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$$- \frac{\left(2 x - 2\right) \sin{\left(x^{2} - 2 x \right)}}{\sin{\left(x \right)}} - \frac{\cos{\left(x \right)} \cos{\left(x^{2} - 2 x \right)}}{\sin^{2}{\left(x \right)}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -43.746407396892$$
$$x_{2} = 82.2688040844508$$
$$x_{3} = 44.2458646465051$$
$$x_{4} = 12.1182456705661$$
$$x_{5} = -86.862938912757$$
$$x_{6} = 67.9161598269257$$
$$x_{7} = -93.8274747711939$$
$$x_{8} = -10.3918849334351$$
$$x_{9} = 27.9585721825056$$
$$x_{10} = 2.12647058182939$$
$$x_{11} = 1.15068338608933$$
$$x_{12} = 20.5226878581487$$
$$x_{13} = 83.5156525433853$$
$$x_{14} = 40.3279310847657$$
$$x_{15} = -60.0226479477986$$
$$x_{16} = 56.0980694636532$$
$$x_{17} = 30.2492822624773$$
$$x_{18} = 60.2729937533348$$
$$x_{19} = -17.8684312582618$$
$$x_{20} = 54.1533782872234$$
$$x_{21} = 64.6683332600713$$
$$x_{22} = -38.0062874048977$$
$$x_{23} = 86.2496507700006$$
$$x_{24} = 3.63517014978846$$
$$x_{25} = 4.21569587397632$$
$$x_{26} = -5.71554427820918$$
$$x_{27} = 23.2311115802212$$
$$x_{28} = 73.9794532556925$$
$$x_{29} = -12.3067547649446$$
$$x_{30} = -78.1142623595997$$
$$x_{31} = -1.70330031224334$$
$$x_{32} = 8.16154001871361$$
$$x_{33} = -67.8825628217311$$
$$x_{34} = -95.908311851051$$
$$x_{35} = -21.7222551495779$$
$$x_{36} = 46.0956442628245$$
$$x_{37} = -58.6166574243715$$
$$x_{38} = 71.8610044638087$$
$$x_{39} = 18.1231121130064$$
$$x_{40} = 24.1997481735108$$
$$x_{41} = -13.7565692847145$$
$$x_{42} = -89.5048721299291$$
$$x_{43} = -65.7518660424607$$
$$x_{44} = -87.805589106547$$
$$x_{45} = 6.95416915725495$$
$$x_{46} = 22.4388432403084$$
$$x_{47} = -98.5310973440415$$
$$x_{48} = -52.3599883380669$$
$$x_{49} = -61.0439297823672$$
$$x_{50} = -83.7877791364698$$
$$x_{51} = -19.9205240925383$$
$$x_{52} = 52.1351943464916$$
$$x_{53} = -16.0275667334468$$
$$x_{54} = -27.8169341998335$$
$$x_{55} = 98.3126732353922$$
$$x_{56} = 93.9875431098235$$
$$x_{57} = 10.2614428037138$$
$$x_{58} = 50.0021899873603$$
$$x_{59} = -81.9521922661175$$
$$x_{60} = -33.5660913623612$$
$$x_{61} = -55.8380811602773$$
$$x_{62} = 38.4451048272178$$
$$x_{63} = 91.7128613674868$$
$$x_{64} = 36.5517451779908$$
$$x_{65} = 34.0802579042839$$
$$x_{66} = 89.2197435503$$
$$x_{67} = -48.7967632702592$$
$$x_{68} = -33.747449900857$$
$$x_{69} = -66.8022366272671$$
$$x_{70} = 16.4831551834765$$
$$x_{71} = -45.9054237458581$$
$$x_{72} = -32.1744106851278$$
$$x_{73} = -7.74023388903111$$
$$x_{74} = 47.5352846373765$$
$$x_{75} = -71.97961111961$$
$$x_{76} = -79.9590101956383$$
$$x_{77} = 63.5983845204038$$
$$x_{78} = -75.7763728408412$$
$$x_{79} = -29.8691263690443$$
$$x_{80} = 70.1556200209534$$
$$x_{81} = 38.7376956077601$$
$$x_{82} = 66.252059467846$$
$$x_{83} = -3.78029331422045$$
$$x_{84} = -20.070261027019$$
$$x_{85} = 79.8755471355912$$
$$x_{86} = -69.7498940133187$$
$$x_{87} = -10.1115836596508$$
$$x_{88} = -69.9051567972536$$
Signos de extremos en los puntos:
(-43.74640739689203, -4.27422408519046)
(82.26880408445076, 1.80434324543325)
(44.24586464650511, -3.83486272958495)
(12.118245670566118, 2.29796324204864)
(-86.86293891275704, -1.12112559433005)
(67.91615982692574, 1.07338048502508)
(-93.82747477119392, 2.45057693971615)
(-10.391884933435051, -1.21414655207492)
(27.9585721825056, -3.21501390016099)
(2.126470581829394, 1.13478887397837)
(1.150683386089327, 0.612531364079008)
(20.522687858148732, -1.00525575821413)
(83.51565254338533, -1.03573351758744)
(40.3279310847657, 2.03781817333236)
(-60.022647947798646, -3.06379386724817)
(56.098069463653175, -2.29578529389099)
(30.249282262477266, -1.08759400903238)
(60.27299375333475, -1.81701466252428)
(-17.86843125826182, -1.20300182705465)
(54.15337828722336, 1.47283146388855)
(64.66833326007126, 1.0363603195982)
(-38.006287404897655, -3.30453558607981)
(86.24965077000058, 1.0104795548432)
(3.635170149788458, -1.99051817959081)
(4.215695873976321, 1.13342728662071)
(-5.715544278209179, 1.84741392469973)
(23.23111158022121, 1.05730576081122)
(73.97945325569248, 1.01166771813396)
(-12.3067547649446, 3.85705307608517)
(-78.11426235959966, -2.42209603160578)
(-1.7033003122433394, -1.00853692094434)
(8.161540018713607, 1.0489768333628)
(-67.88256282173114, 1.06008968452584)
(-95.90831185105095, 1.00403971257254)
(-21.722255149577908, -3.75219810073059)
(46.09564426282445, -1.1676601451892)
(-58.61665742437151, 1.13774212150626)
(71.8610044638087, 2.59442789170666)
(18.12311211300641, 1.50472327413427)
(24.199748173510816, 1.24454240284271)
(-13.756569284714523, 1.07697332840545)
(-89.50487212992914, 1.00046585590311)
(-65.75186604246072, -4.54767861839871)
(-87.80558910654696, 6.31179275908473)
(6.95416915725495, -1.59942224318634)
(22.4388432403084, -2.30734389940937)
(-98.53109734404147, -1.09968062692655)
(-52.35998833806693, -1.15475749532501)
(-61.04392978236721, -1.02404252589112)
(-83.78777913646978, -1.1628161195909)
(-19.92052409253831, 1.1392899528177)
(52.13519434649161, 1.04639644552476)
(-16.02756673344678, 3.17033143088979)
(-27.81693419983353, -2.26300211475842)
(98.31267323539218, -1.25375641121121)
(93.98754310982355, -3.88558428001706)
(10.261442803713807, 1.34536379091395)
(50.002189987360275, -3.83954737213332)
(-81.95219226611745, -3.73762896079986)
(-33.56609136236121, -1.19496294227491)
(-55.83808116027732, 1.53300734039907)
(38.44510482721783, 1.47324820585098)
(91.71286136748682, 1.75390450461571)
(36.551745177990846, -1.09684793605091)
(34.08025790428387, 2.17606838110081)
(89.2197435503, -1.05197013004623)
(-48.79676327025925, -1.00523205474676)
(-33.74744990085705, -1.38044729218729)
(-66.80223662726708, -1.35660970603058)
(16.48315518347654, -1.42809567829834)
(-45.90542374585811, -1.06544117904721)
(-32.174410685127796, -1.45368083101472)
(-7.74023388903111, -1.00648284190223)
(47.53528463737655, 2.49994114887808)
(-71.97961111961, 3.65537448311364)
(-79.95901019563833, 1.01160229098716)
(63.598384520403755, -1.44161847261049)
(-75.77637284084119, -2.70818769790555)
(-29.869126369044338, -1.00028790135275)
(70.1556200209534, 1.15914307834044)
(38.73769560776012, -1.16047963731811)
(66.25205946784597, 3.63475419943604)
(-3.7802933142204544, -1.66102356197216)
(-20.07026102701898, 1.06453567679967)
(79.87554713559122, -1.02827742185185)
(-69.74989401331872, 1.68608877899892)
(-10.111583659650757, -1.57474269499409)
(-69.90515679725364, -1.4075507665956)
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:
$$x_{1} = -43.746407396892$$
$$x_{2} = 44.2458646465051$$
$$x_{3} = -86.862938912757$$
$$x_{4} = -10.3918849334351$$
$$x_{5} = 27.9585721825056$$
$$x_{6} = 1.15068338608933$$
$$x_{7} = 20.5226878581487$$
$$x_{8} = 83.5156525433853$$
$$x_{9} = -60.0226479477986$$
$$x_{10} = 56.0980694636532$$
$$x_{11} = 30.2492822624773$$
$$x_{12} = 60.2729937533348$$
$$x_{13} = -17.8684312582618$$
$$x_{14} = -38.0062874048977$$
$$x_{15} = 3.63517014978846$$
$$x_{16} = -78.1142623595997$$
$$x_{17} = -1.70330031224334$$
$$x_{18} = -21.7222551495779$$
$$x_{19} = 46.0956442628245$$
$$x_{20} = -65.7518660424607$$
$$x_{21} = 6.95416915725495$$
$$x_{22} = 22.4388432403084$$
$$x_{23} = -98.5310973440415$$
$$x_{24} = -52.3599883380669$$
$$x_{25} = -61.0439297823672$$
$$x_{26} = -83.7877791364698$$
$$x_{27} = -27.8169341998335$$
$$x_{28} = 98.3126732353922$$
$$x_{29} = 93.9875431098235$$
$$x_{30} = 50.0021899873603$$
$$x_{31} = -81.9521922661175$$
$$x_{32} = -33.5660913623612$$
$$x_{33} = 36.5517451779908$$
$$x_{34} = 89.2197435503$$
$$x_{35} = -48.7967632702592$$
$$x_{36} = -33.747449900857$$
$$x_{37} = -66.8022366272671$$
$$x_{38} = 16.4831551834765$$
$$x_{39} = -45.9054237458581$$
$$x_{40} = -32.1744106851278$$
$$x_{41} = -7.74023388903111$$
$$x_{42} = 63.5983845204038$$
$$x_{43} = -75.7763728408412$$
$$x_{44} = -29.8691263690443$$
$$x_{45} = 38.7376956077601$$
$$x_{46} = -3.78029331422045$$
$$x_{47} = 79.8755471355912$$
$$x_{48} = -10.1115836596508$$
$$x_{49} = -69.9051567972536$$
Puntos máximos de la función:
$$x_{49} = 82.2688040844508$$
$$x_{49} = 12.1182456705661$$
$$x_{49} = 67.9161598269257$$
$$x_{49} = -93.8274747711939$$
$$x_{49} = 2.12647058182939$$
$$x_{49} = 40.3279310847657$$
$$x_{49} = 54.1533782872234$$
$$x_{49} = 64.6683332600713$$
$$x_{49} = 86.2496507700006$$
$$x_{49} = 4.21569587397632$$
$$x_{49} = -5.71554427820918$$
$$x_{49} = 23.2311115802212$$
$$x_{49} = 73.9794532556925$$
$$x_{49} = -12.3067547649446$$
$$x_{49} = 8.16154001871361$$
$$x_{49} = -67.8825628217311$$
$$x_{49} = -95.908311851051$$
$$x_{49} = -58.6166574243715$$
$$x_{49} = 71.8610044638087$$
$$x_{49} = 18.1231121130064$$
$$x_{49} = 24.1997481735108$$
$$x_{49} = -13.7565692847145$$
$$x_{49} = -89.5048721299291$$
$$x_{49} = -87.805589106547$$
$$x_{49} = -19.9205240925383$$
$$x_{49} = 52.1351943464916$$
$$x_{49} = -16.0275667334468$$
$$x_{49} = 10.2614428037138$$
$$x_{49} = -55.8380811602773$$
$$x_{49} = 38.4451048272178$$
$$x_{49} = 91.7128613674868$$
$$x_{49} = 34.0802579042839$$
$$x_{49} = 47.5352846373765$$
$$x_{49} = -71.97961111961$$
$$x_{49} = -79.9590101956383$$
$$x_{49} = 70.1556200209534$$
$$x_{49} = 66.252059467846$$
$$x_{49} = -20.070261027019$$
$$x_{49} = -69.7498940133187$$
Decrece en los intervalos
$$\left[98.3126732353922, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.5310973440415\right]$$