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$$3 x^{2} \cos{\left(3 x \right)} + 2 x \sin{\left(3 x \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -67.5475318097959$$
$$x_{2} = -95.820895038653$$
$$x_{3} = -25.6649966297225$$
$$x_{4} = -70.6889782741115$$
$$x_{5} = 12.0611776969175$$
$$x_{6} = -93.7265517554948$$
$$x_{7} = 48.6992490006399$$
$$x_{8} = 88.4907042779379$$
$$x_{9} = -69.6418279874538$$
$$x_{10} = -23.5713700221828$$
$$x_{11} = 95.820895038653$$
$$x_{12} = -43.4638107841198$$
$$x_{13} = 73.8304371774155$$
$$x_{14} = 24.6181670049683$$
$$x_{15} = 58.1232872144417$$
$$x_{16} = 9.97063128985059$$
$$x_{17} = -7.88210793986787$$
$$x_{18} = 51.8405651953147$$
$$x_{19} = 78.0190658014538$$
$$x_{20} = -91.6322108409093$$
$$x_{21} = 86.3963700471954$$
$$x_{22} = 16.2452335983018$$
$$x_{23} = -3.72423528944333$$
$$x_{24} = 31.946480380411$$
$$x_{25} = -65.4532419617293$$
$$x_{26} = 3.72423528944333$$
$$x_{27} = -78.0190658014538$$
$$x_{28} = -14.1528569238997$$
$$x_{29} = -46.6050589026417$$
$$x_{30} = -53.9347938783761$$
$$x_{31} = 5.79774798819825$$
$$x_{32} = 29.8525729609081$$
$$x_{33} = -29.8525729609081$$
$$x_{34} = 71.7361299404533$$
$$x_{35} = -36.1344646875895$$
$$x_{36} = 56.0290349994261$$
$$x_{37} = -45.5579709190865$$
$$x_{38} = -100.009588115537$$
$$x_{39} = 23.5713700221828$$
$$x_{40} = -12.0611776969175$$
$$x_{41} = -21.4778930345772$$
$$x_{42} = -89.5378724610825$$
$$x_{43} = -47.6521516999475$$
$$x_{44} = 84.3020388406696$$
$$x_{45} = -1.69566169803409$$
$$x_{46} = 68.5946791436526$$
$$x_{47} = -16.2452335983018$$
$$x_{48} = -41.3696744286036$$
$$x_{49} = -87.4435367981187$$
$$x_{50} = 36.1344646875895$$
$$x_{51} = 0$$
$$x_{52} = 42.4167394139284$$
$$x_{53} = -34.0404477609703$$
$$x_{54} = 62.3118204533444$$
$$x_{55} = -5.79774798819825$$
$$x_{56} = -71.7361299404533$$
$$x_{57} = 90.5850413231642$$
$$x_{58} = 7.88210793986787$$
$$x_{59} = 66.5003860571966$$
$$x_{60} = 38.2285230247401$$
$$x_{61} = 75.9247492611644$$
$$x_{62} = -56.0290349994261$$
$$x_{63} = 80.1133864488351$$
$$x_{64} = 49.7463505204702$$
$$x_{65} = -73.8304371774155$$
$$x_{66} = 14.1528569238997$$
$$x_{67} = 1.69566169803409$$
$$x_{68} = -80.1133864488351$$
$$x_{69} = -38.2285230247401$$
$$x_{70} = 53.9347938783761$$
$$x_{71} = 44.5108880888307$$
$$x_{72} = -27.7587390549925$$
$$x_{73} = 64.4060996042015$$
$$x_{74} = 27.7587390549925$$
$$x_{75} = 18.338069892946$$
$$x_{76} = 93.7265517554948$$
$$x_{77} = -97.9152405384144$$
$$x_{78} = 20.4312249887476$$
$$x_{79} = 82.2077108894621$$
$$x_{80} = 26.7118550646915$$
$$x_{81} = -75.9247492611644$$
$$x_{82} = -51.8405651953147$$
$$x_{83} = -9.97063128985059$$
$$x_{84} = -82.2077108894621$$
$$x_{85} = 22.5246102236197$$
$$x_{86} = -60.2175493662913$$
$$x_{87} = -49.7463505204702$$
$$x_{88} = -6.8391743033139$$
$$x_{89} = -84.3020388406696$$
$$x_{90} = -31.946480380411$$
$$x_{91} = 40.3226163252311$$
$$x_{92} = 100.009588115537$$
$$x_{93} = -58.1232872144417$$
$$x_{94} = 97.9152405384144$$
$$x_{95} = 25.6649966297225$$
$$x_{96} = 60.2175493662913$$
$$x_{97} = 34.0404477609703$$
Signos de extremos en los puntos:
(-67.54753180979594, -4562.44684760667)
(-95.82089503865303, 9181.42171185364)
(-25.664996629722534, -658.469942174557)
(-70.68897827411146, 4996.70944203839)
(12.061177696917529, -145.250293119571)
(-93.72655175549478, 8784.44429018506)
(48.69924900063987, 2371.3946622328)
(88.49070427793794, 7830.38253084235)
(-69.64182798745382, -4849.76199848377)
(-23.571370022182833, -555.38739573202)
(95.82089503865303, -9181.42171185364)
(-43.463810784119765, 1888.8806648591)
(73.83043717741552, 5450.7112451744)
(24.61816700496828, -605.832046611167)
(58.12328721444166, -3378.09431631418)
(9.97063128985059, -99.1920084416929)
(-7.882107939867874, 61.9065885787803)
(51.84056519531469, -2687.22200510667)
(78.01906580145378, 6086.75241847789)
(-91.6322108409093, 8396.2398501923)
(86.39637004719542, 7464.11054503295)
(16.245233598301798, -263.685672729676)
(-3.724235289443328, 13.6529081682129)
(31.946480380411014, 1020.35545902797)
(-65.45324196172929, -4283.90467836732)
(3.724235289443328, -13.6529081682129)
(-78.01906580145378, -6086.75241847789)
(-14.152856923899682, 200.081506013128)
(-46.60505890264171, -2171.80932719426)
(-53.934793878376055, 2908.73979394154)
(5.7977479881982505, -33.3938391842218)
(29.85257296090814, 890.953973249049)
(-29.85257296090814, -890.953973249049)
(71.73612994045328, 5145.85013100463)
(-36.134464687589464, -1305.47737275167)
(56.029034999426095, -3139.03056433799)
(-45.55797091908653, 2075.306527725)
(-100.00958811553703, 10001.6955002229)
(23.571370022182833, 555.38739573202)
(-12.061177696917529, 145.250293119571)
(-21.47789303457719, -461.077827430531)
(-89.53787246108253, 8016.80839187402)
(-47.65215169994751, 2270.50537202856)
(84.30203884066964, 7106.61154089394)
(-1.6956616980340902, 2.67588446922981)
(68.59467914365256, -4705.0078003401)
(-16.245233598301798, 263.685672729676)
(-41.36967442860363, 1711.22778337855)
(-87.44353679811874, 7646.14991522874)
(36.134464687589464, 1305.47737275167)
(0, 0)
(42.416739413928404, 1798.95760144944)
(-34.04044776097026, -1158.52992545043)
(62.31182045334436, -3882.54076506345)
(-5.7977479881982505, 33.3938391842218)
(-71.73612994045328, -5145.85013100463)
(90.5850413231642, 8205.42749832394)
(7.882107939867874, -61.9065885787803)
(66.50038605719662, -4422.07914028269)
(38.22852302474006, 1461.19780110439)
(75.92474926116441, 5764.34534099753)
(-56.029034999426095, 3139.03056433799)
(80.11338644883509, 6417.93247761884)
(49.74635052047024, -2474.47719781134)
(-73.83043717741552, -5450.7112451744)
(14.152856923899682, -200.081506013128)
(1.6956616980340902, -2.67588446922981)
(-80.11338644883509, -6417.93247761884)
(-38.22852302474006, -1461.19780110439)
(53.934793878376055, -2908.73979394154)
(44.51088808883066, 1980.99697361532)
(-27.758739054992514, -770.325467786492)
(64.40609960420149, -4147.92346185966)
(27.758739054992514, 770.325467786492)
(18.33806989294604, -336.062805205876)
(93.72655175549478, -8784.44429018506)
(-97.91524053841435, 9587.17211519922)
(20.431224988747633, -417.212909611885)
(82.20771088946208, 6757.88551842333)
(26.711855064691484, -713.301082535532)
(-75.92474926116441, -5764.34534099753)
(-51.84056519531469, 2687.22200510667)
(-9.97063128985059, 99.1920084416929)
(-82.20771088946208, -6757.88551842333)
(22.524610223619664, -507.13598939687)
(-60.21754936629126, 3625.93104988514)
(-49.74635052047024, 2474.47719781134)
(-6.839174303313896, -46.5536541413815)
(-84.30203884066964, -7106.61154089394)
(-31.946480380411014, -1020.35545902797)
(40.322616325231095, 1625.69121063763)
(100.00958811553703, -10001.6955002229)
(-58.12328721444166, 3378.09431631418)
(97.91524053841435, -9587.17211519922)
(25.664996629722534, 658.469942174557)
(60.21754936629126, -3625.93104988514)
(34.04044776097026, 1158.52992545043)
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:
$$x_{1} = -67.5475318097959$$
$$x_{2} = -25.6649966297225$$
$$x_{3} = 12.0611776969175$$
$$x_{4} = -69.6418279874538$$
$$x_{5} = -23.5713700221828$$
$$x_{6} = 95.820895038653$$
$$x_{7} = 24.6181670049683$$
$$x_{8} = 58.1232872144417$$
$$x_{9} = 9.97063128985059$$
$$x_{10} = 51.8405651953147$$
$$x_{11} = 16.2452335983018$$
$$x_{12} = -65.4532419617293$$
$$x_{13} = 3.72423528944333$$
$$x_{14} = -78.0190658014538$$
$$x_{15} = -46.6050589026417$$
$$x_{16} = 5.79774798819825$$
$$x_{17} = -29.8525729609081$$
$$x_{18} = -36.1344646875895$$
$$x_{19} = 56.0290349994261$$
$$x_{20} = -21.4778930345772$$
$$x_{21} = 68.5946791436526$$
$$x_{22} = -34.0404477609703$$
$$x_{23} = 62.3118204533444$$
$$x_{24} = -71.7361299404533$$
$$x_{25} = 7.88210793986787$$
$$x_{26} = 66.5003860571966$$
$$x_{27} = 49.7463505204702$$
$$x_{28} = -73.8304371774155$$
$$x_{29} = 14.1528569238997$$
$$x_{30} = 1.69566169803409$$
$$x_{31} = -80.1133864488351$$
$$x_{32} = -38.2285230247401$$
$$x_{33} = 53.9347938783761$$
$$x_{34} = -27.7587390549925$$
$$x_{35} = 64.4060996042015$$
$$x_{36} = 18.338069892946$$
$$x_{37} = 93.7265517554948$$
$$x_{38} = 20.4312249887476$$
$$x_{39} = 26.7118550646915$$
$$x_{40} = -75.9247492611644$$
$$x_{41} = -82.2077108894621$$
$$x_{42} = 22.5246102236197$$
$$x_{43} = -6.8391743033139$$
$$x_{44} = -84.3020388406696$$
$$x_{45} = -31.946480380411$$
$$x_{46} = 100.009588115537$$
$$x_{47} = 97.9152405384144$$
$$x_{48} = 60.2175493662913$$
Puntos máximos de la función:
$$x_{48} = -95.820895038653$$
$$x_{48} = -70.6889782741115$$
$$x_{48} = -93.7265517554948$$
$$x_{48} = 48.6992490006399$$
$$x_{48} = 88.4907042779379$$
$$x_{48} = -43.4638107841198$$
$$x_{48} = 73.8304371774155$$
$$x_{48} = -7.88210793986787$$
$$x_{48} = 78.0190658014538$$
$$x_{48} = -91.6322108409093$$
$$x_{48} = 86.3963700471954$$
$$x_{48} = -3.72423528944333$$
$$x_{48} = 31.946480380411$$
$$x_{48} = -14.1528569238997$$
$$x_{48} = -53.9347938783761$$
$$x_{48} = 29.8525729609081$$
$$x_{48} = 71.7361299404533$$
$$x_{48} = -45.5579709190865$$
$$x_{48} = -100.009588115537$$
$$x_{48} = 23.5713700221828$$
$$x_{48} = -12.0611776969175$$
$$x_{48} = -89.5378724610825$$
$$x_{48} = -47.6521516999475$$
$$x_{48} = 84.3020388406696$$
$$x_{48} = -1.69566169803409$$
$$x_{48} = -16.2452335983018$$
$$x_{48} = -41.3696744286036$$
$$x_{48} = -87.4435367981187$$
$$x_{48} = 36.1344646875895$$
$$x_{48} = 42.4167394139284$$
$$x_{48} = -5.79774798819825$$
$$x_{48} = 90.5850413231642$$
$$x_{48} = 38.2285230247401$$
$$x_{48} = 75.9247492611644$$
$$x_{48} = -56.0290349994261$$
$$x_{48} = 80.1133864488351$$
$$x_{48} = 44.5108880888307$$
$$x_{48} = 27.7587390549925$$
$$x_{48} = -97.9152405384144$$
$$x_{48} = 82.2077108894621$$
$$x_{48} = -51.8405651953147$$
$$x_{48} = -9.97063128985059$$
$$x_{48} = -60.2175493662913$$
$$x_{48} = -49.7463505204702$$
$$x_{48} = 40.3226163252311$$
$$x_{48} = -58.1232872144417$$
$$x_{48} = 25.6649966297225$$
$$x_{48} = 34.0404477609703$$
Decrece en los intervalos
$$\left[100.009588115537, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -84.3020388406696\right]$$