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$$\frac{- 5 x^{4} - \tan^{2}{\left(x \right)} - 1}{\left(\left(x^{5} + \tan{\left(x \right)}\right) + 2\right)^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -1580.40202285408$$
$$x_{2} = -1069.32709136831$$
$$x_{3} = 1690.53312491582$$
$$x_{4} = -2449.22821878455$$
$$x_{5} = 1281.67167490765$$
$$x_{6} = 159.404603063071$$
$$x_{7} = 1843.85652666135$$
$$x_{8} = -149.438307759782$$
$$x_{9} = 770.583927304305$$
$$x_{10} = 463.963445210111$$
$$x_{11} = -2398.12773157165$$
$$x_{12} = 1792.75044486226$$
$$x_{13} = 1894.96234069398$$
$$x_{14} = 208.18770854221$$
$$x_{15} = 719.492531300595$$
$$x_{16} = 1639.42851635151$$
$$x_{17} = -507.157323280799$$
$$x_{18} = 617.278028118203$$
$$x_{19} = 515.087306758414$$
$$x_{20} = -404.946298522891$$
$$x_{21} = -609.347965670726$$
$$x_{22} = 1997.18002406314$$
$$x_{23} = 2559.36691022268$$
$$x_{24} = -2193.69632564284$$
$$x_{25} = -1682.61770029578$$
$$x_{26} = 821.706378994404$$
$$x_{27} = -302.736658321913$$
$$x_{28} = 566.174636129376$$
$$x_{29} = 2099.38421052879$$
$$x_{30} = 1332.77482009147$$
$$x_{31} = -1784.83340948569$$
$$x_{32} = -1273.75588606168$$
$$x_{33} = -1529.29455987113$$
$$x_{34} = 2252.71942613499$$
$$x_{35} = -2295.9113197909$$
$$x_{36} = -1733.72395730929$$
$$x_{37} = 2457.15003950443$$
$$x_{38} = 1741.64068843058$$
$$x_{39} = -1222.64897829999$$
$$x_{40} = -813.788192917816$$
$$x_{41} = 1383.8869818917$$
$$x_{42} = 1588.31755707498$$
$$x_{43} = 923.921971569841$$
$$x_{44} = -1120.43390676015$$
$$x_{45} = 1486.10097430463$$
$$x_{46} = -1375.97136898877$$
$$x_{47} = -251.634280324621$$
$$x_{48} = 668.387665822387$$
$$x_{49} = -2500.34359548048$$
$$x_{50} = -2142.5881228129$$
$$x_{51} = -1989.26460690934$$
$$x_{52} = -353.922807947974$$
$$x_{53} = -1835.9411284055$$
$$x_{54} = 975.033466687701$$
$$x_{55} = 2150.50361017737$$
$$x_{56} = 361.801869420894$$
$$x_{57} = -916.019699242469$$
$$x_{58} = 2303.82716389596$$
$$x_{59} = -200.269510444794$$
$$x_{60} = -456.045290542306$$
$$x_{61} = -558.259435157128$$
$$x_{62} = 2354.93536837108$$
$$x_{63} = 1026.13430495239$$
$$x_{64} = 1946.0722416825$$
$$x_{65} = 259.561520031795$$
$$x_{66} = -762.654854311092$$
$$x_{67} = -2244.80401979192$$
$$x_{68} = -660.488493184596$$
$$x_{69} = 1230.564596108$$
$$x_{70} = -2347.0198708417$$
$$x_{71} = -1631.51033495677$$
$$x_{72} = -864.897441360013$$
$$x_{73} = 2201.61354256287$$
$$x_{74} = 2508.25901412971$$
$$x_{75} = -711.577416085388$$
$$x_{76} = -2091.48037367254$$
$$x_{77} = 1537.20994817949$$
$$x_{78} = 310.248577247854$$
$$x_{79} = 872.812922295529$$
$$x_{80} = -1478.19215820444$$
$$x_{81} = -1887.03977668906$$
$$x_{82} = -2551.45149261041$$
$$x_{83} = 1179.45610707982$$
$$x_{84} = -2040.37672817511$$
$$x_{85} = -1427.07909212318$$
$$x_{86} = 2048.28617714866$$
$$x_{87} = 1128.34921223092$$
$$x_{88} = -1171.5373814137$$
$$x_{89} = -1324.85032231434$$
$$x_{90} = -1938.15681414768$$
$$x_{91} = -967.112022094732$$
$$x_{92} = -1018.21856509852$$
$$x_{93} = 412.86158795262$$
$$x_{94} = 1434.99449925465$$
$$x_{95} = 2406.04314140812$$
$$x_{96} = 1077.2480377903$$
Signos de extremos en los puntos:
(-1580.402022854082, -1.01428993414291e-16)
(-1069.327091368315, -7.15232352641115e-16)
(1690.5331249158178, 7.24238445939527e-17)
(-2449.228218784554, -1.13462860264425e-17)
(1281.6716749076547, 2.89145253632263e-16)
(159.4046030630709, 9.71618351667465e-12)
(1843.8565266613466, 4.69206884223659e-17)
(-149.43830775978208, -1.34180774781851e-11)
(770.5839273043048, 3.68044445951254e-15)
(463.9634452101111, 4.65137309694953e-14)
(-2398.1277315716547, -1.2607774830075e-17)
(1792.750444862261, 5.4000876037734e-17)
(1894.9623406939825, 4.09257910454914e-17)
(208.1877085422101, 2.55696377095603e-12)
(719.492531300595, 5.18643034403192e-15)
(1639.4285163515056, 8.44379294433868e-17)
(-507.1573232807994, -2.98048194211988e-14)
(617.2780281182028, 1.11582453589784e-14)
(515.0873067584137, 2.75800952238809e-14)
(-404.946298522891, -9.18359730846713e-14)
(-609.3479656707259, -1.19034587745361e-14)
(1997.1800240631405, 3.14712456006503e-17)
(2559.3669102226845, 9.1062012994865e-18)
(-2193.696325642845, -1.96841860349925e-17)
(-1682.6177002957832, -7.41434403844016e-17)
(821.7063789944035, 2.66941875655762e-15)
(-302.7366583219129, -3.93255618468056e-13)
(566.1746361293762, 1.7188894726428e-14)
(2099.3842105287945, 2.45211236469234e-17)
(1332.7748200914675, 2.37802329478225e-16)
(-1784.8334094856866, -5.5209214279007e-17)
(-1273.7558860616798, -2.98242121841182e-16)
(-1529.2945598711308, -1.19548541449427e-16)
(2252.7194261349923, 1.72371110340238e-17)
(-2295.911319790902, -1.56756101272089e-17)
(-1733.7239573092913, -6.38410832630565e-17)
(2457.1500395044313, 1.11645601592491e-17)
(1741.6406884305818, 6.24032471881268e-17)
(-1222.6489782999936, -3.66008400773553e-16)
(-813.7881929178155, -2.8018383645193e-15)
(1383.8869818916994, 1.97013905411309e-16)
(1588.317557074985, 9.89266533176537e-17)
(923.9219715698408, 1.4853319554486e-15)
(-1120.4339067601463, -5.66328978714028e-16)
(1486.1009743046316, 1.37961658143459e-16)
(-1375.9713689887674, -2.02746335654916e-16)
(-251.63428032462127, -9.91176449721364e-13)
(668.3876658223872, 7.49648825780897e-15)
(-2500.343595480485, -1.02329660650193e-17)
(-2142.5881228129037, -2.21465731085575e-17)
(-1989.2646069093437, -3.21023792756199e-17)
(-353.9228079479741, -1.80076584770844e-13)
(-1835.9411284055036, -4.79409069014455e-17)
(975.0334666877012, 1.13475573784716e-15)
(2150.5036101773703, 2.17419814147252e-17)
(361.8018694208945, 1.61304308386314e-13)
(-916.0196992424692, -1.55051486087227e-15)
(2303.827163895957, 1.5408151032699e-17)
(-200.26951044479372, -3.10402934977843e-12)
(-456.0452905423055, -5.06944159346584e-14)
(-558.2594351571281, -1.84424935762915e-14)
(2354.9353683710765, 1.38071816423773e-17)
(1026.1343049523937, 8.78979943930297e-16)
(1946.0722416825024, 3.58265682400343e-17)
(259.5615200317948, 8.48786485513326e-13)
(-762.6548543110921, -3.87578614165548e-15)
(-2244.8040197919186, -1.75431607586345e-17)
(-660.4884931845959, -7.95561431615118e-15)
(1230.5645961080033, 3.54387112049365e-16)
(-2347.0198708416974, -1.40415861012135e-17)
(-1631.5103349567735, -8.65069208576263e-17)
(-864.8974413600133, -2.06622179537099e-15)
(2201.6135425628745, 1.93327911807176e-17)
(2508.25901412971, 1.00725189146749e-17)
(-711.5774160853884, -5.48137127612023e-15)
(-2091.4803736725353, -2.49879732461819e-17)
(1537.2099481794933, 1.16502184328061e-16)
(310.24857724785414, 3.47897259082543e-13)
(872.8129222955289, 1.97421370503659e-15)
(-1478.1921582044358, -1.41692065068246e-16)
(-1887.0397766890585, -4.17921511984871e-17)
(-2551.451492610408, -9.24833216600398e-18)
(1179.4561070798225, 4.38117986165063e-16)
(-2040.3767281751066, -2.82779676050006e-17)
(-1427.079092123175, -1.68950617303356e-16)
(2048.2861771486632, 2.77361916243546e-17)
(1128.349212230924, 5.46741890143985e-16)
(-1171.5373814137033, -4.53126276003524e-16)
(-1324.8503223143414, -2.44999908028617e-16)
(-1938.1568141476812, -3.65641462910535e-17)
(-967.1120220947322, -1.18199622624959e-15)
(-1018.2185650985182, -9.1368173633525e-16)
(412.86158795261986, 8.33638199430586e-14)
(1434.9944992546502, 1.64342094825423e-16)
(2406.0431414081177, 1.24017493369889e-17)
(1077.2480377902993, 6.89320892327077e-16)
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