«mfenced»«mrow»«mi»#f1s«/mi»«/mrow»«/mfenced»«/mrow»«msup»«mfenced»«mi»#f2s«/mi»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«/math»</span></span></span></span></span></span></span></span></i></font></div> <div style="text-align: center;"><font size="4" face="times new roman, times, serif"><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"»«mo»=«/mo»«mfrac»«mrow»«mi»#f11s«/mi»«mfenced»«mrow»«mi»#f2s«/mi»«/mrow»«/mfenced»«mo»-«/mo»«mfenced»«mrow»«mi»#f22s«/mi»«/mrow»«/mfenced»«mfenced»«mrow»«mi»#f1s«/mi»«/mrow»«/mfenced»«/mrow»«msup»«mfenced»«mi»#f2s«/mi»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«/math»</span><br /><br /></span></span></span></span></span></span></span></i></font></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"»«mo»=«/mo»«mfrac»«mrow»«mi»#f3s«/mi»«/mrow»«msup»«mfenced»«mi»#f2s«/mi»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«/math»</span></span></span></span></span></span></span></span></i></div> <div style="text-align: center;"><font size="4" face="times new roman, times, serif"><i><br /></i></font></div> <div><font size="4" face="times new roman, times, serif"><i>Por lo tanto,</i></font></div> <div style="text-align: center;"><font size="4" face="times new roman, times, serif"><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"»«mfrac»«mrow»«mi»d«/mi»«mi»h«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mi»#f3s«/mi»«msup»«mfenced»«mi»#f2s«/mi»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«/math»</span></span></span></span></span></span></span></span></i></font></div> <div style="text-align: center;"><font size="4" face="times new roman, times, serif"><i><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"></span></span></span></span></i></font></div> <div align="justify" style="text-align: left;"><font size="4"><font face="times new roman, times, serif"><i>Nuevamente, aplicamos la regla de la derivada para la división de funciones y así obtenemos la segunda derivada; pero ahora</i></font><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»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mi»#f3s«/mi»«/math»</span></span></span></span></span></span></span></span> <font size="4">y</font> <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»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mfenced»«mi»#f2s«/mi»«/mfenced»«mn»2«/mn»«/msup»«/math»</span></span></span></span></span></span></span></span><font size="4">, entonces,</font> <font size="4">reemplazamos en nuestra fórmula:</font></i></font></div><font size="4" face="times new roman, times, serif"><i> <div><br /></div></i></font> <div style="text-align: center;"><font size="4" face="times new roman, times, serif"><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"»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mfenced»«mfrac»«mi»#f3s«/mi»«msup»«mfenced»«mi»#f2s«/mi»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«/mfenced»«mo»=«/mo»«mfrac»«mrow»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mfenced»«mi»#f3s«/mi»«/mfenced»«mo»·«/mo»«msup»«mfenced»«mi»#f2s«/mi»«/mfenced»«mn»2«/mn»«/msup»«mo»-«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mfenced»«msup»«mfenced»«mi»#f2s«/mi»«/mfenced»«mn»2«/mn»«/msup»«/mfenced»«mo»·«/mo»«mfenced»«mi»#f3s«/mi»«/mfenced»«/mrow»«msup»«mfenced»«msup»«mfenced»«mi»#f2s«/mi»«/mfenced»«mn»2«/mn»«/msup»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«/math»</span></span></span></span></span></span></span></span></i></font></div> <div style="text-align: center;"><font size="4" face="times new roman, times, serif"><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"»«mo»=«/mo»«mfrac»«mrow»«mi»#fd1s«/mi»«mo»·«/mo»«mo»(«/mo»«mi»#f2s«/mi»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«mi»#fd2s«/mi»«mfenced»«mrow»«mi»#f2s«/mi»«/mrow»«/mfenced»«mo»(«/mo»«mi»#f3s«/mi»«mo»)«/mo»«/mrow»«msup»«mfenced»«msup»«mfenced»«mrow»«mi»#f2s«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«/math»</span><br /><br /></span></span></span></span></span></span></span></i></font></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"»«mtable columnalign="left" rowspacing="0"»«mtr»«mtd»«mo»=«/mo»«mfrac»«mrow»«mi»#fd11s«/mi»«mfenced»«mi»#f2s«/mi»«/mfenced»«mfenced»«mi»#fd22s«/mi»«/mfenced»«/mrow»«mrow»«msup»«mfenced»«mi»#f2s«/mi»«/mfenced»«mn»3«/mn»«/msup»«mfenced»«mi»#f2s«/mi»«/mfenced»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd»«mo»=«/mo»«mfrac»«mrow»«mi»#fd11s«/mi»«mfenced»«mi»#fd22s«/mi»«/mfenced»«/mrow»«msup»«mfenced»«mi»#f2s«/mi»«/mfenced»«mn»3«/mn»«/msup»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»</span></span></span></span></span></span></span></span></i></div> <div style="text-align: left;"><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></span></span></span></span></span></span></i></div> <div style="text-align: left;"><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"><br /></span></span></span></span></span></span></span></i></div> <div><font size="4" face="times new roman, times, serif"><i>Así:</i></font></div> <div style="text-align: center;"><font size="4" face="times new roman, times, serif"><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"»«mfrac»«mrow»«msup»«mi»d«/mi»«mn»2«/mn»«/msup»«mi»h«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«mrow»«mi»d«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mi»#fd11s«/mi»«mfenced»«mi»#fd22s«/mi»«/mfenced»«/mrow»«msup»«mfenced»«mi»#f2s«/mi»«/mfenced»«mn»3«/mn»«/msup»«/mfrac»«/math»</span></span></span></span></span></span></span></span></i></font></div> <div style="text-align: center;"><font size="4" face="times new roman, times, serif"><i><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"></span></span></span></span></i></font></div> <div style="text-align: center;"><font size="4" face="times new roman, times, serif"><i><span class="nolink"><span class="nolink"><span class="nolink"></span></span></span></i></font></div> <div style="text-align: center;"><font size="4" face="times new roman, times, serif"><i><span class="nolink"></span></i></font></div> <div align="justify"><font size="4" face="times new roman, times, serif"><i>Observa que </i></font><i style="font-family: 'times new roman',times,serif; font-size: large;"><font size="4"><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"»«msup»«mfenced»«mi»#f2s«/mi»«/mfenced»«mn»3«/mn»«/msup»«/math»</span></span></span></span></span></span></span></span> siempre es mayor que <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mn»0«/mn»«/math»</span>, por lo tanto:</span></span></span></font><span style="background-color: rgb(51, 255, 102);"><br /></span></i> <div style="text-align: center;"><font size="4" face="times new roman, times, serif"><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"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mfrac»«mrow»«mi»#fd11s«/mi»«mfenced»«mrow»«mi»#fd22s«/mi»«/mrow»«/mfenced»«/mrow»«msup»«mfenced»«mi»#f2s«/mi»«/mfenced»«mn»3«/mn»«/msup»«/mfrac»«mo»§nbsp;«/mo»«mi»#h«/mi»«mo»§nbsp;«/mo»«mn»0«/mn»«/math»</span></span></span></span></span></span></span></span></span></span></span></i></font></div> <div><font size="4" face="times new roman, times, serif"><i>si: </i></font></div> <div style="text-align: center;"><font size="4" face="times new roman, times, serif"><i><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»#fd11s«/mi»«mfenced»«mrow»«mi»#fd22s«/mi»«/mrow»«/mfenced»«mi»#h«/mi»«/math»</span><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mn»0«/mn»«/math»</span></i></font></div> <div style="text-align: left;"><span><font color="#ff0000"><font size="4" face="times new roman, times, serif"><i>Se recomienda que haga una tabla de valores para que analice cuando se cumple que:</i></font></font></span></div> <div style="text-align: center;"><span><font size="4" face="times new roman, times, serif" style="color: rgb(255, 0, 0);"><i> </i></font><i style="font-family: 'times new roman',times,serif; font-size: large; text-align: center;"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»#fd11s«/mi»«mfenced»«mi»#fd22s«/mi»«/mfenced»«mi»#h«/mi»«/math»</span><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mn»0«/mn»«/math»</span></i><font size="4" face="times new roman, times, serif"><i> </i></font></span></div> <div style="text-align: center;"><br /></div> <div><font size="4" face="times new roman, times, serif"><i>Esto sucederá cuando:</i></font></div> <div><br /></div> <div><font size="4" face="times new roman, times, serif"><i><br /></i></font></div> <div style="text-align: center;"><font size="4" face="times new roman, times, serif"><i><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»x«/mi»«mo»§#8712;«/mo»«mi»#n1«/mi»«mi»#k1«/mi»«mi»#kk«/mi»«mi»#k2«/mi»«mi»#n2«/mi»«mi»#n5«/mi»«mi»#n3«/mi»«mi»#k3«/mi»«mi»#kk2«/mi»«mi»#k4«/mi»«mi»#n4«/mi»«/math»</span></i></font></div> <div style="text-align: center;"><br /></div> <div style="text-align: center;"><font size="4" face="times new roman, times, serif"><i><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"></span></span></span></span></i></font></div> <div><font size="4" face="times new roman, times, serif"><i>Finalmente, <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»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mi»#fs«/mi»«/math»</span></span></span></span></span></span></span></span> es <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»#c«/mi»«/math»</span> cuando:</i></font></div> <div><font size="4" face="times new roman, times, serif"><i><br /></i></font></div> <div style="text-align: center;"><font size="4" face="times new roman, times, serif"><i><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»x«/mi»«mo»§#8712;«/mo»«mi»#n1«/mi»«mi»#k1«/mi»«mi»#kk«/mi»«mi»#k2«/mi»«mi»#n2«/mi»«mi»#n5«/mi»«mi»#n3«/mi»«mi»#k3«/mi»«mi»#kk2«/mi»«mi»#k4«/mi»«mi»#n4«/mi»«/math»</span><br /><br /></i></font></div> <div style="text-align: left;"><font size="4" face="times new roman, times, serif"><i>El siguiente paso es determinar los puntos de inflexión de nuestra 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»#fs«/mi»«/math»</span></span></span></span></span></span></span></span>.<br />Recordemos que un punto de inflexión es un punto <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»«/math»</span></span></span></span></span></span></span></span> de una función contínua donde se pasa de cóncava a convexa o viceversa.<br /><br /></i></font></div> <div style="text-align: left;"><font size="4" face="times new roman, times, serif"><i>Como ya determinamos dónde <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»h«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/math»</span> era <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»#c«/mi»«/math»</span>, entonces, </i></font><i style="font-size: large; font-family: 'times new roman',times,serif;"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»h«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/math»</span><font size="4"> será</font> </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»#e«/mi»«/math»</span></span></span></span></span></span></span></span><i style="font-family: 'times new roman',times,serif; font-size: large;"> <font size="4">en:</font></i></div> <div style="text-align: left;"><em><font size="4" face="Times New Roman"></font></em></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»«mo»§#8712;«/mo»«mi»#n11«/mi»«mi»#k11«/mi»«mi»#kkk«/mi»«mi»#k22«/mi»«mi»#n22«/mi»«mi»#n55«/mi»«mi»#n33«/mi»«mi»#k33«/mi»«mi»#kkk2«/mi»«mi»#k44«/mi»«mi»#n44«/mi»«/math»</span></span></span></span></span></span></span></span><br /></i></div> <div align="justify" style="text-align: left;"><em><font size="4" face="Times New Roman"></font></em></div><span style="background-color: rgb(51, 255, 102);"><i style="font-family: 'times new roman',times,serif; font-size: large;"> <div style="text-align: left;"><br /></div></i></span> <div align="justify" 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"><font size="4">Por lo tanto, según lo anterior, los puntos de inflexión son:</font></span></span></span></span></span></span></span></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></span></span></span></span></span></span></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"»«mo»(«/mo»«mi»#l«/mi»«mo»,«/mo»«mi»#ev2«/mi»«mo»)«/mo»«mo»,«/mo»«mo»(«/mo»«mi»#k«/mi»«mo»,«/mo»«mi»#ev1«/mi»«mo»)«/mo»«mo»,«/mo»«mi»#par0«/mi»«mi»#ll«/mi»«mi»#par3«/mi»«mi»#ev3«/mi»«mi»#par1«/mi»«/math»</span></span></span></span></span></span></span></span><br /></i> <div style="text-align: left;"><font size="4" face="times new roman, times, serif"><i><br /></i></font></div> <div align="justify" style="text-align: left;"><font size="4"><font face="times new roman, times, serif"><i>Como podemos apreciar en la gráfica, 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»«/math»</span></span></span></span></span></span></span></span> es </i></font><font face="times new roman, times, serif"><i><font face="times new roman, times, serif"><i><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»#c«/mi»«/math»</span></i></font> en el tramo de color rojo y los puntos negros indican los puntos de inflexión.</i></font></font></div><font size="4" face="times new roman, times, serif"><i> <div style="text-align: left;"><br /></div></i></font> <div style="text-align: center;">#g9</div> <div style="text-align: center;"><br /></div> <div style="text-align: center;"></div></div></div></div> <div style="text-align: center;"></div>
One or multiple answers?
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One answer only
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Number the choices?
a., b., c., ...
A., B., C., ...
1., 2., 3., ...
i., ii., iii., ...
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No numbering
WIRIS variables
Algorithm
Choice 1
Answer
#n1#k1#kk#k2#n2#n5#n3#k3#kk2#k4#n4 Puntos de inflexión: (#l,#ev2), (#k,#ev2), #par0#ll#par3#ev3#par1
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<span style="text-align: justify; font-style: italic; font-family: 'times new roman',times,serif; font-size: large;"><font size="4">¡Muy Bien! Sigue así.</font></span>
Choice 2
Answer
#n11#k11#kkk#k22#n22#n55#n33#k33#kkk2#k44#n44 Puntos de inflexión: No existen en está función
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Feedback
<font size="4" style="font-style: italic; font-family: 'times new roman',times,serif;">Revisa el desarrollo para identificar tus errores. ¡Tú puedes!</font>
Choice 3
Answer
#n19#k1#kk#k2#n29#n5#n39#k3#kk2#k4#n49 Puntos de inflexión: No existen en está función
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Feedback
<font size="4" style="font-style: italic; font-family: 'times new roman',times,serif;">Revisa el desarrollo para identificar tus errores. ¡Tú puedes!</font>
Choice 4
Answer
#n119#k11#kkk#k22#n229#n55#n339#k33#kkk2#k44#n449 Puntos de inflexión: (#l,#ev2), (#k,#ev2) #par0#ll#par3#ev3#par1
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Feedback
<font size="4" style="font-style: italic; font-family: 'times new roman',times,serif;">Revisa el desarrollo para identificar tus errores. ¡Tú puedes!</font>
Choice 5
Answer
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Choice 6
Answer
Grade
None
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Choice 7
Answer
Grade
None
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Feedback
Combined feedback
For any correct response
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Options
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For any incorrect response
Settings for multiple tries
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Hint 1
Hint text
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Clear incorrect responses
Show the number of correct responses
Hint 2
Hint text
Options
Clear incorrect responses
Show the number of correct responses
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