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$$\frac{3 \cos{\left(3 x \right)}}{e^{6 x} - 1} - \frac{6 e^{6 x} \sin{\left(3 x \right)}}{\left(e^{6 x} - 1\right)^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -75.9218224617533$$
$$x_{2} = -98.9601685880785$$
$$x_{3} = -38.2227106186758$$
$$x_{4} = 52.5144267628302$$
$$x_{5} = 58.7976120700097$$
$$x_{6} = -34.0339204138894$$
$$x_{7} = -14.1371669411541$$
$$x_{8} = 26.3344879829152$$
$$x_{9} = 28.4288830853084$$
$$x_{10} = 94.4023288106941$$
$$x_{11} = -62.3082542961976$$
$$x_{12} = 19.004105124539$$
$$x_{13} = 20.0513026757356$$
$$x_{14} = 46.2312414556506$$
$$x_{15} = -97.9129710368819$$
$$x_{16} = -9.94837673636768$$
$$x_{17} = 38.9008585972744$$
$$x_{18} = 96.4967239130873$$
$$x_{19} = -95.8185759344887$$
$$x_{20} = -49.7418836818384$$
$$x_{21} = 42.0424512508642$$
$$x_{22} = -53.9306738866248$$
$$x_{23} = 66.1279949283859$$
$$x_{24} = -43.4586983746588$$
$$x_{25} = -56.025068989018$$
$$x_{26} = 56.7032169676165$$
$$x_{27} = 2.24894412143714$$
$$x_{28} = 10.6265247149662$$
$$x_{29} = -29.845130209103$$
$$x_{30} = -16.2315620435473$$
$$x_{31} = 16.9097100221458$$
$$x_{32} = 70.3167851331723$$
$$x_{33} = -3.66519142900047$$
$$x_{34} = 280.707689618242$$
$$x_{35} = 36.8064634948812$$
$$x_{36} = 22.1456977781288$$
$$x_{37} = 80.7887606451383$$
$$x_{38} = -12.0427718387609$$
$$x_{39} = 88.1191435035145$$
$$x_{40} = 44.1368463532574$$
$$x_{41} = 60.8920071724029$$
$$x_{42} = 12.7209198173594$$
$$x_{43} = 4.34333940778602$$
$$x_{44} = -65.4498469497874$$
$$x_{45} = -5.75958653158129$$
$$x_{46} = 48.3256365580438$$
$$x_{47} = -84.2994028713261$$
$$x_{48} = 74.5055753379587$$
$$x_{49} = 72.4111802355655$$
$$x_{50} = -67.5442420521806$$
$$x_{51} = 8.53212961257305$$
$$x_{52} = -45.553093477052$$
$$x_{53} = 34.712068392488$$
$$x_{54} = -82.2050077689329$$
$$x_{55} = -25.6563400043166$$
$$x_{56} = 76.5999704403519$$
$$x_{57} = -58.1194640914112$$
$$x_{58} = 50.420031660437$$
$$x_{59} = 68.2223900307791$$
$$x_{60} = -78.0162175641465$$
$$x_{61} = 64.0335998259927$$
$$x_{62} = 90.2135386059077$$
$$x_{63} = -1.57074250539597$$
$$x_{64} = 170.654917140517$$
$$x_{65} = 86.0247484011213$$
$$x_{66} = -91.6297857297023$$
$$x_{67} = 6.43773451017986$$
$$x_{68} = -47.6474885794452$$
$$x_{69} = 92.3079337083009$$
$$x_{70} = -60.2138591938044$$
$$x_{71} = -36.1283155162826$$
$$x_{72} = 82.8831557475315$$
$$x_{73} = 78.6943655427451$$
$$x_{74} = -51.8362787842316$$
$$x_{75} = -21.4675497995303$$
$$x_{76} = 30.5232781877016$$
$$x_{77} = -89.5353906273091$$
$$x_{78} = 98.5911190154805$$
$$x_{79} = 54.6088218652234$$
$$x_{80} = -100.007366139275$$
$$x_{81} = 14.8153149197526$$
$$x_{82} = -61.261056745001$$
$$x_{83} = 100.685514117874$$
$$x_{84} = -73.8274273593601$$
$$x_{85} = 24.240092880522$$
$$x_{86} = 58.3010386831513$$
$$x_{87} = 40.9952536996676$$
$$x_{88} = 32.6176732900948$$
$$x_{89} = -23.5619449019235$$
$$x_{90} = -71.733032256967$$
$$x_{91} = -19.3731546971371$$
$$x_{92} = -69.6386371545737$$
$$x_{93} = -27.7507351067098$$
$$x_{94} = -93.7241808320955$$
$$x_{95} = -31.9395253114962$$
$$x_{96} = -80.1106126665397$$
$$x_{97} = -7.85398163397448$$
$$x_{98} = 82.6766335259712$$
Signos de extremos en los puntos:
(-75.92182246175334, 1)
(-98.96016858807849, 1)
(-38.22271061867582, 1)
(52.514426762830155, 6.45892761394037e-138)
(58.79761207000974, 2.73932884908784e-154)
(-34.033920413889426, 1)
(-14.137166941154069, -1)
(26.334487982915213, -1.0690009950662e-69)
(28.428883085308406, -3.72797244892392e-75)
(94.40232881069407, 4.57143037898744e-247)
(-62.30825429619757, -1)
(19.004105124539027, 1.34973050453375e-50)
(20.051302675735627, -2.52054442045611e-53)
(46.23124145565057, 1.52291850377923e-121)
(-97.91297103688188, -1)
(-9.94837673636768, -1)
(38.900858597274386, -1.92285093274627e-102)
(96.49672391308727, 1.59421427891035e-252)
(-95.81857593448869, -1)
(-49.741883681838395, -1)
(42.042451250864175, 1.25223977502893e-110)
(-53.93067388662478, -1)
(66.12799492838593, -2.16957774581824e-173)
(-43.45869837465881, -1)
(-56.02506898901798, -1)
(56.70321696761655, 7.85506144589056e-149)
(2.2489441214371393, 6.17008270406991e-7)
(10.626524714966246, 9.12575331210925e-29)
(-29.845130209103036, 1)
(-16.231562043547264, -1)
(16.909710022145834, 3.8703699455565e-45)
(70.31678513317232, -2.6385442791241e-184)
(-3.6651914290004743, -1.00000000028143)
(280.7076896182416, 6.09599376624673e-733)
(36.80646349488119, -5.51380029931031e-97)
(22.14569777812882, -8.79000131816284e-59)
(80.7887606451383, -1.36093477023162e-211)
(-12.042771838760874, -1)
(88.11914350351448, 1.07787489332943e-230)
(44.136846353257376, 4.36698880758803e-116)
(60.89200717240294, 9.55297752300928e-160)
(12.72091981735944, 3.18246260576331e-34)
(4.343339407786019, 2.15171610686687e-12)
(-65.44984694978736, 1)
(-5.759586531581287, -1)
(48.32563655804376, 5.31093820328375e-127)
(-84.29940287132612, 1)
(74.5055753379587, -3.20888058808552e-195)
(72.41118023556551, -9.20150722332239e-190)
(-67.54424205218055, 1)
(8.532129612573051, 2.61682174560849e-23)
(-45.553093477052, -1)
(34.71206839248799, -1.5810894762005e-91)
(-82.20500776893293, 1)
(-25.656340004316643, 1)
(76.59997044035191, -1.11904651908475e-200)
(-58.119464091411174, -1)
(50.42003166043696, 1.85210597475199e-132)
(68.22239003077912, -7.56606036808039e-179)
(-78.01621756414653, 1)
(64.03359982599274, -6.22129267565441e-168)
(90.21353860590767, 3.75891877020198e-236)
(-1.5707425053959712, -1.00008071906269)
(170.65491714051726, 2.35638943667095e-446)
(86.02474840112129, 3.09082041059244e-225)
(-91.6297857297023, -1)
(6.4377345101798555, 7.50377071797784e-18)
(-47.647488579445195, -1)
(92.30793370830087, 1.31086366408744e-241)
(-60.21385919380437, -1)
(-36.12831551628262, 1)
(82.88315574753149, -4.7460454682663e-217)
(78.6943655427451, -3.90249832457247e-206)
(-51.83627878423159, -1)
(-21.467549799530254, 1)
(30.523278187701603, -1.30007162239126e-80)
(-89.53539062730911, -1)
(98.59111901548046, 5.55957097971723e-258)
(54.608821865223355, 2.25244918437822e-143)
(-100.00736613927508, -1)
(14.815314919752637, 1.10983366421296e-39)
(-61.26105674500097, 1)
(100.68551411787365, 1.93881273599182e-263)
(-73.82742735936014, 1)
(24.240092880522017, -3.06537439079622e-64)
(58.30103868315135, -1.03083020449354e-152)
(40.99525369966758, -6.70563950244203e-108)
(32.6176732900948, -4.53379483487037e-86)
(-23.56194490192345, 1)
(-71.73303225696695, 1)
(-19.373154697137057, 1)
(-69.63863715457374, 1)
(-27.75073510670984, 1)
(-93.7241808320955, -1)
(-31.939525311496233, 1)
(-80.11061266653972, 1)
(-7.853981633974483, -1)
(82.67663352597116, 5.68988856264278e-217)
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:
$$x_{1} = -14.1371669411541$$
$$x_{2} = 26.3344879829152$$
$$x_{3} = 28.4288830853084$$
$$x_{4} = -62.3082542961976$$
$$x_{5} = 20.0513026757356$$
$$x_{6} = -97.9129710368819$$
$$x_{7} = -9.94837673636768$$
$$x_{8} = 38.9008585972744$$
$$x_{9} = -95.8185759344887$$
$$x_{10} = -49.7418836818384$$
$$x_{11} = -53.9306738866248$$
$$x_{12} = 66.1279949283859$$
$$x_{13} = -43.4586983746588$$
$$x_{14} = -56.025068989018$$
$$x_{15} = -16.2315620435473$$
$$x_{16} = 70.3167851331723$$
$$x_{17} = -3.66519142900047$$
$$x_{18} = 36.8064634948812$$
$$x_{19} = 22.1456977781288$$
$$x_{20} = 80.7887606451383$$
$$x_{21} = -12.0427718387609$$
$$x_{22} = -5.75958653158129$$
$$x_{23} = 74.5055753379587$$
$$x_{24} = 72.4111802355655$$
$$x_{25} = -45.553093477052$$
$$x_{26} = 34.712068392488$$
$$x_{27} = 76.5999704403519$$
$$x_{28} = -58.1194640914112$$
$$x_{29} = 68.2223900307791$$
$$x_{30} = 64.0335998259927$$
$$x_{31} = -1.57074250539597$$
$$x_{32} = -91.6297857297023$$
$$x_{33} = -47.6474885794452$$
$$x_{34} = -60.2138591938044$$
$$x_{35} = 82.8831557475315$$
$$x_{36} = 78.6943655427451$$
$$x_{37} = -51.8362787842316$$
$$x_{38} = 30.5232781877016$$
$$x_{39} = -89.5353906273091$$
$$x_{40} = -100.007366139275$$
$$x_{41} = 24.240092880522$$
$$x_{42} = 40.9952536996676$$
$$x_{43} = 32.6176732900948$$
$$x_{44} = -93.7241808320955$$
$$x_{45} = -7.85398163397448$$
Puntos máximos de la función:
$$x_{45} = -75.9218224617533$$
$$x_{45} = -98.9601685880785$$
$$x_{45} = -38.2227106186758$$
$$x_{45} = 52.5144267628302$$
$$x_{45} = 58.7976120700097$$
$$x_{45} = -34.0339204138894$$
$$x_{45} = 94.4023288106941$$
$$x_{45} = 19.004105124539$$
$$x_{45} = 46.2312414556506$$
$$x_{45} = 96.4967239130873$$
$$x_{45} = 42.0424512508642$$
$$x_{45} = 56.7032169676165$$
$$x_{45} = 2.24894412143714$$
$$x_{45} = 10.6265247149662$$
$$x_{45} = -29.845130209103$$
$$x_{45} = 16.9097100221458$$
$$x_{45} = 88.1191435035145$$
$$x_{45} = 44.1368463532574$$
$$x_{45} = 60.8920071724029$$
$$x_{45} = 12.7209198173594$$
$$x_{45} = 4.34333940778602$$
$$x_{45} = -65.4498469497874$$
$$x_{45} = 48.3256365580438$$
$$x_{45} = -84.2994028713261$$
$$x_{45} = -67.5442420521806$$
$$x_{45} = 8.53212961257305$$
$$x_{45} = -82.2050077689329$$
$$x_{45} = -25.6563400043166$$
$$x_{45} = 50.420031660437$$
$$x_{45} = -78.0162175641465$$
$$x_{45} = 90.2135386059077$$
$$x_{45} = 86.0247484011213$$
$$x_{45} = 6.43773451017986$$
$$x_{45} = 92.3079337083009$$
$$x_{45} = -36.1283155162826$$
$$x_{45} = -21.4675497995303$$
$$x_{45} = 98.5911190154805$$
$$x_{45} = 54.6088218652234$$
$$x_{45} = 14.8153149197526$$
$$x_{45} = -61.261056745001$$
$$x_{45} = 100.685514117874$$
$$x_{45} = -73.8274273593601$$
$$x_{45} = -23.5619449019235$$
$$x_{45} = -71.733032256967$$
$$x_{45} = -19.3731546971371$$
$$x_{45} = -69.6386371545737$$
$$x_{45} = -27.7507351067098$$
$$x_{45} = -31.9395253114962$$
$$x_{45} = -80.1106126665397$$
Decrece en los intervalos
$$\left[82.8831557475315, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100.007366139275\right]$$