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$$\frac{366^{- 2 x} \left(- 366^{x} \log{\left(366 \right)} \left(366 - x\right)! + 366^{x} \Gamma\left(367 - x\right) \operatorname{polygamma}{\left(0,367 - x \right)}\right) 366!}{\left(366 - x\right)!^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 131.263547318146$$
$$x_{2} = 142.289419771775$$
$$x_{3} = -221.911397442147$$
$$x_{4} = 148.489926057898$$
$$x_{5} = 132.49252606168$$
$$x_{6} = -156.459578538638$$
$$x_{7} = -148.86488319603$$
$$x_{8} = 131.861663111378$$
$$x_{9} = 147.76452244415$$
$$x_{10} = -163.284693115969$$
$$x_{11} = -150.448442647594$$
$$x_{12} = -166.768216447787$$
$$x_{13} = 136.102993392309$$
$$x_{14} = -164.427010487653$$
$$x_{15} = -155.870374869408$$
$$x_{16} = -147.406921673596$$
$$x_{17} = 133.155048338112$$
$$x_{18} = -156.86283491579$$
$$x_{19} = -169.181379011056$$
$$x_{20} = 132.877429527764$$
$$x_{21} = 133.848239976634$$
$$x_{22} = 146.339438027642$$
$$x_{23} = 224.533385703648$$
$$x_{24} = 143.26919370228$$
$$x_{25} = -161.059389781454$$
$$x_{26} = -158.917104435355$$
$$x_{27} = -167.966055093882$$
$$x_{28} = 139.489098659813$$
$$x_{29} = -566.501331248467$$
$$x_{30} = -148.119595740889$$
$$x_{31} = -152.148818086297$$
$$x_{32} = -151.284527348298$$
$$x_{33} = -149.641544216258$$
$$x_{34} = 134.571187725584$$
$$x_{35} = -153.95838874619$$
$$x_{36} = 137.744197429172$$
$$x_{37} = -159.977566134392$$
$$x_{38} = -154.901976713339$$
$$x_{39} = 140.398771377184$$
$$x_{40} = 138.604024598006$$
$$x_{41} = 151.860845346716$$
$$x_{42} = -162.697268701659$$
$$x_{43} = -165.588362811797$$
$$x_{44} = 141.332414874687$$
$$x_{45} = -157.878640517889$$
$$x_{46} = 149.594703911144$$
$$x_{47} = 147.404667926737$$
$$x_{48} = 136.910288120869$$
$$x_{49} = -153.040394714545$$
$$x_{50} = 0.50011384325594$$
$$x_{51} = 150.718505159764$$
$$x_{52} = 387.185863618224$$
$$x_{53} = -162.161962540824$$
$$x_{54} = 144.271160133495$$
$$x_{55} = 145.294757418136$$
$$x_{56} = 135.323039594095$$
Signos de extremos en los puntos:
(131.26354731814646, 2.55225963773289e-793*366!)
(142.2894197717749, 1.50038819518257e-795*366!)
(-221.91139744214658, 1.98847980773689e-806*366!)
(148.48992605789817, 6.59174078742274e-797*366!)
(132.4925260616803, 1.47772488192058e-793*366!)
(-156.45957853863845, 1.41785157865693e-794*366!)
(-148.86488319603012, 2.01671134415054e-793*366!)
(131.86166311137802, 1.95780927049914e-793*366!)
(147.76452244414992, 9.58727958378125e-797*366!)
(-163.28469311596936, 1.18731904197022e-795*366!)
(-150.44844264759368, 1.17008593976468e-793*366!)
(-166.76821644778653, 3.23658160409487e-796*366!)
(136.10299339230912, 2.85823317844958e-794*366!)
(-164.4270104876529, 7.77241175159263e-796*366!)
(-155.87037486940807, 1.74906333437045e-794*366!)
(-147.4069216735961, 3.31474121742855e-793*366!)
(133.155048338112, 1.09771666886542e-793*366!)
(-156.8628349157902, 1.22761935882012e-794*366!)
(-169.18137901105604, 1.29793222442566e-796*366!)
(132.87742952776432, 1.24362126920477e-793*366!)
(133.84823997663395, 8.02667965590579e-794*366!)
(146.33943802764207, 1.98736492095898e-796*366!)
(224.533385703648, 1.35106337909044e-820*366!)
(143.26919370227972, 9.26301890344295e-796*366!)
(-161.05938978145403, 2.69128386172186e-795*366!)
(-158.91710443535482, 5.86465046094575e-795*366!)
(-167.96605509388237, 2.059199532704e-796*366!)
(139.48909865981287, 5.81590898152898e-795*366!)
(-566.501331248467, 1.28706747785017e-914*366!)
(-148.11959574088874, 2.60126441864714e-793*366!)
(-152.14881808629656, 6.48642986208293e-794*366!)
(-151.2845273482978, 8.76072054720656e-794*366!)
(-149.64154421625796, 1.54508234053974e-793*366!)
(134.57118772558374, 5.77816563510539e-794*366!)
(-153.95838874619, 3.44097198510968e-794*366!)
(137.74419742917203, 1.32942710687008e-794*366!)
(-159.97756613439208, 3.9926440844174e-795*366!)
(-154.90197671333897, 2.46621293063867e-794*366!)
(140.3987713771845, 3.75948125804925e-795*366!)
(138.60402459800554, 8.86061215116374e-795*366!)
(151.86084534671647, 1.11983083855891e-797*366!)
(-162.69726870165925, 1.47494934712904e-795*366!)
(-165.58836281179651, 5.03945639968052e-796*366!)
(141.33241487468746, 2.39327221950849e-795*366!)
(-157.8786405178895, 8.52823310917239e-795*366!)
(149.594703911144, 3.7085382891732e-797*366!)
(147.40466792673706, 1.15349712403773e-796*366!)
(136.91028812086898, 1.96432477692978e-794*366!)
(-153.04039471454522, 4.75001742573446e-794*366!)
(0.5001138432559399, 1.08873475508001e-781*366!)
(150.71850515976377, 2.05398254501303e-797*366!)
(387.1858636182237, -2.14237648190665e-975*366!)
(-162.16196254082405, 1.79631120315423e-795*366!)
(144.27116013349482, 5.63152052166386e-796*366!)
(145.29475741813553, 3.37137403452232e-796*366!)
(135.3230395940954, 4.09542261200787e-794*366!)
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