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$$2 x \cos{\left(x^{2} \right)} - \sin{\left(x \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 70.1519690117632$$
$$x_{2} = -5.45689420149144$$
$$x_{3} = 98.4390519310906$$
$$x_{4} = 60.2503601843957$$
$$x_{5} = 16.000961718966$$
$$x_{6} = -9.29520030332014$$
$$x_{7} = -19.2131827846554$$
$$x_{8} = -68.2452687572932$$
$$x_{9} = 90.2124794543957$$
$$x_{10} = -23.8133956350226$$
$$x_{11} = -85.7673275536261$$
$$x_{12} = 53.4535133009683$$
$$x_{13} = 0$$
$$x_{14} = 123.583369243266$$
$$x_{15} = 63.0281077262879$$
$$x_{16} = 77.5334959297238$$
$$x_{17} = 18.1194601833497$$
$$x_{18} = 42.4836052898255$$
$$x_{19} = -30.1059051977611$$
$$x_{20} = -17.8563208080559$$
$$x_{21} = 6.26667578209062$$
$$x_{22} = 1.07180134799622$$
$$x_{23} = 41.9999210018193$$
$$x_{24} = -53.6295256129204$$
$$x_{25} = 76.2258515383438$$
$$x_{26} = -45.8958343802316$$
$$x_{27} = -79.6717869621148$$
$$x_{28} = -6.008807510337$$
$$x_{29} = -5.74727419627096$$
$$x_{30} = 28.0528972678513$$
$$x_{31} = 56.1199638171434$$
$$x_{32} = -40.4375336170006$$
$$x_{33} = 1.07180134799621$$
$$x_{34} = 3.77031009470098$$
$$x_{35} = -47.87258851768$$
$$x_{36} = 36.0860637747281$$
$$x_{37} = -77.8367347661307$$
$$x_{38} = -7.83102368526207$$
$$x_{39} = -31.9780665870872$$
$$x_{40} = 31.0306977554749$$
$$x_{41} = 54.1252365169635$$
$$x_{42} = -4.53090580308408$$
$$x_{43} = -35.6917795239445$$
$$x_{44} = 80.1827636412082$$
$$x_{45} = -1.07180134799621$$
$$x_{46} = -81.8117494870919$$
$$x_{47} = 32.1248043487951$$
$$x_{48} = 92.1079558673844$$
$$x_{49} = -21.744058273147$$
$$x_{50} = 20.2473087856324$$
$$x_{51} = -97.8789272264631$$
$$x_{52} = 34.2087454964102$$
$$x_{53} = 10.2570609435673$$
$$x_{54} = 22.3147834698915$$
$$x_{55} = -77.6144900088749$$
$$x_{56} = 4.144499701712$$
$$x_{57} = -93.7978280387191$$
$$x_{58} = -65.8556585728539$$
$$x_{59} = -47.8069247537141$$
$$x_{60} = -3.77031009470098$$
$$x_{61} = 2.21234304383249$$
$$x_{62} = 84.3080687600362$$
$$x_{63} = 58.1813944632285$$
$$x_{64} = -62.8784063450366$$
$$x_{65} = 7.61942227709894$$
$$x_{66} = -75.7711223256984$$
$$x_{67} = -90.2299374902237$$
$$x_{68} = -33.7931822913434$$
$$x_{69} = -45.9642370531503$$
$$x_{70} = 28.1646978075122$$
$$x_{71} = 94.2155500525576$$
$$x_{72} = 62.2255375583191$$
$$x_{73} = -16.000961718966$$
$$x_{74} = -71.61456434202$$
$$x_{75} = 40.1648554721824$$
$$x_{76} = -97.7504590719906$$
Signos de extremos en los puntos:
(70.15196901176319, 1.50884608091918)
(-5.45689420149144, -0.320119040910535)
(98.43905193109056, 0.502141413380089)
(60.250360184395696, -1.84719239330664)
(16.00096171896595, -1.95734142734859)
(-9.295200303320142, -1.99159239923997)
(-19.2131827846554, -0.0653441436752971)
(-68.24526875729316, 1.64498694165504)
(90.2124794543957, 0.373463955522474)
(-23.813395635022637, 1.24860252790097)
(-85.76732755362612, -1.58627794456955)
(53.45351330096833, -1.99892184674017)
(0, 1)
(123.58336924326606, -1.48776999626358)
(63.028107726287885, 1.98080259093646)
(77.53349592972378, -1.53495826506863)
(18.1194601833497, 1.74494118614731)
(42.483605289825455, 1.07197310293439)
(-30.105905197761075, 1.25770066208084)
(-17.856320808055948, -0.453742435610938)
(6.2666757820906245, 1.99986285339417)
(1.0718013479962192, 1.39079925561018)
(41.999921001819274, -1.39999819474236)
(-53.629525612920446, -1.97535763224338)
(76.22585153834383, -0.32336399397888)
(-45.8958343802316, 0.663877523219909)
(-79.6717869621148, 0.575107136506196)
(-6.008807510337004, -0.0371518399924072)
(-5.747274196270963, 1.85881655289124)
(28.052897267851268, 0.0244094074002162)
(56.11996381714341, 1.90949841419268)
(-40.43753361700063, 0.0801666661315266)
(1.0718013479962063, 1.39079925561018)
(3.7703100947009753, 0.188171529732733)
(-47.87258851768004, -1.73254998445829)
(36.0860637747281, 0.957665004259692)
(-77.83673476613066, 0.237138019972482)
(-7.8310236852620685, -0.97500473999545)
(-31.978066587087238, -0.153848867127473)
(31.030697755474943, 1.92669415719388)
(54.125236516963454, 0.246964727214996)
(-4.530905803084081, 0.813603394896721)
(-35.69177952394449, -1.42272226306776)
(80.18276364120818, 1.07206904934775)
(-1.0718013479962063, 1.39079925561018)
(-81.8117494870919, 1.99151738113696)
(32.124804348795124, 1.75904154691855)
(92.10795586738436, 0.46117657150365)
(-21.74405827314698, 0.03035599441004)
(20.247308785632367, 1.17188523451614)
(-97.8789272264631, -1.8825393310492)
(34.2087454964102, 0.0601950205241431)
(10.257060943567257, -1.6725394180635)
(22.314783469891452, 0.0518888788102883)
(-77.61449000887494, -1.60156070469618)
(4.144499701711996, -1.53266836501194)
(-93.7978280387191, 1.90046547962862)
(-65.85565857285391, 0.00692849222149217)
(-47.806924753714064, -1.77563899825343)
(-3.7703100947009753, 0.188171529732733)
(2.2123430438324894, -1.58190583537133)
(84.30806876003615, 0.129669958874587)
(58.18139446322855, -1.0618540057132)
(-62.87840634503662, 1.99891652359167)
(7.619422277098935, 1.23037554740468)
(-75.77112232569843, -0.0687218727206863)
(-90.22993749022372, -1.64002827614011)
(-33.79318229134336, -1.72178899512071)
(-45.964237053150306, 0.600292418836843)
(28.164697807512244, 0.00600213023250085)
(94.21555005255756, -0.000519312533249106)
(62.22553755831907, 1.82174268895309)
(-16.00096171896595, -1.95734142734859)
(-71.61456434202, 0.199131431632696)
(40.16485547218239, -1.7801457943983)
(-97.75045907199062, -1.93551178280402)
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:
$$x_{1} = -5.45689420149144$$
$$x_{2} = 60.2503601843957$$
$$x_{3} = 16.000961718966$$
$$x_{4} = -9.29520030332014$$
$$x_{5} = -19.2131827846554$$
$$x_{6} = -85.7673275536261$$
$$x_{7} = 53.4535133009683$$
$$x_{8} = 0$$
$$x_{9} = 123.583369243266$$
$$x_{10} = 77.5334959297238$$
$$x_{11} = -17.8563208080559$$
$$x_{12} = 41.9999210018193$$
$$x_{13} = -53.6295256129204$$
$$x_{14} = 76.2258515383438$$
$$x_{15} = -6.008807510337$$
$$x_{16} = -47.87258851768$$
$$x_{17} = -7.83102368526207$$
$$x_{18} = -31.9780665870872$$
$$x_{19} = -35.6917795239445$$
$$x_{20} = -97.8789272264631$$
$$x_{21} = 10.2570609435673$$
$$x_{22} = -77.6144900088749$$
$$x_{23} = 4.144499701712$$
$$x_{24} = -47.8069247537141$$
$$x_{25} = 2.21234304383249$$
$$x_{26} = 58.1813944632285$$
$$x_{27} = -75.7711223256984$$
$$x_{28} = -90.2299374902237$$
$$x_{29} = -33.7931822913434$$
$$x_{30} = 94.2155500525576$$
$$x_{31} = -16.000961718966$$
$$x_{32} = 40.1648554721824$$
$$x_{33} = -97.7504590719906$$
Puntos máximos de la función:
$$x_{33} = 70.1519690117632$$
$$x_{33} = 98.4390519310906$$
$$x_{33} = -68.2452687572932$$
$$x_{33} = 90.2124794543957$$
$$x_{33} = -23.8133956350226$$
$$x_{33} = 63.0281077262879$$
$$x_{33} = 18.1194601833497$$
$$x_{33} = 42.4836052898255$$
$$x_{33} = -30.1059051977611$$
$$x_{33} = 6.26667578209062$$
$$x_{33} = 1.07180134799622$$
$$x_{33} = -45.8958343802316$$
$$x_{33} = -79.6717869621148$$
$$x_{33} = -5.74727419627096$$
$$x_{33} = 28.0528972678513$$
$$x_{33} = 56.1199638171434$$
$$x_{33} = -40.4375336170006$$
$$x_{33} = 1.07180134799621$$
$$x_{33} = 3.77031009470098$$
$$x_{33} = 36.0860637747281$$
$$x_{33} = -77.8367347661307$$
$$x_{33} = 31.0306977554749$$
$$x_{33} = 54.1252365169635$$
$$x_{33} = -4.53090580308408$$
$$x_{33} = 80.1827636412082$$
$$x_{33} = -1.07180134799621$$
$$x_{33} = -81.8117494870919$$
$$x_{33} = 32.1248043487951$$
$$x_{33} = 92.1079558673844$$
$$x_{33} = -21.744058273147$$
$$x_{33} = 20.2473087856324$$
$$x_{33} = 34.2087454964102$$
$$x_{33} = 22.3147834698915$$
$$x_{33} = -93.7978280387191$$
$$x_{33} = -65.8556585728539$$
$$x_{33} = -3.77031009470098$$
$$x_{33} = 84.3080687600362$$
$$x_{33} = -62.8784063450366$$
$$x_{33} = 7.61942227709894$$
$$x_{33} = -45.9642370531503$$
$$x_{33} = 28.1646978075122$$
$$x_{33} = 62.2255375583191$$
$$x_{33} = -71.61456434202$$
Decrece en los intervalos
$$\left[123.583369243266, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -97.8789272264631\right]$$