Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada$$\frac{1 + \frac{4}{\left(9 - x\right)^{2}}}{\left(x + \frac{4}{9 - x}\right) \log{\left(x^{2} \right)}} - \frac{2 \log{\left(x + \frac{4}{9 - x} \right)}}{x \log{\left(x^{2} \right)}^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 24588.5361462439$$
$$x_{2} = 22559.0808554189$$
$$x_{3} = 13821.1001684757$$
$$x_{4} = 16499.0036697859$$
$$x_{5} = 27980.0437640936$$
$$x_{6} = 23911.5742761268$$
$$x_{7} = 30019.9361475354$$
$$x_{8} = 25943.832100175$$
$$x_{9} = 29339.5787266134$$
$$x_{10} = 27300.884689026$$
$$x_{11} = 9823.4917876132$$
$$x_{12} = 26622.1440992107$$
$$x_{13} = 10488.2492187904$$
$$x_{14} = 30700.6750966907$$
$$x_{15} = 19860.1644815706$$
$$x_{16} = 20534.1022610809$$
$$x_{17} = 28659.6115768734$$
$$x_{18} = 15158.7717833455$$
$$x_{19} = 25265.9591656405$$
$$x_{20} = 11153.5649228019$$
$$x_{21} = 11819.4857814118$$
$$x_{22} = 23235.0851765489$$
$$x_{23} = 12486.0400268801$$
$$x_{24} = 7831.61646129693$$
$$x_{25} = 17170.0527933322$$
$$x_{26} = 13153.2428314584$$
$$x_{27} = 17841.7070875579$$
$$x_{28} = 14489.6115176182$$
$$x_{29} = 9159.21887065643$$
$$x_{30} = 21883.5737012967$$
$$x_{31} = 18513.9530261346$$
$$x_{32} = 15828.5726690929$$
$$x_{33} = 5.58864244271429$$
$$x_{34} = 21208.5764699667$$
$$x_{35} = 8495.31748448982$$
$$x_{36} = 31381.7871417024$$
$$x_{37} = 19186.7767914446$$
Signos de extremos en los puntos:
(24588.536146243852, 0.499999999672681)
(22559.08085541885, 0.499999999607785)
(13821.100168475743, 0.499999998901109)
(16499.00366978589, 0.499999999243019)
(27980.04376409364, 0.499999999750423)
(23911.574276126797, 0.499999999652923)
(30019.936147535376, 0.499999999784673)
(25943.832100175045, 0.499999999707544)
(29339.578726613367, 0.499999999774067)
(27300.884689026036, 0.499999999737218)
(9823.491787613204, 0.499999997743366)
(26622.144099210716, 0.499999999722963)
(10488.249218790406, 0.499999998034472)
(30700.675096690655, 0.499999999794565)
(19860.164481570624, 0.499999999487397)
(20534.10226108094, 0.499999999522111)
(28659.61157687338, 0.499999999762676)
(15158.771783345506, 0.499999999095312)
(25265.959165640492, 0.499999999690832)
(11153.564922801941, 0.499999998273527)
(11819.48578141178, 0.499999998472165)
(23235.085176548884, 0.499999999631366)
(12486.040026880091, 0.499999998638953)
(7831.616461296932, 0.499999996358909)
(17170.052793332154, 0.499999999303905)
(13153.242831458361, 0.499999998780307)
(17841.707087557854, 0.499999999357867)
(14489.611517618196, 0.499999999005128)
(9159.218870656434, 0.499999997384074)
(21883.57370129671, 0.499999999581924)
(18513.95302613462, 0.499999999405908)
(15828.572669092884, 0.499999999173988)
(5.588642442714286, 0.555343578019234)
(21208.57646996672, 0.499999999553483)
(8495.31748448982, 0.499999996933708)
(31381.7871417024, 0.499999999803803)
(19186.77679144461, 0.499999999448855)
Intervalos de crecimiento y decrecimiento de la función:Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
$$x_{1} = 22559.0808554189$$
$$x_{2} = 16499.0036697859$$
$$x_{3} = 23911.5742761268$$
$$x_{4} = 29339.5787266134$$
$$x_{5} = 25265.9591656405$$
$$x_{6} = 7831.61646129693$$
$$x_{7} = 17170.0527933322$$
$$x_{8} = 5.58864244271429$$
$$x_{9} = 31381.7871417024$$
$$x_{10} = 19186.7767914446$$
Puntos máximos de la función:
$$x_{10} = 27980.0437640936$$
$$x_{10} = 30019.9361475354$$
$$x_{10} = 26622.1440992107$$
$$x_{10} = 15158.7717833455$$
$$x_{10} = 17841.7070875579$$
Decrece en los intervalos
$$\left[31381.7871417024, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, 5.58864244271429\right]$$