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$$\frac{2 x \cos{\left(x^{2} \right)}}{x^{2}} - \frac{2 \sin{\left(x^{2} \right)}}{x^{3}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -27.487300879733$$
$$x_{2} = -81.8117534286295$$
$$x_{3} = 56.1199279909343$$
$$x_{4} = -51.294088034684$$
$$x_{5} = 7.19840138414632$$
$$x_{6} = 46.0326381269472$$
$$x_{7} = 70.1296163961195$$
$$x_{8} = -77.5739683255625$$
$$x_{9} = 82.2139674644798$$
$$x_{10} = -97.7504677801482$$
$$x_{11} = 30.4686650830675$$
$$x_{12} = -16.0011216977146$$
$$x_{13} = -63.4751315873709$$
$$x_{14} = -57.7477298421119$$
$$x_{15} = -53.6295415482154$$
$$x_{16} = -12.8424122001834$$
$$x_{17} = 9.29420367942992$$
$$x_{18} = -111.755990954751$$
$$x_{19} = 92.1079304031337$$
$$x_{20} = 68.5896113576236$$
$$x_{21} = 67.6209131009439$$
$$x_{22} = 6.26453768135879$$
$$x_{23} = -17.856889938083$$
$$x_{24} = -47.8726582137852$$
$$x_{25} = -38.5686848262369$$
$$x_{26} = -21.7441389653189$$
$$x_{27} = -84.4756034159993$$
$$x_{28} = -33.7930178276353$$
$$x_{29} = -7.8259012426924$$
$$x_{30} = -85.7673542930979$$
$$x_{31} = 147.556117595223$$
$$x_{32} = 27.9968943148725$$
$$x_{33} = 86.3878070945521$$
$$x_{34} = 26.0194492595522$$
$$x_{35} = -3.75049248937143$$
$$x_{36} = -90.2995216474216$$
$$x_{37} = 98.4230706903319$$
$$x_{38} = -91.8859626824559$$
$$x_{39} = -93.7978150743204$$
$$x_{40} = 22.3145788067872$$
$$x_{41} = 78.6997443140484$$
$$x_{42} = -52.3549350628188$$
$$x_{43} = -23.812931580963$$
$$x_{44} = 23.6806356632024$$
$$x_{45} = 68.1300442553852$$
$$x_{46} = 58.1813182114996$$
$$x_{47} = 32.1249469684196$$
$$x_{48} = 16.0989917566665$$
$$x_{49} = 54.1251772109928$$
$$x_{50} = -87.848290956425$$
$$x_{51} = -47.7083162370527$$
$$x_{52} = 4.14978978647482$$
$$x_{53} = -225.808811928315$$
$$x_{54} = 84.2148756127912$$
$$x_{55} = 339.12512890907$$
$$x_{56} = 80.1827238875859$$
$$x_{57} = 18.1188682387635$$
$$x_{58} = 34.2088060145132$$
$$x_{59} = 94.332184745798$$
$$x_{60} = -65.8556635962687$$
$$x_{61} = -2.11976636870884$$
$$x_{62} = 261.64865047635$$
$$x_{63} = 51.5994127711782$$
$$x_{64} = 20.2478493210409$$
$$x_{65} = 62.2254986905353$$
$$x_{66} = -68.0146672875878$$
$$x_{67} = -42.0000441440065$$
$$x_{68} = 42.1120943325309$$
$$x_{69} = -98.1833843061319$$
$$x_{70} = 37.828491311387$$
$$x_{71} = 10.2583552061545$$
$$x_{72} = 40.164750811318$$
$$x_{73} = -29.8961868990962$$
$$x_{74} = 60.2503944862544$$
$$x_{75} = -76.902847810038$$
$$x_{76} = 36.0423055930793$$
$$x_{77} = 2.11976636870884$$
$$x_{78} = -69.7703204185698$$
Signos de extremos en los puntos:
(-27.487300879732977, 0.001323534989533)
(-81.81175342862946, 0.000149406190911279)
(56.11992799093429, 0.000317516111949359)
(-51.29408803468397, -0.000380071533342589)
(7.198401384146321, 0.019295099487588)
(46.032638126947205, 0.000471919824505443)
(70.12961639611953, -0.00020332794172061)
(-77.57396832556249, -0.000166175876052604)
(82.21396746447982, -0.000147947892013696)
(-97.75046778014821, -0.000104655560719047)
(30.468665083067464, -0.00107719144114341)
(-16.001121697714638, -0.00390567256417978)
(-63.475131587370946, 0.000248194850672783)
(-57.74772984211192, -0.000299868017423585)
(-53.62954154821543, -0.000347689683696048)
(-12.842412200183375, 0.00606315689591026)
(9.294203679429923, -0.0115756804584678)
(-111.75599095475077, -8.00678876126555e-5)
(92.10793040313365, 0.000117870723345681)
(68.58961135762361, -0.000212560863689315)
(67.62091310094392, -0.000218694533696834)
(6.264537681358792, 0.0254730530928808)
(-17.856889938083004, -0.00313607341346806)
(-47.8726582137852, -0.000436339844394369)
(-38.568684826236904, -0.000672249119554206)
(-21.74413896531894, 0.00211502058560655)
(-84.4756034159993, -0.000140132022588208)
(-33.79301782763527, -0.000875680902961545)
(-7.825901242692397, -0.0163257593209978)
(-85.76735429309788, -0.000135942724375418)
(147.5561175952228, 4.59288487871757e-5)
(27.996894314872534, -0.00127579216525722)
(86.3878070945521, -0.000133997006542391)
(26.01944925955225, -0.00147707764928763)
(-3.7504924893714255, 0.0709134594504622)
(-90.29952164742163, -0.000122639140272602)
(98.42307069033188, -0.000103230059309167)
(-91.88596268245591, -0.0001184408887131)
(-93.79781507432041, 0.000113661806191721)
(22.314578806787196, 0.00200826831595226)
(78.69974431404836, -0.000161455688730729)
(-52.35493506281884, 0.000364825108729964)
(-23.81293158096305, 0.00176349241629226)
(23.680635663202445, 0.00178325149744762)
(68.13004425538523, -0.000215438168237391)
(58.18131821149962, -0.000295415220485631)
(32.124946968419636, 0.000968980321516034)
(16.098991756666486, 0.00385833036946338)
(54.125177210992774, 0.000341351104266613)
(-87.848290956425, 0.000129578623592958)
(-47.70831623705273, 0.000439351162050216)
(4.149789786474824, -0.0579718023461539)
(-225.80881192831538, 1.9611834893873e-5)
(84.21487561279123, -0.000141001058404429)
(339.1251289090697, -8.69520962086523e-6)
(80.18272388758591, 0.000155538670918207)
(18.118868238763504, 0.00304604175008073)
(34.20880601451324, 0.000854523496376895)
(94.33218474579803, 0.000112377718691516)
(-65.85566359626866, 0.000230575801988624)
(-2.1197663687088406, -0.217233628211222)
(261.64865047634953, -1.46070663431957e-5)
(51.5994127711782, -0.000375586912843737)
(20.247849321040928, 0.00243916347187379)
(62.22549869053529, 0.000258263607950864)
(-68.01466728758777, 0.000216169707043122)
(-42.00004414400653, -0.000566892141283866)
(42.112094332530894, 0.000563879427437908)
(-98.18338430613193, 0.000103734687272752)
(37.828491311387026, -0.000698814410331)
(10.258355206154523, -0.00950221661878354)
(40.16475081131796, -0.000619883052268501)
(-29.896186899096215, 0.00111884037052573)
(60.25039448625436, -0.000275473732809509)
(-76.902847810038, 0.000169088919380717)
(36.04230559307926, -0.000769794390559548)
(2.1197663687088406, -0.217233628211222)
(-69.77032041856978, -0.0002054274881576)
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:
$$x_{1} = -51.294088034684$$
$$x_{2} = 70.1296163961195$$
$$x_{3} = -77.5739683255625$$
$$x_{4} = 82.2139674644798$$
$$x_{5} = -97.7504677801482$$
$$x_{6} = 30.4686650830675$$
$$x_{7} = -16.0011216977146$$
$$x_{8} = -57.7477298421119$$
$$x_{9} = -53.6295415482154$$
$$x_{10} = 9.29420367942992$$
$$x_{11} = -111.755990954751$$
$$x_{12} = 68.5896113576236$$
$$x_{13} = 67.6209131009439$$
$$x_{14} = -17.856889938083$$
$$x_{15} = -47.8726582137852$$
$$x_{16} = -38.5686848262369$$
$$x_{17} = -84.4756034159993$$
$$x_{18} = -33.7930178276353$$
$$x_{19} = -7.8259012426924$$
$$x_{20} = -85.7673542930979$$
$$x_{21} = 27.9968943148725$$
$$x_{22} = 86.3878070945521$$
$$x_{23} = 26.0194492595522$$
$$x_{24} = -90.2995216474216$$
$$x_{25} = 98.4230706903319$$
$$x_{26} = -91.8859626824559$$
$$x_{27} = 78.6997443140484$$
$$x_{28} = 68.1300442553852$$
$$x_{29} = 58.1813182114996$$
$$x_{30} = 4.14978978647482$$
$$x_{31} = 84.2148756127912$$
$$x_{32} = 339.12512890907$$
$$x_{33} = -2.11976636870884$$
$$x_{34} = 261.64865047635$$
$$x_{35} = 51.5994127711782$$
$$x_{36} = -42.0000441440065$$
$$x_{37} = 37.828491311387$$
$$x_{38} = 10.2583552061545$$
$$x_{39} = 40.164750811318$$
$$x_{40} = 60.2503944862544$$
$$x_{41} = 36.0423055930793$$
$$x_{42} = 2.11976636870884$$
$$x_{43} = -69.7703204185698$$
Puntos máximos de la función:
$$x_{43} = -27.487300879733$$
$$x_{43} = -81.8117534286295$$
$$x_{43} = 56.1199279909343$$
$$x_{43} = 7.19840138414632$$
$$x_{43} = 46.0326381269472$$
$$x_{43} = -63.4751315873709$$
$$x_{43} = -12.8424122001834$$
$$x_{43} = 92.1079304031337$$
$$x_{43} = 6.26453768135879$$
$$x_{43} = -21.7441389653189$$
$$x_{43} = 147.556117595223$$
$$x_{43} = -3.75049248937143$$
$$x_{43} = -93.7978150743204$$
$$x_{43} = 22.3145788067872$$
$$x_{43} = -52.3549350628188$$
$$x_{43} = -23.812931580963$$
$$x_{43} = 23.6806356632024$$
$$x_{43} = 32.1249469684196$$
$$x_{43} = 16.0989917566665$$
$$x_{43} = 54.1251772109928$$
$$x_{43} = -87.848290956425$$
$$x_{43} = -47.7083162370527$$
$$x_{43} = -225.808811928315$$
$$x_{43} = 80.1827238875859$$
$$x_{43} = 18.1188682387635$$
$$x_{43} = 34.2088060145132$$
$$x_{43} = 94.332184745798$$
$$x_{43} = -65.8556635962687$$
$$x_{43} = 20.2478493210409$$
$$x_{43} = 62.2254986905353$$
$$x_{43} = -68.0146672875878$$
$$x_{43} = 42.1120943325309$$
$$x_{43} = -98.1833843061319$$
$$x_{43} = -29.8961868990962$$
$$x_{43} = -76.902847810038$$
Decrece en los intervalos
$$\left[339.12512890907, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -111.755990954751\right]$$