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Test-Lidia
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<i>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.<br><br><br>obs: Concava=conava hacia arriba; Convexa=concava hacia abajo <br></i>
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<div> <div><i style="font-size: large;">Solución:</i></div> <div><br> <div align="justify"><em></em></div> <div align="justify"><em></em></div> <div align="justify"><em></em></div> <div align="justify"></div> <div align="justify"><em></em></div><i> <div align="justify"></div> <div align="justify">Para resolver, debemos recordar lo siguiente:<br></div> <div align="justify"><i style="text-align: start;"> <div align="justify"> <p align="justify"><em>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:</em></p> <p align="justify"><em>- 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></em><i><em><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></i></p></div></i></div></i></div> <div style="text-align: center;"><br></div> <div align="justify"><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></div> <div align="justify"><i>Para obtener</i><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;">mos la regla de la derivada de cuociente (división) de funciones 2 veces:</i></div><i> <div><br></div></i> <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;"><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></div></i></div> <div><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></div> <div><i><br></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¨»«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></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¨»«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></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;"><i><br></i></div> <div><i>Por lo tanto,</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¨»«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></div> <div style="text-align: center;"><i><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"></span></span></span></span></i></div> <div style="text-align: left;" align="justify"><i>Nuevamente, aplicamos la regla de la derivada para la división de funciones y así obtenemos la segunda derivada; pero ahora</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¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mi»#f3s«/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»«msup»«mfenced»«mi»#f2s«/mi»«/mfenced»«mn»2«/mn»«/msup»«/math»</span></span></span></span></span></span></span></span>, entonces, reemplazamos en nuestra fórmula:</i></div><i> <div><br></div></i> <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¨»«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></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¨»«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></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><i>Así:</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¨»«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></div> <div style="text-align: center;"><i><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"></span></span></span></span></i></div> <div style="text-align: center;"><i><span class="nolink"><span class="nolink"><span class="nolink"></span></span></span></i></div> <div style="text-align: center;"><i><span class="nolink"></span></i></div> <div align="justify"><i>Observa que </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¨»«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><span style="background-color: rgb(51, 255, 102);"><br></span></i> <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"><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></div> <div><i>si: </i></div> <div style="text-align: center;"><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></div> <div style="text-align: left;"><span><i>Se recomienda que haga una tabla de valores para que analice cuando se cumple que:</i></span></div> <div style="text-align: center;"><span><i> </i><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><i> </i></span></div> <div style="text-align: center;"><br></div> <div><i>Esto sucederá cuando:</i></div> <div><br></div> <div><i><br></i></div> <div style="text-align: center;"><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></div> <div style="text-align: center;"><br></div> <div style="text-align: center;"><i><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"></span></span></span></span></i></div> <div><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></div> <div><i><br></i></div> <div style="text-align: center;"><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></div> <div style="text-align: left;"><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></div> <div style="text-align: left;"><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><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> será </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;"> en:</i></div> <div style="text-align: left;"><em></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 style="text-align: left;" align="justify"><em></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 style="text-align: center;" align="justify"><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">Por lo tanto, según lo anterior, los puntos de inflexión son:</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;"><i><br></i></div> <div style="text-align: left;" align="justify"><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><i><i><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»#c«/mi»«/math»</span></i> en el tramo de color rojo y los puntos negros indican los puntos de inflexión.</i></div><i> <div style="text-align: left;"><br></div></i> <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?
Multiple answers allowed
One answer only
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Number the choices?
a., b., c., ...
A., B., C., ...
1., 2., 3., ...
i., ii., iii., ...
I., II., III., ...
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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Feedback
<span style="text-align: justify; font-style: italic; font-family: 'times new roman',times,serif; font-size: large;">¡Muy Bien! Sigue así.</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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83.33333%
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Feedback
Revisa el desarrollo para identificar tus errores. ¡Tú puedes!
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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83.33333%
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Feedback
Revisa el desarrollo para identificar tus errores. ¡Tú puedes!
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
Revisa el desarrollo para identificar tus errores. ¡Tú puedes!
Choice 5
Answer
Grade
None
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83.33333%
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Feedback
Choice 6
Answer
Grade
None
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Feedback
Choice 7
Answer
Grade
None
100%
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83.33333%
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70%
66.66667%
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Feedback
Combined feedback
For any correct response
For any partially correct response
Options
Show the number of correct responses once the question has finished
For any incorrect response
Settings for multiple tries
Penalty for each incorrect try
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Hint 1
Hint text
Options
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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Wednesday, 15 February 2017, 3:41 PM
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