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<i><font size="4" face="times new roman,times,serif">El </font></i><i><font size="4" face="times new roman,times,serif"><i><font size="4" face="times new roman,times,serif">siguiente </font></i>gráfico representa la segunda derivada de una función <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»«/math»</span></span></span></span></span></span></span><br />#grafdd<br /><br />El gráfico que representa a la función <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> corresponde a:<br /></font></i>
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<div><i><font size="4" face="times new roman,times,serif"><br /></font></i></div><i><font size="4" face="times new roman,times,serif">Solución:<br /><br />Recordemos que una función es cóncava hacia arriba si la recta tangente en el punto <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mfenced»«mrow»«mi»a«/mi»«mo»,«/mo»«mi»f«/mi»«mfenced»«mi»a«/mi»«/mfenced»«/mrow»«/mfenced»«/math»</span></span></span></span> queda por abajo de la gráfica de la función <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»f«/mi»«/math»</span> (Figura 1) y que es cóncava hacia abajo si la recta tangente en el punto <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mfenced»«mrow»«mi»a«/mi»«mo»,«/mo»«mi»f«/mi»«mfenced»«mi»a«/mi»«/mfenced»«/mrow»«/mfenced»«/math»</span></span></span></span> queda por arriba de la gráfica de la función <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»f«/mi»«/math»</span> (Figura 2).</font></i><div><font face="times new roman, times, serif" size="4"><i><table width="100%" border="1"><tbody><tr><td width="50%" valign="top"><img 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align="baseline" border="0" hspace="0" vspace="0" /><br />Figura 1</td><td width="50%" valign="top"><img 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" alt="hacia abajo" title="hacia abajo" align="baseline" border="0" hspace="0" vspace="0" /><br /><br /><br /><br /><br /><br />Figura 2</td></tr></tbody></table></i></font><br /><br /><br /><font size="4"><i><font face="times new roman,times,serif">En términos analíticos lo anterior se traduce en el siguiente teorema:<br /><br /></font></i></font> <ul> <li><font size="4"><i><font face="times new roman,times,serif">Si <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»«/mrow»«mrow»«mi»d«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mfenced»«mi»x«/mi»«/mfenced»«mo»§gt;«/mo»«mn»0«/mn»«/math»</span></span></span></span></span></span></span>, entonces <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»f«/mi»«/math»</span> es cóncava hacia arriba.</font></i></font> </li> <li><font size="4"><i><font face="times new roman,times,serif">Si <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»«/mrow»«mrow»«mi»d«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mfenced»«mi»x«/mi»«/mfenced»«mo»§lt;«/mo»«mn»0«/mn»«/math»</span></span></span></span></span></span></span>, entonces <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»f«/mi»«/math»</span> es cóncava hacia abajo (convexa).</font></i></font> </li> </ul> <p><font size="4"><i><font face="times new roman,times,serif">De esto, obtenemos que cuando la segunda derivada cambia de signo la función pasa de ser cóncava hacia arriba a cóncava hacia abajo (o viceversa). El punto donde la función cambia de concavidad se denomina Punto de Inflexión y ocurre cuando la segunda derivada cambia de signo.</font></i></font></p> <p><font size="4"><i><font face="times new roman,times,serif">Aplicado esta información a nuestro ejercicio, se ve en la gráfica de <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»«/mrow»«mrow»«mi»d«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mfenced»«mi»x«/mi»«/mfenced»«/math»</span></span></span></span></span></span></span> que está por sobre el eje X y bajo el eje X, es decir, es positiva y negativa. Dibujando esto en el gráfico del enunciado tenemos que:<br /></font></i></font></p> <p><font size="4"><i><font face="times new roman,times,serif">#grafMm<br /></font></i></font></p> <p><font size="4"><i><font face="times new roman,times,serif">En el gráfico, está pintado en rojo donde es positiva y en azul donde es negativa. Por lo tanto, es cóncava hacia abajo en <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"»«mfenced close="]" open="["»«mrow»«mi»a«/mi»«mo»,«/mo»«mi»b«/mi»«/mrow»«/mfenced»«/math»</span></span></span></span></span></span></span> y cóncava hacia arriba en el resto. <br /></font></i></font></p> <p><font size="4"><i><font face="times new roman,times,serif">Debemos notar también, que la segunda derivada cambia dos veces de signo, o sea, tiene dos puntos de inflexión o cambios de concavidad. Así, la gráfica de la función que posee todas las características mencionadas anteriormente, corresponde a la alternativa con el gráfico que se muestra a continuación:<br /></font></i></font></p> <p><font size="4"><i><font face="times new roman,times,serif">#graf</font></i></font><br /></p></div>
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WIRIS variables
Algorithm
Choice 1
Answer
#graf
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<i><font size="4" face="times new roman,times,serif">¡Excelente! </font></i>
Choice 2
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#grafr1
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<i><font size="4" face="times new roman,times,serif">Revisa el desarrollo para identificar tus errores.</font></i> <em><font size="4" face="times new roman,times,serif">¡Tú puedes!</font></em>
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#grafr2
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<font size="4" face="times new roman,times,serif"><em>Revisa el desarrollo para identificar tus errores.</em><font size="3" face="Trebuchet MS"> </font><font size="4" face="times new roman,times,serif"><em>¡Tú puedes!</em></font></font>
Choice 4
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#grafr3
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<font size="4" face="times new roman,times,serif"><em>Revisa el desarrollo para identificar tus errores.</em><font size="3" face="Trebuchet MS"> </font><font size="4" face="times new roman,times,serif"><em>¡Tú puedes!</em></font></font>
Choice 5
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Choice 6
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Hint 1
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on
Wednesday, 15 February 2017, 3:41 PM
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