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$$- 2 \frac{1}{2 x} \sin{\left(2 x \right)} - \frac{\cos{\left(2 x \right)}}{2 x^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -31.4079674444867$$
$$x_{2} = 370.707258737426$$
$$x_{3} = 89.5325983772724$$
$$x_{4} = -58.1151623942039$$
$$x_{5} = 64.3987674147151$$
$$x_{6} = 45.5476049347036$$
$$x_{7} = -21.9797764444477$$
$$x_{8} = -23.5513313851812$$
$$x_{9} = 87.9617521867404$$
$$x_{10} = -14.1194682876301$$
$$x_{11} = -56.5442465635306$$
$$x_{12} = -50.2605085373433$$
$$x_{13} = 6.24322719761189$$
$$x_{14} = -75.3949078609596$$
$$x_{15} = -20.4081046633173$$
$$x_{16} = -87.9617521867404$$
$$x_{17} = -81.678348244891$$
$$x_{18} = 100.528478077876$$
$$x_{19} = -53.4023938097352$$
$$x_{20} = 51.8314556079832$$
$$x_{21} = 36.1213948523486$$
$$x_{22} = -97.3868052009628$$
$$x_{23} = 34.5502838639905$$
$$x_{24} = -7.82206418516651$$
$$x_{25} = 95.8159667896962$$
$$x_{26} = -15.6920370089449$$
$$x_{27} = -86.3909041828623$$
$$x_{28} = 56.5442465635306$$
$$x_{29} = 86.3909041828623$$
$$x_{30} = -59.6860719342923$$
$$x_{31} = 15.6920370089449$$
$$x_{32} = 119.378426667934$$
$$x_{33} = 23.5513313851812$$
$$x_{34} = -37.6924796092674$$
$$x_{35} = 4.65893323089553$$
$$x_{36} = -64.3987674147151$$
$$x_{37} = 12.546455206056$$
$$x_{38} = -36.1213948523486$$
$$x_{39} = -42.405605649659$$
$$x_{40} = 58.1151623942039$$
$$x_{41} = 824.667768414974$$
$$x_{42} = 81.678348244891$$
$$x_{43} = -243.472403844238$$
$$x_{44} = 94.2451269755362$$
$$x_{45} = 67.54054063921$$
$$x_{46} = 65.9696561768633$$
$$x_{47} = -65.9696561768633$$
$$x_{48} = -28.2654900969093$$
$$x_{49} = 50.2605085373433$$
$$x_{50} = -39.263541283971$$
$$x_{51} = 48.6895517393056$$
$$x_{52} = -45.5476049347036$$
$$x_{53} = 155.507228715927$$
$$x_{54} = 37.6924796092674$$
$$x_{55} = 70.6822978112741$$
$$x_{56} = 7.82206418516651$$
$$x_{57} = -61.2569756677764$$
$$x_{58} = -6.24322719761189$$
$$x_{59} = -51.8314556079832$$
$$x_{60} = 20.4081046633173$$
$$x_{61} = -17.264282877731$$
$$x_{62} = 80.1074919003334$$
$$x_{63} = 14.1194682876301$$
$$x_{64} = 78.5366331548744$$
$$x_{65} = -161.790476456414$$
$$x_{66} = 28.2654900969093$$
$$x_{67} = 42.405605649659$$
$$x_{68} = 21.9797764444477$$
$$x_{69} = 43.9766125553363$$
$$x_{70} = 29.8367520652203$$
$$x_{71} = -43.9766125553363$$
$$x_{72} = 92.6742856871438$$
$$x_{73} = -89.5325983772724$$
$$x_{74} = -10.9728064399905$$
$$x_{75} = -94.2451269755362$$
$$x_{76} = -9.39820218310508$$
$$x_{77} = -73.8240409804225$$
$$x_{78} = 31.4079674444867$$
$$x_{79} = 72.2531710320437$$
$$x_{80} = -95.8159667896962$$
$$x_{81} = -1.39919302289194$$
$$x_{82} = -72.2531710320437$$
$$x_{83} = 26.6941733108628$$
$$x_{84} = -80.1074919003334$$
$$x_{85} = -29.8367520652203$$
$$x_{86} = -83.2492023244609$$
$$x_{87} = 73.8240409804225$$
$$x_{88} = -67.54054063921$$
$$x_{89} = 59.6860719342923$$
Signos de extremos en los puntos:
(-31.40796744448671, -0.015917510583426)
(370.7072587374256, 0.00134877193100099)
(89.53259837727238, -0.00558447104654696)
(-58.11516239420393, 0.00860328827998151)
(64.39876741471514, -0.00776388975051473)
(45.54760493470357, -0.0109768642483425)
(-21.979776444447747, -0.0227423004725314)
(-23.551331385181175, 0.0212254394164143)
(87.96175218674041, 0.00568419693765964)
(-14.119468287630136, 0.0353899155541688)
(-56.54424656353064, -0.00884228694331555)
(-50.26050853734329, -0.00994767611536293)
(6.2432271976118905, 0.0798311807800032)
(-75.39490786095963, -0.00663160178225336)
(-20.40810466331729, 0.0244927205346957)
(-87.96175218674041, -0.00568419693765964)
(-81.67834824489104, -0.00612145865571533)
(100.5284780778763, 0.00497365348819037)
(-53.402393809735244, -0.00936246579806236)
(51.83145560798318, -0.00964620289525996)
(36.121394852348644, -0.0138408859131547)
(-97.38680520096284, -0.00513409808632722)
(34.55028386399049, 0.0144701459746764)
(-7.822064185166514, 0.0637915530395936)
(95.81596678969623, -0.00521826590213634)
(-15.69203700894493, -0.0318471321112693)
(-86.39090418286231, 0.00578754940380675)
(56.54424656353064, 0.00884228694331555)
(86.39090418286231, -0.00578754940380675)
(-59.68607193429229, -0.00837686985425208)
(15.69203700894493, 0.0318471321112693)
(119.37842666793355, 0.00418832471176385)
(23.551331385181175, -0.0212254394164143)
(-37.69247960926737, -0.0132640786518247)
(4.658933230895533, -0.106707947715237)
(-64.39876741471514, 0.00776388975051473)
(12.546455206056049, 0.0398202855500511)
(-36.121394852348644, 0.0138408859131547)
(-42.40560564965901, 0.0117900744410766)
(58.11516239420393, -0.00860328827998151)
(824.6677684149739, -0.000606304656552379)
(81.67834824489104, 0.00612145865571533)
(-243.4724038442376, 0.00205361649932032)
(94.24512697553622, 0.00530523942833593)
(67.54054063920995, -0.00740275832666827)
(65.96965617686325, 0.00757902448438246)
(-65.96965617686325, -0.00757902448438246)
(-28.26549009690932, -0.0176866485521696)
(50.26050853734329, 0.00994767611536293)
(-39.263541283970966, 0.0127334276777468)
(48.6895517393056, -0.0102686022030809)
(-45.54760493470357, 0.0109768642483425)
(155.50722871592723, -0.00321526799515938)
(37.69247960926737, 0.0132640786518247)
(70.68229781127408, -0.00707372999905127)
(7.822064185166514, -0.0637915530395936)
(-61.25697566777638, 0.00816206382131534)
(-6.2432271976118905, -0.0798311807800032)
(-51.83145560798318, 0.00964620289525996)
(20.40810466331729, -0.0244927205346957)
(-17.26428287773103, 0.0289493889114503)
(80.10749190033339, -0.00624149188782456)
(14.119468287630136, -0.0353899155541688)
(78.53663315487437, 0.00636632673547799)
(-161.79047645641373, 0.00309040200201688)
(28.26549009690932, 0.0176866485521696)
(42.40560564965901, -0.0117900744410766)
(21.979776444447747, 0.0227423004725314)
(43.976612555336274, 0.0113689449158811)
(29.836752065220264, -0.0167555036571887)
(-43.976612555336274, -0.0113689449158811)
(92.67428568714384, -0.00539516133626855)
(-89.53259837727238, 0.00558447104654696)
(-10.972806439990523, 0.0455199604051285)
(-94.24512697553622, -0.00530523942833593)
(-9.398202183105079, -0.0531265325613881)
(-73.8240409804225, 0.00677270609738254)
(31.40796744448671, 0.015917510583426)
(72.25317103204372, 0.00691994581417642)
(-95.81596678969623, 0.00521826590213634)
(-1.3991930228919436, 0.336508416918395)
(-72.25317103204372, -0.00691994581417642)
(26.694173310862816, -0.0187273944640866)
(-80.10749190033339, 0.00624149188782456)
(-29.836752065220264, 0.0167555036571887)
(-83.24920232446088, 0.00600595522940177)
(73.8240409804225, -0.00677270609738254)
(-67.54054063920995, 0.00740275832666827)
(59.68607193429229, 0.00837686985425208)
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:
$$x_{1} = -31.4079674444867$$
$$x_{2} = 89.5325983772724$$
$$x_{3} = 64.3987674147151$$
$$x_{4} = 45.5476049347036$$
$$x_{5} = -21.9797764444477$$
$$x_{6} = -56.5442465635306$$
$$x_{7} = -50.2605085373433$$
$$x_{8} = -75.3949078609596$$
$$x_{9} = -87.9617521867404$$
$$x_{10} = -81.678348244891$$
$$x_{11} = -53.4023938097352$$
$$x_{12} = 51.8314556079832$$
$$x_{13} = 36.1213948523486$$
$$x_{14} = -97.3868052009628$$
$$x_{15} = 95.8159667896962$$
$$x_{16} = -15.6920370089449$$
$$x_{17} = 86.3909041828623$$
$$x_{18} = -59.6860719342923$$
$$x_{19} = 23.5513313851812$$
$$x_{20} = -37.6924796092674$$
$$x_{21} = 4.65893323089553$$
$$x_{22} = 58.1151623942039$$
$$x_{23} = 824.667768414974$$
$$x_{24} = 67.54054063921$$
$$x_{25} = -65.9696561768633$$
$$x_{26} = -28.2654900969093$$
$$x_{27} = 48.6895517393056$$
$$x_{28} = 155.507228715927$$
$$x_{29} = 70.6822978112741$$
$$x_{30} = 7.82206418516651$$
$$x_{31} = -6.24322719761189$$
$$x_{32} = 20.4081046633173$$
$$x_{33} = 80.1074919003334$$
$$x_{34} = 14.1194682876301$$
$$x_{35} = 42.405605649659$$
$$x_{36} = 29.8367520652203$$
$$x_{37} = -43.9766125553363$$
$$x_{38} = 92.6742856871438$$
$$x_{39} = -94.2451269755362$$
$$x_{40} = -9.39820218310508$$
$$x_{41} = -72.2531710320437$$
$$x_{42} = 26.6941733108628$$
$$x_{43} = 73.8240409804225$$
Puntos máximos de la función:
$$x_{43} = 370.707258737426$$
$$x_{43} = -58.1151623942039$$
$$x_{43} = -23.5513313851812$$
$$x_{43} = 87.9617521867404$$
$$x_{43} = -14.1194682876301$$
$$x_{43} = 6.24322719761189$$
$$x_{43} = -20.4081046633173$$
$$x_{43} = 100.528478077876$$
$$x_{43} = 34.5502838639905$$
$$x_{43} = -7.82206418516651$$
$$x_{43} = -86.3909041828623$$
$$x_{43} = 56.5442465635306$$
$$x_{43} = 15.6920370089449$$
$$x_{43} = 119.378426667934$$
$$x_{43} = -64.3987674147151$$
$$x_{43} = 12.546455206056$$
$$x_{43} = -36.1213948523486$$
$$x_{43} = -42.405605649659$$
$$x_{43} = 81.678348244891$$
$$x_{43} = -243.472403844238$$
$$x_{43} = 94.2451269755362$$
$$x_{43} = 65.9696561768633$$
$$x_{43} = 50.2605085373433$$
$$x_{43} = -39.263541283971$$
$$x_{43} = -45.5476049347036$$
$$x_{43} = 37.6924796092674$$
$$x_{43} = -61.2569756677764$$
$$x_{43} = -51.8314556079832$$
$$x_{43} = -17.264282877731$$
$$x_{43} = 78.5366331548744$$
$$x_{43} = -161.790476456414$$
$$x_{43} = 28.2654900969093$$
$$x_{43} = 21.9797764444477$$
$$x_{43} = 43.9766125553363$$
$$x_{43} = -89.5325983772724$$
$$x_{43} = -10.9728064399905$$
$$x_{43} = -73.8240409804225$$
$$x_{43} = 31.4079674444867$$
$$x_{43} = 72.2531710320437$$
$$x_{43} = -95.8159667896962$$
$$x_{43} = -1.39919302289194$$
$$x_{43} = -80.1074919003334$$
$$x_{43} = -29.8367520652203$$
$$x_{43} = -83.2492023244609$$
$$x_{43} = -67.54054063921$$
$$x_{43} = 59.6860719342923$$
Decrece en los intervalos
$$\left[824.667768414974, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -97.3868052009628\right]$$