Prueba de la primera derivada

<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> Prueba de la primera derivada</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> <B>Funciones crecientes y decrecientes</B></div> <div>  <A NAME="funciones_crecientes_1_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#0099FF" FACE="Times New Roman" ><B><IMG SRC="1f2f69550.gif" border=0 width="617" height="85" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <bR> <div>  <A NAME="funciones_decrecientes"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#0099FF" FACE="Times New Roman" ><B><IMG SRC="1f3064550.gif" border=0 width="612" height="85" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></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= 248 WIDTH="306"><div> <IMG SRC="1f3127470.gif" border=0 width="295" height="71" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0">.</div> <bR> <div> <FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B>U</B></FONT>na función que siempre es creciente o decreciente en un intervalo, se dice que es <FONT SIZE="3" COLOR="#0099FF" FACE="Times New Roman" ><I>monótona</I></FONT> en ese intervalo.</div> </tD> <tD VALIGN=TOP><NOBR><div> <B><IMG SRC="1f3219f10.gif" border=0 width="281" height="241" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> </NOBR></tD> </tR> </TABLE></div> <div>  <A NAME="teorema"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B><IMG SRC="1f336f450.gif" border=0 width="623" height="69" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div>  <A NAME="prueba_de_la_primera_derivada"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#0099FF" FACE="Times New Roman" ><B><IMG SRC="1f3485a50.gif" border=0 width="645" height="165" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="594" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 249 ><NOBR><div> <IMG SRC="1f3502d70.gif" border=0 width="258" height="215" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="313"><div> En la figura de la izquierda se esboza la interpretación geométrica del teorema:  "Prueba de la primera derivada". </div> <div> En la parte izquierda de la figura se tiene un valor máximo relativo en <I>c</I>, y se observa que <I>f</I> '(<I>x</I>)>0 para <I>x<c</I> (en algún intervaloque tiene a <I>c</I> como su extremo derecho) y  <I>f</I> '(<I>x</I>)<0 para <I>x>c </I>(en algún intervalo que tiene a <I>c</I> como su extremo izquierdo); en la parte derecha se tiene un valor mínimo relativo en <I>c</I>, y se observa que <I> f</I> '(<I>x</I>)<0 para <I>x<c</I> (en algún intervalo que tiene a <I>c</I> como su extremo derecho) y  <I>f</I> '(<I>x</I>)>0 para <I>x>c </I>(en algún intervalo que tiene a <I>c</I> como su extremo izquierdo)</div> </tD> </tR> </TABLE></div> <DIV ALIGN="LEFT"></DIV> <div> <TABLE BORDER="2" CELLPADDING="2" CELLSPACING="1" WIDTH="508" BORDERCOLORLIGHT="#009900" BORDERCOLORDARK="#33FFCC" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 132 WIDTH="496"><div>  <A NAME="p_r_o_c_e_d_i_m_i_e_n_t_o_"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#FF9900" FACE="Times New Roman" ><B>P r o c e d i m i e n t o </B></FONT></div> <div> Para determinar los valores extremos relativos de una función se procede de la siguiente manera:</div> <div> <FONT SIZE="3" COLOR="#FF9900" FACE="Times New Roman" ><B>1.</B></FONT>  Se halla la derivada de la función:  <I>f</I> '(<I>x</I>)</div> <div> <FONT SIZE="3" COLOR="#FF9933" FACE="Times New Roman" ><B>2.</B></FONT>  Se hallan los #s críticos de la función, esto es los valores de <I>x</I> para los cuales </div> <div> <I>      f</I> '(<I>x</I>) = 0  o para los cuales  <I>f</I> ' no existe.</div> <div> <B>3.</B><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >  Se aplica el criterio de la primera derivada</FONT></div> </tD> </tR> </TABLE></div> <bR> <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= 96 WIDTH="600"><div>  <A NAME="ejercicios_resueltos_1"><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="#0099FF" FACE="Times New Roman" ><B>   </B></FONT><FONT SIZE="3" COLOR="#990000" 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" >14</FONT>, proceda a lo siguiente: (<FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >a</FONT>)  obtenga los extremos relativos de  <I>f</I>  aplicando la prueba de la primera derivada; (<FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >b</FONT>)  determine los valores <I>x</I> en los que ocurren extremos relativos; (<FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >c</FONT>)  determine los intervalos en los que  <I>f</I>  es creciente; (<FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >d</FONT>)  determine los intervalos en los cuales  <I>f</I>  es decreciente; (<FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >e</FONT>)  trace la gráfica correspondiente.</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 31 ><NOBR><div> <A HREF="#1" TITLE="Prueba de la primera derivada"><U><IMG SRC="1f3689180.gif" border=0 width="137" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#2" TITLE="Prueba de la primera derivada"><U><IMG SRC="1f378c180.gif" border=0 width="140" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="198"><div> <A HREF="#3" TITLE="Prueba de la primera derivada"><U><IMG SRC="1f387b180.gif" border=0 width="123" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 51 WIDTH="198"><div> <A HREF="#4" TITLE="Prueba de la primera derivada"><U><IMG SRC="1f39812c0.gif" border=0 width="129" height="44" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="198"><div> <A HREF="#5" TITLE="Prueba de la primera derivada"><U><IMG SRC="1f3a69280.gif" border=0 width="105" height="40" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="198"><div> <A HREF="#6" TITLE="Prueba de la primera derivada"><U><IMG SRC="1f3b70290.gif" border=0 width="112" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 32 ><NOBR><div> <A HREF="#7" TITLE="Prueba de la primera derivada"><U><IMG SRC="1f3ca1180.gif" border=0 width="161" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="198"><div> <A HREF="#8" TITLE="Prueba de la primera derivada"><U><IMG SRC="1f3d81190.gif" border=0 width="129" height="25" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="198"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="32" ALIGN="bottom" WIDTH="198" HSPACE="0" VSPACE="0"></tD> </tR> </TABLE></div> <bR> <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="1f3e89180.gif" border=0 width="137" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="1f3f77740.gif" border=0 width="631" height="116" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="613" BORDERCOLORLIGHT="#EEEEEE" BORDERCOLORDARK="#CC9900" FRAME="BOX" RULES="NONE" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP BGCOLOR=#FFCC99 HEIGHT= 20 WIDTH="138"><div> <I><B>x</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="140"><div> <I><B>f</B></I><B> (</B><I><B>x</B></I><B>)</B></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="139"><div> <I><B>f</B></I><B> '(</B><I><B>x</B></I><B>)</B></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="179"><div> <B>Conclusión</B></div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="138"><div> <I><IMG SRC="1f4025130.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="140"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="140" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="139"><div> <IMG SRC="1f410d0b0.gif" border=0 width="13" height="11" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="179"><div> <I>f</I>  decrece</div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="138"><div> <IMG SRC="1f4227130.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="140"><div> <IMG SRC="1f4315130.gif" border=0 width="21" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="139"><div> 0</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF><NOBR><div> <I>f </I> tiene un mínimo relativo</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="138"><div> <IMG SRC="1f4427130.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="140"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="140" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="139"><div> +</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="179"><div> <I>f</I>  crece</div> </tD> </tR> </TABLE></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="608" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 245 ><NOBR><div> <IMG SRC="1f453fd50.gif" border=0 width="319" height="213" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1f460aee0.gif" border=0 width="266" height="238" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <bR> <div>  <A NAME="2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1f478c180.gif" border=0 width="140" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="1f4873400.gif" border=0 width="627" height="320" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="613" BORDERCOLORLIGHT="#EEEEEE" BORDERCOLORDARK="#CC9900" FRAME="BOX" RULES="NONE" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP BGCOLOR=#FFCC99 HEIGHT= 20 WIDTH="138"><div> <I><B>x</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="140"><div> <I><B>f</B></I><B> (</B><I><B>x</B></I><B>)</B></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="139"><div> <I><B>f</B></I><B> '(</B><I><B>x</B></I><B>)</B></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="179"><div> <B>Conclusión</B></div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="138"><div> <I><IMG SRC="1f493d130.gif" border=0 width="61" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="140"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="140" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="139"><div> +</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="179"><div> <I>f</I>  crece</div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="138"><div> <IMG SRC="1f4a3f130.gif" border=0 width="63" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="140"><div> <IMG SRC="1f4b25130.gif" border=0 width="37" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="139"><div> 0</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF><NOBR><div> <I>f </I> tiene un máximo relativo</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="138"><div> <IMG SRC="1f4c54130.gif" border=0 width="84" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="140"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="140" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="139"><div> <IMG SRC="1f4d0d0b0.gif" border=0 width="13" height="11" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="179"><div> <I>f</I>  decrece</div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="138"><div> <IMG SRC="1f4e23130.gif" border=0 width="35" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="140"><div> <IMG SRC="1f4f15110.gif" border=0 width="21" height="17" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="139"><div> 0</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF><NOBR><div> <I>f  </I>tiene un mínimo relativo</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="138"><div> <IMG SRC="1f5023130.gif" border=0 width="35" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="140"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="140" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="139"><div> +</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="179"><div> <I>f  </I>crece</div> </tD> </tR> </TABLE></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="603" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 226 ><NOBR><div> <IMG SRC="1f51f5db0.gif" border=0 width="245" height="219" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="296"><div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="159" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=2 BGCOLOR=#FFFBF0 HEIGHT= 20 WIDTH="149"><div> Tabla de valores</div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFBF0 HEIGHT= 20 WIDTH="73"><div> <I><B>x</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFBF0 WIDTH="73"><div> <I><B>y</B></I></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> -1</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> -1</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> -1/3</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 5/27</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> 0</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 0</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> 1</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> -1</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> 2</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 2</div> </tD> </tR> </TABLE></div> <bR> </tD> </tR> </TABLE></div> <bR> <div>  <A NAME="3"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1f527b180.gif" border=0 width="123" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="1f5344880.gif" border=0 width="580" height="392" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="607" BORDERCOLORLIGHT="#EEEEEE" BORDERCOLORDARK="#CC9900" FRAME="BOX" RULES="NONE" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP BGCOLOR=#FFCC99 HEIGHT= 20 WIDTH="192"><div> <I><B>x</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="109"><div> <I><B>f</B></I><B> (</B><I><B>x</B></I><B>)</B></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="110"><div> <I><B>f</B></I><B> '(</B><I><B>x</B></I><B>)</B></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="179"><div> <B>Conclusión</B></div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 34 ><NOBR><div> <I><IMG SRC="1f54bc1b0.gif" border=0 width="188" height="27" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></I></div> </NOBR></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="109"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="34" ALIGN="bottom" WIDTH="109" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="110"><div> +</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="179"><div> <I>f</I>  crece</div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 34 ><NOBR><div> <IMG SRC="1f55bf1b0.gif" border=0 width="191" height="27" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="109"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="34" ALIGN="bottom" WIDTH="109" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="110"><div> <IMG SRC="1f560d0b0.gif" border=0 width="13" height="11" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="179"><div> <I>f </I> decrece</div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 28 WIDTH="192"><div> <IMG SRC="1f576f150.gif" border=0 width="111" height="21" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="109"><div> 4</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="110"><div> 0</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF><NOBR><div> <I>f  </I>tiene un máximo relativo</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 28 WIDTH="192"><div> <IMG SRC="1f5870150.gif" border=0 width="112" height="21" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="109"><div> <IMG SRC="1f5915110.gif" border=0 width="21" height="17" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="110"><div> 0</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF><NOBR><div> <I>f  </I>tiene un mínimo relativo</div> </NOBR></tD> </tR> </TABLE></div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="1f5af4f50.gif" border=0 width="500" height="245" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="4"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1f5b812c0.gif" border=0 width="129" height="44" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="618" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 237 ><NOBR><div> <IMG SRC="1f5c4de50.gif" border=0 width="333" height="229" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1f5d0ce60.gif" border=0 width="268" height="230" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div> <IMG SRC="1f5ec9700.gif" border=0 width="457" height="112" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="5"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1f5f69280.gif" border=0 width="105" height="40" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="1f60309d0.gif" border=0 width="560" height="157" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="618" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 287 ><NOBR><div> <IMG SRC="1f611b080.gif" border=0 width="283" height="264" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1f621d990.gif" border=0 width="285" height="153" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="315" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=7 BGCOLOR=#FFFBF0 HEIGHT= 20 WIDTH="295"><div> Tabla de valores</div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFBF0 HEIGHT= 20 WIDTH="38"><div> <I><B>x</B></I></div> </tD> <tD VALIGN=TOP WIDTH="38"><div> -6</div> </tD> <tD VALIGN=TOP WIDTH="38"><div> -2.5</div> </tD> <tD VALIGN=TOP WIDTH="38"><div> -1.5</div> </tD> <tD VALIGN=TOP WIDTH="38"><div> 0</div> </tD> <tD VALIGN=TOP WIDTH="38"><div> 2</div> </tD> <tD VALIGN=TOP WIDTH="49"><div> 6</div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFBF0 HEIGHT= 20 WIDTH="38"><div> <B>y</B></div> </tD> <tD VALIGN=TOP WIDTH="38"><div> 2</div> </tD> <tD VALIGN=TOP WIDTH="38"><div> 9</div> </tD> <tD VALIGN=TOP WIDTH="38"><div> -7</div> </tD> <tD VALIGN=TOP WIDTH="38"><div> -1</div> </tD> <tD VALIGN=TOP WIDTH="38"><div> 0</div> </tD> <tD VALIGN=TOP WIDTH="49"><div> 0.5</div> </tD> </tR> </TABLE></div> <bR> </NOBR></tD> </tR> </TABLE></div> <div>  <A NAME="6"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1f6370290.gif" border=0 width="112" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="623" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 253 ><NOBR><div> <IMG SRC="1f6414f00.gif" border=0 width="276" height="240" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1f652ef60.gif" border=0 width="302" height="246" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div> Aplicando el criterio de la primera derivada, se resumen los resultados en la siguiente tabla:</div> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="613" BORDERCOLORLIGHT="#EEEEEE" BORDERCOLORDARK="#CC9900" FRAME="BOX" RULES="NONE" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP BGCOLOR=#FFCC99 HEIGHT= 20 WIDTH="138"><div> <I><B>x</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="140"><div> <I><B>f</B></I><B> (</B><I><B>x</B></I><B>)</B></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="139"><div> <I><B>f</B></I><B> '(</B><I><B>x</B></I><B>)</B></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="179"><div> <B>Conclusión</B></div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="138"><div> <I><IMG SRC="1f663d130.gif" border=0 width="61" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="140"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="140" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="139"><div> +</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="179"><div> <I>f</I>  crece</div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="138"><div> <IMG SRC="1f673f130.gif" border=0 width="63" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="140"><div> No existe</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="139"><div> No existe</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="179"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="179" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 30 WIDTH="138"><div> <IMG SRC="1f684b170.gif" border=0 width="75" height="23" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="140"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="30" ALIGN="bottom" WIDTH="140" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="139"><div> <IMG SRC="1f690d0b0.gif" border=0 width="13" height="11" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="179"><div> <I>f</I>  decrece</div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 30 WIDTH="138"><div> <IMG SRC="1f6a33170.gif" border=0 width="51" height="23" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="140"><div> <IMG SRC="1f6b17150.gif" border=0 width="23" height="21" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="139"><div> 0</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF><NOBR><div> <I>f  </I>tiene un mínimo relativo</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 30 WIDTH="138"><div> <IMG SRC="1f6c33170.gif" border=0 width="51" height="23" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="140"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="30" ALIGN="bottom" WIDTH="140" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="139"><div> +</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="179"><div> <I>f  </I>crece</div> </tD> </tR> </TABLE></div> <bR> <div>  <A NAME="7"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1f6da1180.gif" border=0 width="161" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="604" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 285 ><NOBR><div> <IMG SRC="1f6e2c130.gif" border=0 width="300" height="275" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="284"><div> <IMG SRC="1f6f01930.gif" border=0 width="257" height="147" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <bR> <bR> <bR> <bR> <div> Aplicando el criterio de la primera derivada, se resumen los resultados en la siguiente tabla:</div> </tD> </tR> </TABLE></div> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="613" BORDERCOLORLIGHT="#EEEEEE" BORDERCOLORDARK="#CC9900" FRAME="BOX" RULES="NONE" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP BGCOLOR=#FFCC99 HEIGHT= 20 WIDTH="138"><div> <I><B>x</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="140"><div> <I><B>f</B></I><B> (</B><I><B>x</B></I><B>)</B></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="139"><div> <I><B>f</B></I><B> '(</B><I><B>x</B></I><B>)</B></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="179"><div> <B>Conclusión</B></div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="138"><div> <I><IMG SRC="1f703d130.gif" border=0 width="61" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="140"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="140" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="139"><div> +</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="179"><div> <I>f</I>  crece</div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="138"><div> <IMG SRC="1f713f130.gif" border=0 width="63" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="140"><div> 0</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="139"><div> 0</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF><NOBR><div> No hay un extremo relativo</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="138"><div> <IMG SRC="1f7254130.gif" border=0 width="84" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="140"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="140" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="139"><div> +</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="179"><div> <I>f</I>  crece</div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 28 WIDTH="138"><div> <IMG SRC="1f7334130.gif" border=0 width="52" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="140"><div> <IMG SRC="1f7417150.gif" border=0 width="23" height="21" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="139"><div> 0</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF><NOBR><div> <I>f  </I>tiene un máximo relativo</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="138"><div> <IMG SRC="1f7549130.gif" border=0 width="73" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="140"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="140" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="139"><div> -</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="179"><div> <I>f  </I>decrece</div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="138"><div> <IMG SRC="1f7623130.gif" border=0 width="35" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="140"><div> 0</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="139"><div> 0</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF><NOBR><div> <I>f  </I>tiene un mínimo relativo</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 26 WIDTH="138"><div> <IMG SRC="1f7723130.gif" border=0 width="35" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="140"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="140" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="139"><div> +</div> </tD> <tD VALIGN=TOP BGCOLOR=#FFFFFF WIDTH="179"><div> <I>f </I>crece</div> </tD> </tR> </TABLE></div> <bR> <div>  <A NAME="8"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1f7881190.gif" border=0 width="129" height="25" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <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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