Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d t} f{\left(t \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d t} f{\left(t \right)} = $$
primera derivada$$4 \left(- t \sin{\left(t \right)} + \cos{\left(t \right)}\right) \sin{\left(t \cos{\left(t \right)} \right)} \cos{\left(t \cos{\left(t \right)} \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$t_{1} = 9.52933440536196$$
$$t_{2} = 47.1662993169327$$
$$t_{3} = 37.7256128277765$$
$$t_{4} = -68.2746898137795$$
$$t_{5} = 24.1862248303366$$
$$t_{6} = -1.5707963267949$$
$$t_{7} = 28.3449225622678$$
$$t_{8} = -30.5110645458325$$
$$t_{9} = 80.1498192134838$$
$$t_{10} = 34.5575191894877$$
$$t_{11} = -21.9911485751286$$
$$t_{12} = 68.1864393033044$$
$$t_{13} = -72.2843013230124$$
$$t_{14} = -51.3245530867134$$
$$t_{15} = 87.9873258258086$$
$$t_{16} = -75.4114834888481$$
$$t_{17} = -11.7292528690793$$
$$t_{18} = 54.2490766358244$$
$$t_{19} = -53.6691739907764$$
$$t_{20} = -15.7712848748159$$
$$t_{21} = -89.7828454291688$$
$$t_{22} = -37.7521014243949$$
$$t_{23} = 10.1053547868089$$
$$t_{24} = 72.566281542806$$
$$t_{25} = -46.7765007581068$$
$$t_{26} = -31.4159265358979$$
$$t_{27} = -40.5854022598267$$
$$t_{28} = -15.8344387416857$$
$$t_{29} = 100.540910786842$$
$$t_{30} = -59.7237510632525$$
$$t_{31} = -15.707963267949$$
$$t_{32} = -85.7902078153718$$
$$t_{33} = 76.5461072364502$$
$$t_{34} = 63.0720690226406$$
$$t_{35} = 25.1327412287183$$
$$t_{36} = 12.1303742531355$$
$$t_{37} = 0$$
$$t_{38} = 8.24489690812664$$
$$t_{39} = 75.398223686155$$
$$t_{40} = 0.86033358901938$$
$$t_{41} = 67.4976815189735$$
$$t_{42} = -83.9660742234445$$
$$t_{43} = -50.2654824574367$$
$$t_{44} = 20.1847144072068$$
$$t_{45} = 66.0037493532594$$
$$t_{46} = 76.3033501534943$$
$$t_{47} = 59.088081511794$$
$$t_{48} = 97.4099047756117$$
$$t_{49} = 91.3032950363969$$
$$t_{50} = -51.1753017001883$$
$$t_{51} = -39.5928836844695$$
$$t_{52} = 131.946891450771$$
$$t_{53} = 28.2743338823081$$
$$t_{54} = -9.63314724838335$$
$$t_{55} = -96.6873970931048$$
$$t_{56} = 59.7237510632525$$
$$t_{57} = -3.69672292256781$$
$$t_{58} = 4.34224753568081$$
$$t_{59} = 40.8407044966673$$
$$t_{60} = -66.0037493532594$$
$$t_{61} = 2.31611213068295$$
$$t_{62} = -28.309642854452$$
$$t_{63} = 56.3291859523472$$
$$t_{64} = 31.999500727538$$
$$t_{65} = -81.7058882556996$$
$$t_{66} = -135.088484104361$$
$$t_{67} = -87.9873258258086$$
$$t_{68} = 18.9551661778348$$
$$t_{69} = -30.9938864935603$$
$$t_{70} = -91.1171613944647$$
$$t_{71} = 40.8896264144125$$
$$t_{72} = -44.0050179208308$$
$$t_{73} = 94.2477796076938$$
$$t_{74} = 78.2036923334948$$
$$t_{75} = -57.0980924408068$$
$$t_{76} = 72.2566310325652$$
$$t_{77} = -18.90240995686$$
$$t_{78} = 16.2203662438486$$
$$t_{79} = -69.1439655522063$$
$$t_{80} = 48.8237352274238$$
$$t_{81} = 44.0050179208308$$
$$t_{82} = 6.28318530717959$$
$$t_{83} = 64.9078784744433$$
$$t_{84} = -78.100545462825$$
$$t_{85} = 21.9911485751286$$
$$t_{86} = -79.4759164084994$$
$$t_{87} = 50.2654824574367$$
$$t_{88} = -94.2689962866132$$
$$t_{89} = 100.034645727999$$
$$t_{90} = -97.4099047756117$$
$$t_{91} = 53.4070751110265$$
$$t_{92} = -49.9287025648298$$
$$t_{93} = 81.4148630970183$$
$$t_{94} = -37.6991118430775$$
$$t_{95} = -7.85398163397448$$
$$t_{96} = -63.4009836613048$$
$$t_{97} = 14.2476410900056$$
$$t_{98} = -63.9168256049229$$
$$t_{99} = -34.5575191894877$$
Signos de extremos en los puntos:
(9.529334405361963, 0.0055108542232601)
(47.166299316932744, 5.8097384599231e-29)
(37.7256128277765, 0.000351336059243502)
(-68.2746898137795, 2)
(24.18622483033663, 2)
(-1.5707963267948966, 1.85025446894204e-32)
(28.344922562267822, 1.20102098488764e-29)
(-30.511064545832454, 1.07982704351654e-30)
(80.14981921348375, 1.99547578460227e-25)
(34.55751918948773, 9.72936212698185e-30)
(-21.991148575128552, 1.46976458700862e-30)
(68.18643930330437, 3.37263900646877e-26)
(-72.28430132301244, 1.22976782797384e-28)
(-51.32455308671338, 2.70879499801747e-28)
(87.98732582580864, 2.35162333921379e-29)
(-75.41148348884815, 8.79227763509786e-5)
(-11.729252869079271, 2)
(54.24907663582436, 2)
(-53.669173990776365, 2)
(-15.771284874815882, 0.00201084252147885)
(-89.78284542916876, 5.69808665686428e-25)
(-37.75210142439493, 4.31930817406615e-30)
(10.105354786808867, 2)
(72.56628154280597, 5.49827422663806e-27)
(-46.776500758106835, 4.61251041218787e-27)
(-31.41592653589793, 2.99951956532372e-30)
(-40.585402259826694, 2)
(-15.83443874168566, 7.49879891330929e-31)
(100.54091078684232, 4.94638540860084e-5)
(-59.72375106325252, 6.91380423041194e-29)
(-15.707963267948966, 7.49879891330929e-31)
(-85.79020781537176, 2)
(76.54610723645023, 9.4995845168189e-26)
(63.0720690226406, 2)
(25.132741228718345, 1.91969252180718e-30)
(12.130374253135477, 2)
(0, 0)
(8.244896908126641, 7.21104632442734e-30)
(75.39822368615503, 1.72772326962646e-29)
(0.8603335890193797, 0.566292228465965)
(67.49768151897352, 2.64390428762807e-25)
(-83.9660742234445, 2)
(-50.26548245743669, 7.67877008722871e-30)
(20.184714407206833, 2)
(66.00374935325938, 1.92454699589753e-30)
(76.30335015349426, 2.30309745424817e-25)
(59.08808151179398, 2)
(97.40990477561171, 1.08019092984206e-28)
(91.30329503639689, 2)
(-51.1753017001883, 4.70708922874983e-27)
(-39.59288368446951, 5.95734224048085e-27)
(131.94689145077132, 7.69818798359012e-30)
(28.274333882308138, 2.42961084791221e-30)
(-9.633147248383352, 2.69956760879134e-31)
(-96.6873970931048, 2)
(59.72375106325252, 6.91380423041194e-29)
(-3.6967229225678127, 2.99951956532372e-32)
(4.342247535680812, 2)
(40.840704496667314, 7.68847597031319e-30)
(-66.00374935325938, 1.92454699589753e-30)
(2.316112130682953, 2)
(-28.30964285445201, 0.000623944646835302)
(56.32918595234723, 2)
(31.999500727538045, 2)
(-81.70588825569959, 2.11747992378503e-28)
(-135.0884841043611, 1.55538760012425e-28)
(-87.98732582580864, 2.35162333921379e-29)
(18.95516617783476, 1.07982704351654e-30)
(-30.99388649356027, 4.33361097810928e-29)
(-91.11716139446474, 6.02246101568735e-5)
(40.889626414412525, 7.68847597031319e-30)
(-44.005017920830845, 0.000258216645514578)
(94.2477796076938, 2.35502055154817e-29)
(78.20369233349476, 2)
(-57.098092440806816, 2)
(72.25663103256524, 8.11385462699113e-29)
(-18.902409956860023, 0.00139970121808476)
(16.220366243848606, 2)
(-69.14396555220625, 3.89174485079274e-29)
(48.823735227423796, 4.00248611286014e-26)
(44.005017920830845, 0.000258216645514578)
(6.283185307179586, 1.19980782612949e-31)
(64.9078784744433, 1.58922847564534e-25)
(-78.10054546282495, 2)
(21.991148575128552, 1.46976458700862e-30)
(-79.47591640849937, 1.55626109895089e-28)
(50.26548245743669, 7.67877008722871e-30)
(-94.26899628661319, 2.35502055154817e-29)
(100.03464572799854, 4.2433374167186e-27)
(-97.40990477561171, 1.08019092984206e-28)
(53.40707511102649, 4.32658835265357e-30)
(-49.92870256482977, 3.34837242674972e-27)
(81.41486309701827, 2.56265515794991e-26)
(-37.69911184307752, 4.31930817406615e-30)
(-7.853981633974483, 1.15640904308877e-29)
(-63.4009836613048, 1.05197766907004e-26)
(14.247641090005631, 2)
(-63.9168256049229, 2)
(-34.55751918948773, 9.72936212698185e-30)
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:
$$t_{1} = 47.1662993169327$$
$$t_{2} = -1.5707963267949$$
$$t_{3} = 28.3449225622678$$
$$t_{4} = -30.5110645458325$$
$$t_{5} = 80.1498192134838$$
$$t_{6} = 34.5575191894877$$
$$t_{7} = -21.9911485751286$$
$$t_{8} = 68.1864393033044$$
$$t_{9} = -72.2843013230124$$
$$t_{10} = -51.3245530867134$$
$$t_{11} = 87.9873258258086$$
$$t_{12} = -89.7828454291688$$
$$t_{13} = -37.7521014243949$$
$$t_{14} = 72.566281542806$$
$$t_{15} = -46.7765007581068$$
$$t_{16} = -31.4159265358979$$
$$t_{17} = -15.8344387416857$$
$$t_{18} = -59.7237510632525$$
$$t_{19} = -15.707963267949$$
$$t_{20} = 76.5461072364502$$
$$t_{21} = 25.1327412287183$$
$$t_{22} = 0$$
$$t_{23} = 8.24489690812664$$
$$t_{24} = 75.398223686155$$
$$t_{25} = 67.4976815189735$$
$$t_{26} = -50.2654824574367$$
$$t_{27} = 66.0037493532594$$
$$t_{28} = 76.3033501534943$$
$$t_{29} = 97.4099047756117$$
$$t_{30} = -51.1753017001883$$
$$t_{31} = -39.5928836844695$$
$$t_{32} = 131.946891450771$$
$$t_{33} = 28.2743338823081$$
$$t_{34} = -9.63314724838335$$
$$t_{35} = 59.7237510632525$$
$$t_{36} = -3.69672292256781$$
$$t_{37} = 40.8407044966673$$
$$t_{38} = -66.0037493532594$$
$$t_{39} = -81.7058882556996$$
$$t_{40} = -135.088484104361$$
$$t_{41} = -87.9873258258086$$
$$t_{42} = 18.9551661778348$$
$$t_{43} = -30.9938864935603$$
$$t_{44} = 40.8896264144125$$
$$t_{45} = 94.2477796076938$$
$$t_{46} = 72.2566310325652$$
$$t_{47} = -69.1439655522063$$
$$t_{48} = 48.8237352274238$$
$$t_{49} = 6.28318530717959$$
$$t_{50} = 64.9078784744433$$
$$t_{51} = 21.9911485751286$$
$$t_{52} = -79.4759164084994$$
$$t_{53} = 50.2654824574367$$
$$t_{54} = -94.2689962866132$$
$$t_{55} = 100.034645727999$$
$$t_{56} = -97.4099047756117$$
$$t_{57} = 53.4070751110265$$
$$t_{58} = -49.9287025648298$$
$$t_{59} = 81.4148630970183$$
$$t_{60} = -37.6991118430775$$
$$t_{61} = -7.85398163397448$$
$$t_{62} = -63.4009836613048$$
$$t_{63} = -34.5575191894877$$
Puntos máximos de la función:
$$t_{63} = 9.52933440536196$$
$$t_{63} = 37.7256128277765$$
$$t_{63} = -68.2746898137795$$
$$t_{63} = 24.1862248303366$$
$$t_{63} = -75.4114834888481$$
$$t_{63} = -11.7292528690793$$
$$t_{63} = 54.2490766358244$$
$$t_{63} = -53.6691739907764$$
$$t_{63} = -15.7712848748159$$
$$t_{63} = 10.1053547868089$$
$$t_{63} = -40.5854022598267$$
$$t_{63} = 100.540910786842$$
$$t_{63} = -85.7902078153718$$
$$t_{63} = 63.0720690226406$$
$$t_{63} = 12.1303742531355$$
$$t_{63} = 0.86033358901938$$
$$t_{63} = -83.9660742234445$$
$$t_{63} = 20.1847144072068$$
$$t_{63} = 59.088081511794$$
$$t_{63} = 91.3032950363969$$
$$t_{63} = -96.6873970931048$$
$$t_{63} = 4.34224753568081$$
$$t_{63} = 2.31611213068295$$
$$t_{63} = -28.309642854452$$
$$t_{63} = 56.3291859523472$$
$$t_{63} = 31.999500727538$$
$$t_{63} = -91.1171613944647$$
$$t_{63} = -44.0050179208308$$
$$t_{63} = 78.2036923334948$$
$$t_{63} = -57.0980924408068$$
$$t_{63} = -18.90240995686$$
$$t_{63} = 16.2203662438486$$
$$t_{63} = 44.0050179208308$$
$$t_{63} = -78.100545462825$$
$$t_{63} = 14.2476410900056$$
$$t_{63} = -63.9168256049229$$
Decrece en los intervalos
$$\left[131.946891450771, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -135.088484104361\right]$$