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$$\frac{2 x \sin{\left(x^{2} \right)}}{x^{2} - \sin{\left(x^{2} \right)}} + \frac{\left(1 - \cos{\left(x^{2} \right)}\right) \left(2 x \cos{\left(x^{2} \right)} - 2 x\right)}{\left(x^{2} - \sin{\left(x^{2} \right)}\right)^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -55.8533814804374$$
$$x_{2} = -43.8480866628973$$
$$x_{3} = 36.7115823747944$$
$$x_{4} = -67.7485497660991$$
$$x_{5} = 82.1853040708499$$
$$x_{6} = 56.273665198017$$
$$x_{7} = 23.9117097455063$$
$$x_{8} = 97.9350905159343$$
$$x_{9} = -59.0522467437495$$
$$x_{10} = 78.2292943160867$$
$$x_{11} = 27.9687585943629$$
$$x_{12} = 6.13996024767893$$
$$x_{13} = -83.8878271505535$$
$$x_{14} = 64.9550458834008$$
$$x_{15} = -52.639193767251$$
$$x_{16} = 36.9249402390575$$
$$x_{17} = 8.11866849179377$$
$$x_{18} = 86.2148955714351$$
$$x_{19} = 47.2617243224204$$
$$x_{20} = 88.9234963506805$$
$$x_{21} = -58.4103554329069$$
$$x_{22} = 16.244807875181$$
$$x_{23} = 55.3732284583772$$
$$x_{24} = -89.1528429921457$$
$$x_{25} = -363.849851480352$$
$$x_{26} = -30.1316423581595$$
$$x_{27} = -33.8626317675647$$
$$x_{28} = 20.2864220127995$$
$$x_{29} = 18.2485292908913$$
$$x_{30} = -5.86868899703005$$
$$x_{31} = 64.2499240996983$$
$$x_{32} = -19.8163794777508$$
$$x_{33} = -43.7763808313316$$
$$x_{34} = -77.4625197650871$$
$$x_{35} = -11.6214831047628$$
$$x_{36} = -13.4986063691111$$
$$x_{37} = 66.2243410266297$$
$$x_{38} = -53.5562573203185$$
$$x_{39} = -86.4150750849244$$
$$x_{40} = -47.5599275712721$$
$$x_{41} = 28.2482660354898$$
$$x_{42} = -27.9687585943629$$
$$x_{43} = 77.826670467913$$
$$x_{44} = 2.506628274631$$
$$x_{45} = 22.137941502317$$
$$x_{46} = 31.8055472277386$$
$$x_{47} = -64.2743601386076$$
$$x_{48} = -95.8600963720283$$
$$x_{49} = -39.7914902637393$$
$$x_{50} = 42.2424505354389$$
$$x_{51} = -25.5626929713338$$
$$x_{52} = -95.5975553641563$$
$$x_{53} = -2.506628274631$$
$$x_{54} = 60.0806861715646$$
$$x_{55} = -7.72160589396959$$
$$x_{56} = 46.1519210773927$$
$$x_{57} = -50.3202114343071$$
$$x_{58} = 94.1238082693386$$
$$x_{59} = -91.5348410700472$$
$$x_{60} = 61.5018500121916$$
$$x_{61} = 52.2498090981313$$
$$x_{62} = -89.7498945058111$$
$$x_{63} = 3.54490770181103$$
$$x_{64} = -3.93071287095298$$
$$x_{65} = 9.86652362607564$$
$$x_{66} = -15.7534028698413$$
$$x_{67} = -41.8313856112806$$
$$x_{68} = -9.86652362607564$$
$$x_{69} = -47.0952695441314$$
$$x_{70} = -53.9070665877391$$
$$x_{71} = -87.731989612085$$
Signos de extremos en los puntos:
(-55.85338148043736, 0.000641107788550345)
(-43.848086662897316, 0)
(36.71158237479443, 0.001483965441261)
(-67.74854976609909, 0.000435742568955222)
(82.1853040708499, 0)
(56.273665198017014, 0)
(23.91170974550633, 0)
(97.9350905159343, 0.000208522699000623)
(-59.05224674374948, 0)
(78.22929431608671, 0)
(27.968758594362903, 0.00255672262583981)
(6.139960247678931, 0)
(-83.88782715055348, 0)
(64.95504588340077, 0.000474028231882199)
(-52.63919376725101, 0)
(36.924940239057534, 0)
(8.118668491793771, 0.0303431300920638)
(86.21489557143505, 0)
(47.26172432242044, 0.000895387176873559)
(88.92349635068051, 0.000252928014737302)
(-58.41035543290689, 0)
(16.244807875181, 0)
(55.373228458377234, 0)
(-89.15284299214566, 0)
(-363.84985148035156, 0)
(-30.131642358159514, 0.00220284727191354)
(-33.86263176756467, 0.00174416906587125)
(20.286422012799463, 0.00485980769285212)
(18.24852929089131, 0)
(-5.868688997030047, 0.0580694623556938)
(64.24992409969826, 0)
(-19.81637947775082, 0.0050930902869368)
(-43.776380831331565, 0)
(-77.46251976508707, 0)
(-11.621483104762843, 0.0148083573229974)
(-13.498606369111146, 0)
(66.22434102662972, 0)
(-53.55625732031849, 0.000697283769874073)
(-86.41507508492442, 0.000267824912929084)
(-47.559927571272134, 0)
(28.248266035489817, 0)
(-27.968758594362903, 0.00255672262583981)
(77.826670467913, 0)
(2.5066282746310007, 0)
(22.137941502316963, 0)
(31.805547227738646, 0)
(-64.27436013860758, 0.000484121613364657)
(-95.86009637202834, 0.000217647795734912)
(-39.791490263739256, 0)
(42.242450535438905, 0)
(-25.56269297133377, 0)
(-95.59755536415626, 0.00021884489613519)
(-2.5066282746310007, 0)
(60.0806861715646, 0.000554064375806586)
(-7.72160589396959, 0.033543990948432)
(46.15192107739265, 0)
(-50.320211434307126, 0)
(94.12380826933862, 0)
(-91.53484107004724, 0.000238702590416757)
(61.501850012191596, 0)
(52.249809098131294, 0.000732589314192338)
(-89.74989450581114, 0)
(3.544907701811032, 0)
(-3.9307128709529793, 0.129445618228078)
(9.866523626075638, 0.0205447885274479)
(-15.753402869841345, 0.00805900149795483)
(-41.83138561128058, 0.00114294542908271)
(-9.866523626075638, 0.0205447885274479)
(-47.09526954413143, 0)
(-53.90706658773913, 0.00068823791774717)
(-87.73198961208502, 0)
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:
$$x_{1} = -43.8480866628973$$
$$x_{2} = 82.1853040708499$$
$$x_{3} = 56.273665198017$$
$$x_{4} = 23.9117097455063$$
$$x_{5} = -59.0522467437495$$
$$x_{6} = 78.2292943160867$$
$$x_{7} = 6.13996024767893$$
$$x_{8} = -83.8878271505535$$
$$x_{9} = -52.639193767251$$
$$x_{10} = 36.9249402390575$$
$$x_{11} = 86.2148955714351$$
$$x_{12} = -58.4103554329069$$
$$x_{13} = 16.244807875181$$
$$x_{14} = 55.3732284583772$$
$$x_{15} = -89.1528429921457$$
$$x_{16} = -363.849851480352$$
$$x_{17} = 18.2485292908913$$
$$x_{18} = 64.2499240996983$$
$$x_{19} = -43.7763808313316$$
$$x_{20} = -77.4625197650871$$
$$x_{21} = -13.4986063691111$$
$$x_{22} = 66.2243410266297$$
$$x_{23} = -47.5599275712721$$
$$x_{24} = 28.2482660354898$$
$$x_{25} = 77.826670467913$$
$$x_{26} = 2.506628274631$$
$$x_{27} = 22.137941502317$$
$$x_{28} = 31.8055472277386$$
$$x_{29} = -39.7914902637393$$
$$x_{30} = 42.2424505354389$$
$$x_{31} = -25.5626929713338$$
$$x_{32} = -2.506628274631$$
$$x_{33} = 46.1519210773927$$
$$x_{34} = -50.3202114343071$$
$$x_{35} = 94.1238082693386$$
$$x_{36} = 61.5018500121916$$
$$x_{37} = -89.7498945058111$$
$$x_{38} = 3.54490770181103$$
$$x_{39} = -47.0952695441314$$
$$x_{40} = -87.731989612085$$
Puntos máximos de la función:
$$x_{40} = -55.8533814804374$$
$$x_{40} = 36.7115823747944$$
$$x_{40} = -67.7485497660991$$
$$x_{40} = 97.9350905159343$$
$$x_{40} = 27.9687585943629$$
$$x_{40} = 64.9550458834008$$
$$x_{40} = 8.11866849179377$$
$$x_{40} = 47.2617243224204$$
$$x_{40} = 88.9234963506805$$
$$x_{40} = -30.1316423581595$$
$$x_{40} = -33.8626317675647$$
$$x_{40} = 20.2864220127995$$
$$x_{40} = -5.86868899703005$$
$$x_{40} = -19.8163794777508$$
$$x_{40} = -11.6214831047628$$
$$x_{40} = -53.5562573203185$$
$$x_{40} = -86.4150750849244$$
$$x_{40} = -27.9687585943629$$
$$x_{40} = -64.2743601386076$$
$$x_{40} = -95.8600963720283$$
$$x_{40} = -95.5975553641563$$
$$x_{40} = 60.0806861715646$$
$$x_{40} = -7.72160589396959$$
$$x_{40} = -91.5348410700472$$
$$x_{40} = 52.2498090981313$$
$$x_{40} = -3.93071287095298$$
$$x_{40} = 9.86652362607564$$
$$x_{40} = -15.7534028698413$$
$$x_{40} = -41.8313856112806$$
$$x_{40} = -9.86652362607564$$
$$x_{40} = -53.9070665877391$$
Decrece en los intervalos
$$\left[94.1238082693386, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -363.849851480352\right]$$