2008-06-15T11:22:16Z https://www.matematicasies.com/Estudio-Global-de-Funciones#comment273 2008-06-15T11:22:16Z

<p>Estudio global de <img src='https://www.matematicasies.com/local/cache-TeX/56c3b72ddbeec0aa5ffb5b2025a977a9.png' style="vertical-align:middle;" width="140" height="47" alt="f(x) = x^4-3x^2" title="f(x) = x^4-3x^2"></p> <p><strong>Dominio</strong></p> <p><img src='https://www.matematicasies.com/local/cache-TeX/9989a79b687634fb89b8d62e7e2e744b.png' style="vertical-align:middle;" width="118" height="42" alt="Dom(f) = R" title="Dom(f) = R"> <br>El dominio de cualquier función polinómica es "todos los números reales"</p> <p><strong>Asíntotas</strong></p> <p>Una función polinómica <strong>no tiene asíntotas</strong>.</p> <p><strong>Simetrías</strong></p> <p><img src='https://www.matematicasies.com/local/cache-TeX/24102d3d1bfe295090b8d3d6e0dfa82e.png' style="vertical-align:middle;" width="380" height="47" alt="f(-x) = (-x)^4-3(-x)^2 = x^4-3x^2 = f(x)" title="f(-x) = (-x)^4-3(-x)^2 = x^4-3x^2 = f(x)"> <br>Es una función <strong>par</strong> (simétrica respecto al eje OY)</p> <p><strong>Corte con los ejes</strong></p> <p><span class="spip-puce ltr"><b>–</b></span> Si <img src='https://www.matematicasies.com/local/cache-TeX/3dad28281778d5ef4b7a78c7bc7a6b09.png' style="vertical-align:middle;" width="50" height="38" alt="x = 0" title="x = 0"> entonces <img src='https://www.matematicasies.com/local/cache-TeX/596093e0539c4bb5b3d58f7dbabcf754.png' style="vertical-align:middle;" width="50" height="38" alt="y = 0" title="y = 0">. Punto:<img src='https://www.matematicasies.com/local/cache-TeX/5c16f757233856dcf311176b7410d2d5.png' style="vertical-align:middle;" width="47" height="42" alt="(0,0)" title="(0,0)"> <br><span class="spip-puce ltr"><b>–</b></span> Si <img src='https://www.matematicasies.com/local/cache-TeX/fab37d6c4a697fe660387d3ff8e889a4.png' style="vertical-align:middle;" width="50" height="38" alt="y=0" title="y=0"> entonces <img src='https://www.matematicasies.com/local/cache-TeX/f62d5f29478c27993038c7852ec692a1.png' style="vertical-align:middle;" width="112" height="47" alt="x^4-3x^2=0" title="x^4-3x^2=0">. Resolvemos la ecuación: <br><img src='https://www.matematicasies.com/local/cache-TeX/9a6f9aeb8a295373d0574244e48dc8b8.png' style="vertical-align:middle;" width="127" height="47" alt="x^2(x^2-3)=0" title="x^2(x^2-3)=0"> de donde: <br><img src='https://www.matematicasies.com/local/cache-TeX/74c75dcc62c43f0a20edfe8c06fbcb79.png' style="vertical-align:middle;" width="77" height="38" alt="\rightarrow x=0" title="\rightarrow x=0"> <br><img src='https://www.matematicasies.com/local/cache-TeX/88aaed55a8944ea7cae9c1b24bc38587.png' style="vertical-align:middle;" width="240" height="50" alt="\rightarrow x^2-3=0 \Longrightarrow x=\pm \sqrt{3}" title="\rightarrow x^2-3=0 \Longrightarrow x=\pm \sqrt{3}"> <br>Puntos: <img src='https://www.matematicasies.com/local/cache-TeX/5c16f757233856dcf311176b7410d2d5.png' style="vertical-align:middle;" width="47" height="42" alt="(0,0)" title="(0,0)"> , <img src='https://www.matematicasies.com/local/cache-TeX/c18246d349eeea9ddf93676e6e5118e7.png' style="vertical-align:middle;" width="63" height="50" alt="(\sqrt{3},0)" title="(\sqrt{3},0)"> , <img src='https://www.matematicasies.com/local/cache-TeX/87cf12421e6caf215bdc7a09a496e972.png' style="vertical-align:middle;" width="80" height="50" alt="(-\sqrt{3},0)" title="(-\sqrt{3},0)"></p> <p><strong>Monotonía</strong></p> <p><img src='https://www.matematicasies.com/local/cache-TeX/07466ec186140964b25b92a11026e06d.png' style="vertical-align:middle;" alt="f'(x)=4x^3-6x" title="f'(x)=4x^3-6x"> <br><img src='https://www.matematicasies.com/local/cache-TeX/8ccbd57497e3fdee00ce6273939c8a6f.png' style="vertical-align:middle;" alt="f'(x)=0 \Longrightarrow 4x^3-6x=0 \Longrightarrow x(4x^2-6)=0" title="f'(x)=0 \Longrightarrow 4x^3-6x=0 \Longrightarrow x(4x^2-6)=0"> <br>De donde: <br><img src='https://www.matematicasies.com/local/cache-TeX/74c75dcc62c43f0a20edfe8c06fbcb79.png' style="vertical-align:middle;" width="77" height="38" alt="\rightarrow x=0" title="\rightarrow x=0"> <br><img src='https://www.matematicasies.com/local/cache-TeX/4f3e82a1584a85e732abc93eb06a1b76.png' style="vertical-align:middle;" width="253" height="80" alt="\rightarrow 4x^2-6=0 \Longrightarrow x=\pm \sqrt{\frac{3}{2}}" title="\rightarrow 4x^2-6=0 \Longrightarrow x=\pm \sqrt{\frac{3}{2}}"></p> <p>Intervalos:<br class="autobr"> <img src='https://www.matematicasies.com/local/cache-TeX/7ca5a7bc472c5115c7fb8f847336fe27.png' style="vertical-align:middle;" width="108" height="80" alt="(-\infty, -\sqrt{\frac{3}{2}})" title="(-\infty, -\sqrt{\frac{3}{2}})"> , <img src='https://www.matematicasies.com/local/cache-TeX/4786a4e60533cb78907df84ebf73c5e6.png' style="vertical-align:middle;" width="83" height="80" alt="(-\sqrt{\frac{3}{2}} , 0)" title="(-\sqrt{\frac{3}{2}} , 0)"> , <img src='https://www.matematicasies.com/local/cache-TeX/9303c9ed492666009aa44c003a741017.png' style="vertical-align:middle;" width="83" height="80" alt="(0, +\sqrt{\frac{3}{2}})" title="(0, +\sqrt{\frac{3}{2}})"> , <img src='https://www.matematicasies.com/local/cache-TeX/84a4648baea7d537e44e3c46db4ff2b8.png' style="vertical-align:middle;" width="108" height="80" alt="(+\sqrt{\frac{3}{2}} , +\infty)" title="(+\sqrt{\frac{3}{2}} , +\infty)"></p> <p>Tomamos un punto de cada intervalo para ver el signo de la derivada (positivo significa creciente y negativo decreciente)</p> <p><span class="spip-puce ltr"><b>–</b></span> <img src='https://www.matematicasies.com/local/cache-TeX/2bf1cec190edc9a42d8ff4a564698131.png' style="vertical-align:middle;" alt="f'(-2) = 4(-2)^3-6(-2)=-32+12=-20 < 0\Longrightarrow" title="f'(-2) = 4(-2)^3-6(-2)=-32+12=-20 < 0\Longrightarrow"> DECRECIENTE <br><span class="spip-puce ltr"><b>–</b></span> <img src='https://www.matematicasies.com/local/cache-TeX/7be1dcf7c0063299e1505473e49257c7.png' style="vertical-align:middle;" alt="f'(-1) = 4(-1)^3-6(-1)=-4+6=2 > 0\Longrightarrow" title="f'(-1) = 4(-1)^3-6(-1)=-4+6=2 > 0\Longrightarrow"> CRECIENTE <br><span class="spip-puce ltr"><b>–</b></span> <img src='https://www.matematicasies.com/local/cache-TeX/91b64080494369f14fa0e7edeb846ce0.png' style="vertical-align:middle;" alt="f'(1) = 4(1)^3-6(1)=4-6=-2 < 0\Longrightarrow" title="f'(1) = 4(1)^3-6(1)=4-6=-2 < 0\Longrightarrow"> DECRECIENTE <br><span class="spip-puce ltr"><b>–</b></span> <img src='https://www.matematicasies.com/local/cache-TeX/c0c2d7c3def7ea838d8c262989e5a6c7.png' style="vertical-align:middle;" alt="f'(2) = 4(2)^3-6(2)=32-12=20 > 0\Longrightarrow" title="f'(2) = 4(2)^3-6(2)=32-12=20 > 0\Longrightarrow"> CRECIENTE</p> <center><img src="http://matematicasies.com/images/ej_2257a.png" alt="ej_2257a.png"></center> <p><strong>Extremos</strong></p> <p>Aplicamos la derivada segunda a los puntos 0 , <img src='https://www.matematicasies.com/local/cache-TeX/518ba6ae98fbbf998d4ec14d6ce16e8d.png' style="vertical-align:middle;" width="65" height="57" alt="\pm \sqrt{3/2}" title="\pm \sqrt{3/2}"></p> <p><img src='https://www.matematicasies.com/local/cache-TeX/4703d9a945e09a5aca38dcfad46ec692.png' style="vertical-align:middle;" alt="f''(x) = 12x^2-6" title="f''(x) = 12x^2-6"></p> <p><span class="spip-puce ltr"><b>–</b></span> <img src='https://www.matematicasies.com/local/cache-TeX/1b045cd1a3337508cc5422e24d5d781c.png' style="vertical-align:middle;" alt="f''(0) = -6 < 0 \Longrightarrow" title="f''(0) = -6 < 0 \Longrightarrow"> MÁXIMO en <img src='https://www.matematicasies.com/local/cache-TeX/e11729b0b65ecade3fc272548a3883fc.png' style="vertical-align:middle;" width="50" height="38" alt="x=0" title="x=0"> <br><span class="spip-puce ltr"><b>–</b></span> <img src='https://www.matematicasies.com/local/cache-TeX/6fa40c2a9f83e65407276b2bb19b974e.png' style="vertical-align:middle;" alt="f''(-\sqrt{3/2}) = 12\frac{3}{2}-6=12 > 0 \Longrightarrow" title="f''(-\sqrt{3/2}) = 12\frac{3}{2}-6=12 > 0 \Longrightarrow"> MÍNIMO en <img src='https://www.matematicasies.com/local/cache-TeX/61f4e4b4b034c4497e8cea131a082c1c.png' style="vertical-align:middle;" width="102" height="57" alt="x=-\sqrt{3/2}" title="x=-\sqrt{3/2}"> <br><span class="spip-puce ltr"><b>–</b></span> <img src='https://www.matematicasies.com/local/cache-TeX/6fa40c2a9f83e65407276b2bb19b974e.png' style="vertical-align:middle;" alt="f''(-\sqrt{3/2}) = 12\frac{3}{2}-6=12 > 0 \Longrightarrow" title="f''(-\sqrt{3/2}) = 12\frac{3}{2}-6=12 > 0 \Longrightarrow"> MÍNIMO en <img src='https://www.matematicasies.com/local/cache-TeX/6fc6e4234b5f215d868560a0ed4f848d.png' style="vertical-align:middle;" width="102" height="57" alt="x=+\sqrt{3/2}" title="x=+\sqrt{3/2}"></p> <p>Calculemos la imagen de los extremos para conocer la segunda coordenada:</p> <p><span class="spip-puce ltr"><b>–</b></span> <img src='https://www.matematicasies.com/local/cache-TeX/4439cb2120403fe94c6755d922dbd3f3.png' style="vertical-align:middle;" width="327" height="47" alt="f(0) = 0^4-3 \cdot 0^2 = 0 \Longrightarrow MAX(0,0)" title="f(0) = 0^4-3 \cdot 0^2 = 0 \Longrightarrow MAX(0,0)"> <br><span class="spip-puce ltr"><b>–</b></span> <img src='https://www.matematicasies.com/local/cache-TeX/4cdef4c817ab792e223ee41d6643f5c2.png' style="vertical-align:middle;" width="482" height="68" alt="f\left(-\sqrt{3/2} \right) = \left( -\sqrt{3/2}\right)^4-3 \cdot \left( -\sqrt{3/2} \right)^2 = -2.25 \Longrightarrow" title="f\left(-\sqrt{3/2} \right) = \left( -\sqrt{3/2}\right)^4-3 \cdot \left( -\sqrt{3/2} \right)^2 = -2.25 \Longrightarrow">MIN<img src='https://www.matematicasies.com/local/cache-TeX/31d6856c8c36e36801afe752d69ff186.png' style="vertical-align:middle;" width="140" height="57" alt="(-\sqrt{3/2}, -2.25)" title="(-\sqrt{3/2}, -2.25)"> <br><span class="spip-puce ltr"><b>–</b></span> <img src='https://www.matematicasies.com/local/cache-TeX/7a11329143208d1dcec4d7a03450e10c.png' style="vertical-align:middle;" width="482" height="68" alt="f\left(+\sqrt{3/2} \right) = \left( +\sqrt{3/2}\right)^4-3 \cdot \left( +\sqrt{3/2} \right)^2 = -2.25 \Longrightarrow" title="f\left(+\sqrt{3/2} \right) = \left( +\sqrt{3/2}\right)^4-3 \cdot \left( +\sqrt{3/2} \right)^2 = -2.25 \Longrightarrow"> MIN<img src='https://www.matematicasies.com/local/cache-TeX/819be48398f2943aca9457bd778965e3.png' style="vertical-align:middle;" width="138" height="57" alt="(+\sqrt{3/2}, -2.25)" title="(+\sqrt{3/2}, -2.25)"></p> <p><strong>Gráfica</strong></p> <p>Con los datos obtenidos anteriormente: <br></p> <center><img src="http://matematicasies.com/images/ej_2257b.png" alt="ej_2257b.png"></center> <p><br>podemos ya dibujar la gráfica:</p> <center><img src="http://matematicasies.com/images/ej_2257c.png" alt="ej_2257c.png"></center>