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$$\left(- 22 x \log{\left(\operatorname{atan}{\left(4 x \right)} \right)} + \frac{4 \left(4 - 11 x^{2}\right)}{\left(16 x^{2} + 1\right) \operatorname{atan}{\left(4 x \right)}}\right) \operatorname{atan}^{4 - 11 x^{2}}{\left(4 x \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 16.3768764899306$$
$$x_{2} = -48$$
$$x_{3} = 48.25$$
$$x_{4} = 22.2546156273675$$
$$x_{5} = -82$$
$$x_{6} = 50.25$$
$$x_{7} = -76$$
$$x_{8} = 72.25$$
$$x_{9} = -66$$
$$x_{10} = 76.25$$
$$x_{11} = 40.25$$
$$x_{12} = -90$$
$$x_{13} = -92$$
$$x_{14} = 42.25$$
$$x_{15} = 24.25$$
$$x_{16} = 94.25$$
$$x_{17} = -72$$
$$x_{18} = 36.25$$
$$x_{19} = 96.25$$
$$x_{20} = 18.3629150829958$$
$$x_{21} = 82.25$$
$$x_{22} = 6.57936873064899$$
$$x_{23} = 46.25$$
$$x_{24} = 8.50016513308312$$
$$x_{25} = 20.3517163340659$$
$$x_{26} = -62$$
$$x_{27} = 60.25$$
$$x_{28} = -64$$
$$x_{29} = -44$$
$$x_{30} = 74.25$$
$$x_{31} = 66.25$$
$$x_{32} = 80.25$$
$$x_{33} = 32.25$$
$$x_{34} = 98.25$$
$$x_{35} = 52.25$$
$$x_{36} = 3.0954610507573$$
$$x_{37} = -74$$
$$x_{38} = 4.72871770847854$$
$$x_{39} = 64.25$$
$$x_{40} = -88$$
$$x_{41} = 30.25$$
$$x_{42} = 58.25$$
$$x_{43} = -80$$
$$x_{44} = -54$$
$$x_{45} = -68$$
$$x_{46} = -94$$
$$x_{47} = 68.25$$
$$x_{48} = 100.25$$
$$x_{49} = -84$$
$$x_{50} = 54.25$$
$$x_{51} = 14.3947606429418$$
$$x_{52} = 62.25$$
$$x_{53} = -52$$
$$x_{54} = 78.25$$
$$x_{55} = -56$$
$$x_{56} = 26.25$$
$$x_{57} = -70$$
$$x_{58} = -98$$
$$x_{59} = 90.25$$
$$x_{60} = -42$$
$$x_{61} = -100$$
$$x_{62} = 70.25$$
$$x_{63} = 44.25$$
$$x_{64} = -78$$
$$x_{65} = -86$$
$$x_{66} = -58$$
$$x_{67} = -46$$
$$x_{68} = -60$$
$$x_{69} = 56.25$$
$$x_{70} = -96$$
$$x_{71} = 92.25$$
$$x_{72} = -50$$
$$x_{73} = 38.25$$
$$x_{74} = 12.4184799166039$$
$$x_{75} = 88.25$$
$$x_{76} = 34.25$$
$$x_{77} = 28.25$$
$$x_{78} = 84.25$$
$$x_{79} = 86.25$$
$$x_{80} = 10.4514187219475$$
Signos de extremos en los puntos:
(16.376876489930602, 4.80584434052083e-566)
(-48, 7.45201827350769e-4934)
(48.25, 1.41874792567526e-4985)
(22.254615627367475, 2.00463126282422e-1051)
(-82, 2.38369999645287e-14443)
(50.25, 4.79745425894621e-5409)
(-76, 8.97689067556117e-12403)
(72.25, 2.72936152183187e-11206)
(-66, 5.51850375666237e-9347)
(76.25, 1.07220442487673e-12484)
(40.25, 2.81679604450246e-3464)
(-90, 9.76502541880758e-17406)
(-92, 1.7635892036574e-18189)
(42.25, 1.03076266781901e-3818)
(24.25, 4.2928660481011e-1250)
(94.25, 8.05098307586828e-19092)
(-72, 1.10731376909646e-11128)
(36.25, 3.52524497985739e-2807)
(96.25, 3.07715888123986e-19912)
(18.36291508299578, 2.24097785335232e-713)
(82.25, 9.60385448185444e-14532)
(6.579368730648988, 2.6134745865422e-88)
(46.25, 2.31328933855839e-4579)
(8.500165133083115, 2.52089230468082e-149)
(20.351716334065912, 5.75207435669298e-878)
(-62, 1.76767973626342e-8245)
(60.25, 2.80585490984441e-7785)
(-64, 1.33012187543054e-8787)
(-44, 5.29307081500082e-4143)
(74.25, 7.28528799155251e-11837)
(66.25, 4.03246182693459e-9418)
(80.25, 3.89293049395639e-13832)
(32.25, 4.07596760213282e-2219)
(98.25, 6.48483866512397e-20750)
(52.25, 8.94435696811467e-5850)
(3.095461050757303, 2.70803476522693e-18)
(-74, 4.2459584627383e-11757)
(4.728717708478544, 1.3789022126837e-44)
(64.25, 1.39623990196238e-8856)
(-88, 2.98124040350267e-16639)
(30.25, 5.67322385261986e-1951)
(58.25, 1.62846619452272e-7275)
(-80, 6.7261026336694e-13746)
(-54, 1.43170387344564e-6249)
(-68, 1.26239154611231e-9923)
(-94, 1.75618134533181e-18990)
(68.25, 6.42130654847512e-9997)
(100.25, 7.53523556687799e-21605)
(-84, 4.65785923710286e-15158)
(54.25, 9.19437243789439e-6308)
(14.394760642941765, 5.6692146967844e-436)
(62.25, 2.66558159824221e-8312)
(-52, 9.69526462198995e-5794)
(78.25, 8.7006982630586e-13150)
(-56, 1.16568965417864e-6722)
(26.25, 1.84124077971675e-1466)
(-70, 1.59224658793556e-10517)
(-98, 2.91915773478334e-20644)
(90.25, 9.23827659304813e-17503)
(-42, 1.82625719920824e-3773)
(-100, 4.8727503580324e-21497)
(70.25, 5.63793593117373e-10593)
(44.25, 2.07961148787324e-4190)
(-78, 1.04645780498545e-13065)
(-86, 5.0184323321562e-15890)
(-58, 5.23299605221553e-7213)
(-46, 8.45818601035469e-4530)
(-60, 1.2952598596477e-7720)
(56.25, 5.2111247623332e-6783)
(-96, 9.64249332432367e-19809)
(92.25, 1.1614378842688e-18288)
(-50, 3.61993345868591e-5355)
(38.25, 4.24392368593481e-3127)
(12.418479916603895, 3.67357219329963e-323)
(88.25, 4.05166596770093e-16734)
(34.25, 1.61440825676553e-2504)
(28.25, 4.3530761901904e-1700)
(84.25, 1.306354928916e-15248)
(86.25, 9.79769213094386e-15983)
(10.451418721947539, 1.30403455716412e-227)
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:
La función no tiene puntos mínimos
La función no tiene puntos máximos
Decrece en todo el eje numérico