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$$- \frac{10 \log{\left(x - 4 \right)} \operatorname{acot}{\left(5 x \right)}}{25 x^{2} + 1} + \frac{\operatorname{acot}^{2}{\left(5 x \right)}}{x - 4} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 9611.84053835698$$
$$x_{2} = 42413.5683484109$$
$$x_{3} = 29493.9398872642$$
$$x_{4} = 48833.3664888115$$
$$x_{5} = 15408.0280671425$$
$$x_{6} = 54165.7543471659$$
$$x_{7} = 26245.9453016179$$
$$x_{8} = 27329.3794864507$$
$$x_{9} = 14337.7301145638$$
$$x_{10} = 49901.0501089827$$
$$x_{11} = 43485.2360945014$$
$$x_{12} = 32734.6864836606$$
$$x_{13} = 40268.0740407141$$
$$x_{14} = 34891.0686096907$$
$$x_{15} = 38119.6029796687$$
$$x_{16} = 31655.262374402$$
$$x_{17} = 39194.2178946115$$
$$x_{18} = 56294.6607802403$$
$$x_{19} = 19734.6989562826$$
$$x_{20} = 13278.7443201098$$
$$x_{21} = 45626.4866169943$$
$$x_{22} = 35968.0411974386$$
$$x_{23} = 37044.21520896$$
$$x_{24} = 28412.0554959634$$
$$x_{25} = 25161.8043242596$$
$$x_{26} = 12239.4561489534$$
$$x_{27} = 20820.296559896$$
$$x_{28} = 18649.7686333039$$
$$x_{29} = 53100.457664144$$
$$x_{30} = 7.12721878090273$$
$$x_{31} = 21906.0860007187$$
$$x_{32} = 24077.0325904846$$
$$x_{33} = 33813.286614885$$
$$x_{34} = 44556.2039659933$$
$$x_{35} = 46696.0985008394$$
$$x_{36} = 10313.4454621241$$
$$x_{37} = 16485.077701001$$
$$x_{38} = 47765.0538198603$$
$$x_{39} = 52034.5829422324$$
$$x_{40} = 41341.1859405396$$
$$x_{41} = 55230.4848734819$$
$$x_{42} = 30575.012374393$$
$$x_{43} = 10608.6135636748$$
$$x_{44} = 17566.2079670701$$
$$x_{45} = 50968.1179506774$$
$$x_{46} = 22991.7403596272$$
$$x_{47} = 11237.0503173276$$
Signos de extremos en los puntos:
(9611.840538356983, 3.97037993011726e-9)
(42413.56834841089, 2.36924018742695e-10)
(29493.939887264165, 4.73244893157346e-10)
(48833.366488811494, 1.81089406203148e-10)
(15408.028067142517, 1.6246144312175e-9)
(54165.754347165945, 1.48602512289146e-10)
(26245.945301617925, 5.90846751656658e-10)
(27329.379486450718, 5.47095451511591e-10)
(14337.730114563796, 1.86220699206495e-9)
(49901.0501089827, 1.73770577247695e-10)
(43485.23609450142, 2.25918099752669e-10)
(32734.686483660567, 3.88072390146042e-10)
(40268.074040714106, 2.61562715438744e-10)
(34891.06860969066, 3.43682801235254e-10)
(38119.602979668685, 2.90368151234317e-10)
(31655.262374402013, 4.13651016560067e-10)
(39194.21789461153, 2.75387941797556e-10)
(56294.66078024027, 1.38062193845164e-10)
(19734.698956282617, 1.01576265643813e-9)
(13278.74432010984, 2.153663032117e-9)
(45626.48661699426, 2.06134657201003e-10)
(35968.041197438615, 3.24349538938298e-10)
(37044.21520896002, 3.06637316314249e-10)
(28412.05549596342, 5.08119806492439e-10)
(25161.80432425957, 6.40193398145826e-10)
(12239.456148953417, 2.51317060558657e-9)
(20820.296559896007, 9.1754014168478e-10)
(18649.76863330391, 1.13087778972143e-9)
(53100.45766414402, 1.54343018381941e-10)
(7.127218780902727, 0.000897328830949463)
(21906.08600071868, 8.33075593289527e-10)
(24077.032590484556, 6.9613845983863e-10)
(33813.28661488504, 3.6484355763632e-10)
(44556.20396599328, 2.15678393390042e-10)
(46696.09850083937, 1.9722456195479e-10)
(10313.44546212413, 3.47506355209164e-9)
(16485.07770100099, 1.42920919527646e-9)
(47765.053819860346, 1.88892642249023e-10)
(52034.58294223241, 1.60431314724117e-10)
(41341.18594053956, 2.48775547875599e-10)
(55230.48487348186, 1.43183514210459e-10)
(30575.012374392958, 4.41911032636565e-10)
(10608.61356367477, 3.29441036288958e-9)
(17566.207967070062, 1.26693482125914e-9)
(50968.11795067739, 1.66896446061025e-10)
(22991.740359627212, 7.59919333674734e-10)
(11237.05031732756, 2.9544688756137e-9)
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
Puntos máximos de la función:
$$x_{47} = 7.12721878090273$$
Decrece en los intervalos
$$\left(-\infty, 7.12721878090273\right]$$
Crece en los intervalos
$$\left[7.12721878090273, \infty\right)$$