Concavidad y puntos de inflexión

<body topmargin=0 leftmargin=0 marginheight=0 marginwidth=0> <table cellpadding=0 cellspacing=0 border=0><tr> <td valign="top" colspan=3 height=24 BGCOLOR="#006666" TEXT="#080000"> <div> <B>C á l c u l o   2 1</B><FONT SIZE="4" COLOR="#FFCC66" FACE="Bell MT" ><B> </B></FONT><FONT SIZE="4" COLOR="#FFFFCC" FACE="Bell MT" >   </FONT><B>  </B>|   <A HREF="index.htm" TARGET="_top" TITLE="Cálculo 21"><U>Home</U></A>  |  <A HREF="id98.htm" TARGET="_top" TITLE="Contenido"><U>Contenido</U></A>  |  <A HREF="id307.htm" TARGET="_top" TITLE="Introducción al cálculo diferencial"><U>Cálculo diferencial</U></A>  |</div> <div> </div> </td> </tr><tr> <td valign="top" rowspan="2" width=160 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B><IMG BORDER="0" SRC="Wing0a73007700c800660000.gif" HEIGHT="13" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0"> <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:160px;height:600px" data-ad-client="ca-pub-1578463498418701" data-ad-slot="3073775660"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </B></div> </td> <td valign="top" height=432 colspan=2 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> Concavidad y puntos de inflexión</div> <bR> <div>  <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:block" data-ad-client="ca-pub-1578463498418701" data-ad-slot="3926688482" data-ad-format="auto" data-full-width-responsive="true"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> <bR> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="639" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 614 WIDTH="385"><div> <IMG SRC="1f7b80cb0.gif" border=0 width="384" height="203" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="concavidad_hacia_arriba"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1f7c7c650.gif" border=0 width="380" height="101" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="concavidad_hacia_abajo"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1f7d77650.gif" border=0 width="375" height="101" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="teorema"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1f7e7c7b0.gif" border=0 width="380" height="123" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="243"><div> <IMG SRC="1f7feaa30.gif" border=0 width="234" height="163" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> (<FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" >fig.1</FONT>)</div> <bR> <div> <IMG SRC="1f80f39f0.gif" border=0 width="243" height="159" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0">(<FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" >fig.2</FONT>)</div> <bR> <div> <IMG SRC="1f81ebb80.gif" border=0 width="235" height="184" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0">(<FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" >fig.3</FONT>)</div> </tD> </tR> </TABLE></div> <div>  <A NAME="punto_de_inflexi_n"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1f82783d0.gif" border=0 width="632" height="61" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="teorema_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1f83652b0.gif" border=0 width="613" height="43" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <FONT SIZE="3" COLOR="#9999FF" FACE="Times New Roman" ><B><IMG BORDER="0" SRC="Symb075c00b700f0ff999900.gif" HEIGHT="16" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">L</B></FONT>os posibles puntos de inflexión se identifican despejando a <I>x</I> de la ecuación que resulta una vez se ha igualado la segunda derivada de la función a cero; o para los valores de <I>x</I> para los cuales la segunda derivada no existe. </div> <bR> <DIV ALIGN="LEFT"></DIV> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="612" BORDERCOLORLIGHT="#006600" BORDERCOLORDARK="#009900" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=3 BGCOLOR=#FFCC99 HEIGHT= 77 WIDTH="600"><div>  <A NAME="ejercicios_resueltos"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><B>Ejercicios resueltos</B></FONT></div> <div> <FONT SIZE="3" COLOR="#CCFF00" FACE="Times New Roman" ><B>E</B></FONT>n los ejercicios <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >1</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >7</FONT>, halle los puntos de inflexión de la gráfica de la función que se indica, si los hay. Determine dónde la gráfica es cóncava hacia arriba y dónde lo es hacia abajo. Trace la gráfica y muestre un segmento de cada tangente de inflexión.</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 31 WIDTH="198"><div> <A HREF="#1" TITLE="Concavidad y puntos de inflexión"><U><IMG SRC="1f8475180.gif" border=0 width="117" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="198"><div> <A HREF="#2" TITLE="Concavidad y puntos de inflexión"><U><IMG SRC="1f857c180.gif" border=0 width="124" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="198"><div> <A HREF="#3" TITLE="Concavidad y puntos de inflexión"><U><IMG SRC="1f8680180.gif" border=0 width="128" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> </TABLE></div> <div>  </div> <div> <B>S o l u c i o n e s </B></div> <bR> <div>  <A NAME="1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1f8775180.gif" border=0 width="117" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <TABLE BORDER="0" CELLPADDING="1" CELLSPACING="1" WIDTH="640" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 304 WIDTH="298"><div> <IMG SRC="1f880cf00.gif" border=0 width="268" height="240" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> En la siguiente tabla se resumen los resultados obtenidos:</div> </tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1f894e210.gif" border=0 width="334" height="289" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="606" BORDERCOLORLIGHT="#EEEEEE" BORDERCOLORDARK="#CC9900" FRAME="BOX" RULES="NONE" ALIGN="LEFT" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP BGCOLOR=#FFCC99 HEIGHT= 20 WIDTH="73"><div> <I><B>x</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="73"><div> <I><B>f</B></I><B> (</B><I><B>x</B></I><B>)</B></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="73"><div> <I><B>f</B></I><B> '(</B><I><B>x</B></I><B>)</B></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="73"><div> <I><B>f</B></I><B> ''(</B><I><B>x)</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="294"><div> <B>Conclusión</B></div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="73"><div> <I><IMG SRC="1f8a25130.gif" border=0 width="37" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="73" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="73" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><div> -</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF><NOBR><div> la gráfica de <I>f</I> es cóncava hacia abajo</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="73"><div> <IMG SRC="1f8b27130.gif" border=0 width="39" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><div> 0</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><div> 9</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><div> 0</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="294"><div> <I>f </I> tiene un punto de inflexión</div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="73"><div> <IMG SRC="1f8c27130.gif" border=0 width="39" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="73" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="73" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><div> +</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF><NOBR><div> la gráfica de <I>f</I> es cóncava hacia arriba</div> </NOBR></tD> </tR> </TABLE></div> <bR> <bR> <bR> <bR> <bR> <bR> <bR> <bR> <div>  <A NAME="2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1f8d7c180.gif" border=0 width="124" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="610" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 344 ><NOBR><div> <IMG SRC="1f8e2dab0.gif" border=0 width="301" height="171" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="1f8f2ca30.gif" border=0 width="300" height="163" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1f9009370.gif" border=0 width="265" height="311" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> fig.2</div> </NOBR></tD> </tR> </TABLE></div> <div> <IMG SRC="1f915b810.gif" border=0 width="347" height="129" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="1f9273530.gif" border=0 width="371" height="83" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="606" BORDERCOLORLIGHT="#EEEEEE" BORDERCOLORDARK="#CC9900" FRAME="BOX" RULES="NONE" ALIGN="LEFT" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP BGCOLOR=#FFCC99 HEIGHT= 20 WIDTH="73"><div> <I><B>x</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="73"><div> <I><B>f</B></I><B> (</B><I><B>x</B></I><B>)</B></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="73"><div> <I><B>f</B></I><B> '(</B><I><B>x</B></I><B>)</B></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="73"><div> <I><B>f</B></I><B> ''(</B><I><B>x)</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="294"><div> <B>Conclusión</B></div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="73"><div> <I><IMG SRC="1f9325130.gif" border=0 width="37" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="73" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="73" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><div> +</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF><NOBR><div> la gráfica de <I>f</I> es cóncava hacia arriba</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="73"><div> <IMG SRC="1f9427130.gif" border=0 width="39" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><div> 0</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><div> 0</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><div> 0</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="294"><div> <I>f </I> tiene un punto de inflexión</div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 ><NOBR><div> <IMG SRC="1f953f130.gif" border=0 width="63" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="73" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="73" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><div> -</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF><NOBR><div> la gráfica de <I>f</I> es cóncava hacia abajo</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="73"><div> <IMG SRC="1f9627130.gif" border=0 width="39" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><div> -256</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><div> -128</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><div> 0</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="294"><div> <I>f </I> tiene un punto de inflexión</div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="73"><div> <IMG SRC="1f9727130.gif" border=0 width="39" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="73" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="73" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><div> +</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF><NOBR><div> la gráfica de <I>f</I> es cóncava hacia arriba</div> </NOBR></tD> </tR> </TABLE></div> <bR> <bR> <bR> <bR> <bR> <bR> <bR> <bR> <bR> <bR> <bR> <div>  <A NAME="3"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1f9880180.gif" border=0 width="128" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="613" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 210 ><NOBR><div> <IMG SRC="1f9935cb0.gif" border=0 width="309" height="203" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1f9aeac00.gif" border=0 width="234" height="192" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div> <IMG SRC="1f9b0d530.gif" border=0 width="525" height="83" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> En la tabla que sigue se resumen los resultados obtenidos:</div> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="606" BORDERCOLORLIGHT="#EEEEEE" BORDERCOLORDARK="#CC9900" FRAME="BOX" RULES="NONE" ALIGN="LEFT" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP BGCOLOR=#FFCC99 HEIGHT= 20 WIDTH="73"><div> <I><B>x</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="73"><div> <I><B>f</B></I><B> (</B><I><B>x</B></I><B>)</B></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="73"><div> <I><B>f</B></I><B> '(</B><I><B>x</B></I><B>)</B></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="73"><div> <I><B>f</B></I><B> ''(</B><I><B>x)</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="294"><div> <B>Conclusión</B></div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="73"><div> <I><IMG SRC="1f9c2f130.gif" border=0 width="47" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="73" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="73" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><div> +</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF><NOBR><div> la gráfica de <I>f</I> es cóncava hacia arriba</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="73"><div> <IMG SRC="1f9d2f130.gif" border=0 width="47" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><div> 0</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><div> no existe</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><div> no existe</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="294"><div> <I>f </I> tiene un punto de inflexión</div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="73"><div> <IMG SRC="1f9e2f130.gif" border=0 width="47" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="73" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="73" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="73"><div> -</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF><NOBR><div> la gráfica de <I>f</I> es cóncava hacia abajo</div> </NOBR></tD> </tR> </TABLE></div> <bR> <bR> <bR> <bR> <bR> <bR> <bR> <bR> <bR> <bR> </td> </tr><tr> <td valign="top" colspan=2 height=24 BGCOLOR="#006666" TEXT="#080000"> <div> <B> <form action="http://www.google.com.co" 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