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$$2 \sin{\left(\frac{1}{x} \right)} - \frac{2 \cos{\left(\frac{1}{x} \right)}}{x} + \frac{\sin{\left(\frac{1}{x} \right)}}{x^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -23667.7807761837$$
$$x_{2} = -24515.3684606503$$
$$x_{3} = 15323.23216915$$
$$x_{4} = -14344.4453659261$$
$$x_{5} = -16039.5703456219$$
$$x_{6} = 21256.2554233482$$
$$x_{7} = -36381.6770245262$$
$$x_{8} = 22103.839219867$$
$$x_{9} = -37229.2741705613$$
$$x_{10} = -10954.2729524476$$
$$x_{11} = 25494.1879976264$$
$$x_{12} = 14475.6712322997$$
$$x_{13} = -39772.0674218958$$
$$x_{14} = -30448.5080552154$$
$$x_{15} = 41598.4964374195$$
$$x_{16} = 33970.119473448$$
$$x_{17} = -41467.2642484886$$
$$x_{18} = 27189.3687618399$$
$$x_{19} = 42446.0952348827$$
$$x_{20} = -34686.483772001$$
$$x_{21} = 11933.0253359143$$
$$x_{22} = -32143.6969032752$$
$$x_{23} = 28036.9604004731$$
$$x_{24} = 13628.1156823495$$
$$x_{25} = -26210.5470412737$$
$$x_{26} = -21972.6092384625$$
$$x_{27} = -38924.4693889889$$
$$x_{28} = -33838.8877175908$$
$$x_{29} = -31296.1022134924$$
$$x_{30} = -29600.9144741046$$
$$x_{31} = 34817.7155904103$$
$$x_{32} = 40750.8978661988$$
$$x_{33} = 35665.3120932869$$
$$x_{34} = -17734.7112890337$$
$$x_{35} = 22951.4245261489$$
$$x_{36} = 17018.3669835115$$
$$x_{37} = -13496.8907426424$$
$$x_{38} = -18582.2863816239$$
$$x_{39} = 30579.7395065666$$
$$x_{40} = 38208.1036605632$$
$$x_{41} = -12649.3427697718$$
$$x_{42} = -11801.8028806562$$
$$x_{43} = 28884.5527784603$$
$$x_{44} = -32991.2920835956$$
$$x_{45} = -27905.7292537638$$
$$x_{46} = 29732.1458325661$$
$$x_{47} = 37360.5061517164$$
$$x_{48} = -20277.4438767813$$
$$x_{49} = 36512.9089552013$$
$$x_{50} = 12780.5665913212$$
$$x_{51} = -19429.8640171832$$
$$x_{52} = 16170.7976456873$$
$$x_{53} = -35534.0802167403$$
$$x_{54} = -21125.025692681$$
$$x_{55} = 31427.3337503038$$
$$x_{56} = 18713.51514406$$
$$x_{57} = 24646.5990449056$$
$$x_{58} = -40619.6657136875$$
$$x_{59} = 17865.9396329837$$
$$x_{60} = -27058.1377362019$$
$$x_{61} = 39903.299535638$$
$$x_{62} = -16887.1391223858$$
$$x_{63} = 33122.5237720275$$
$$x_{64} = 39055.7014614063$$
$$x_{65} = -28753.321521205$$
$$x_{66} = -22820.1943213495$$
$$x_{67} = 26341.7779339352$$
$$x_{68} = 20408.6733247176$$
$$x_{69} = 23799.0111808714$$
$$x_{70} = 11085.4937140326$$
$$x_{71} = -38076.8716322562$$
$$x_{72} = 32274.9285188887$$
$$x_{73} = -42314.8630116956$$
$$x_{74} = -25362.9572514602$$
$$x_{75} = -15192.0055262454$$
$$x_{76} = 19561.0931447107$$
Signos de extremos en los puntos:
(-23667.780776183714, 2.99999999851234)
(-24515.368460650297, 2.99999999861343)
(15323.23216915003, 2.9999999964509)
(-14344.44536592608, 2.99999999595004)
(-16039.57034562194, 2.99999999676083)
(21256.255423348193, 2.99999999815564)
(-36381.67702452621, 2.99999999937042)
(22103.839219867034, 2.99999999829438)
(-37229.27417056133, 2.99999999939876)
(-10954.272952447594, 2.99999999305533)
(25494.187997626384, 2.99999999871786)
(14475.671232299745, 2.99999999602313)
(-39772.06742189575, 2.99999999947318)
(-30448.508055215378, 2.99999999910115)
(41598.49643741948, 2.99999999951843)
(33970.11947344803, 2.99999999927785)
(-41467.26424848864, 2.99999999951537)
(27189.368761839905, 2.99999999887275)
(42446.09523488272, 2.99999999953747)
(-34686.48377200103, 2.99999999930737)
(11933.025335914264, 2.99999999414782)
(-32143.69690327515, 2.99999999919346)
(28036.96040047314, 2.99999999893988)
(13628.11568234953, 2.99999999551309)
(-26210.547041273716, 2.99999999878698)
(-21972.609238462534, 2.99999999827394)
(-38924.46938898889, 2.99999999944999)
(-33838.887717590784, 2.99999999927224)
(-31296.102213492435, 2.99999999914918)
(-29600.91447410458, 2.99999999904894)
(34817.71559041035, 2.99999999931259)
(40750.897866198764, 2.99999999949818)
(35665.312093286906, 2.99999999934487)
(-17734.71128903371, 2.99999999735046)
(22951.42452614892, 2.99999999841803)
(17018.36698351151, 2.99999999712271)
(-13496.890742642408, 2.99999999542542)
(-18582.286381623937, 2.99999999758665)
(30579.739506566602, 2.99999999910885)
(38208.10366056323, 2.99999999942917)
(-12649.342769771787, 2.99999999479186)
(-11801.80288065619, 2.99999999401696)
(28884.55277846027, 2.99999999900118)
(-32991.2920835956, 2.99999999923437)
(-27905.72925376384, 2.99999999892988)
(29732.14583256606, 2.99999999905732)
(37360.50615171638, 2.99999999940297)
(-20277.44387678132, 2.99999999797329)
(36512.90895520135, 2.99999999937493)
(12780.566591321194, 2.99999999489826)
(-19429.864017183212, 2.99999999779261)
(16170.797645687257, 2.99999999681319)
(-35534.08021674026, 2.99999999934002)
(-21125.025692681, 2.99999999813266)
(31427.33375030377, 2.99999999915627)
(18713.51514405997, 2.99999999762038)
(24646.59904490561, 2.99999999862816)
(-40619.66571368752, 2.99999999949494)
(17865.939632983664, 2.99999999738924)
(-27058.13773620189, 2.99999999886179)
(39903.29953563803, 2.99999999947664)
(-16887.139122385786, 2.99999999707782)
(33122.52377202749, 2.99999999924042)
(39055.701461406345, 2.99999999945368)
(-28753.321521205013, 2.99999999899204)
(-22820.194321349496, 2.99999999839978)
(26341.777933935235, 2.99999999879904)
(20408.673324717554, 2.99999999799927)
(23799.011180871425, 2.9999999985287)
(11085.493714032567, 2.99999999321877)
(-38076.87163225615, 2.99999999942523)
(32274.92851888869, 2.9999999992)
(-42314.86301169557, 2.99999999953459)
(-25362.95725146023, 2.99999999870456)
(-15192.005526245362, 2.99999999638932)
(19561.093144710703, 2.99999999782213)
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:
$$x_{1} = -10954.2729524476$$
$$x_{2} = -32143.6969032752$$
$$x_{3} = -42314.8630116956$$
Puntos máximos de la función:
$$x_{3} = -21972.6092384625$$
$$x_{3} = 30579.7395065666$$
$$x_{3} = -20277.4438767813$$
$$x_{3} = 23799.0111808714$$
$$x_{3} = -25362.9572514602$$
Decrece en los intervalos
$$\left[-32143.6969032752, -25362.9572514602\right] \cup \left[-10954.2729524476, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -42314.8630116956\right]$$