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$$\frac{\left(3 - \cos{\left(x \right)}\right) \cos{\left(3 x - \sin{\left(x \right)} \right)}}{x} - \frac{\sin{\left(3 x - \sin{\left(x \right)} \right)}}{x^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -99.7779466705706$$
$$x_{2} = -49.510493153168$$
$$x_{3} = -79.7927195468083$$
$$x_{4} = -29.5252949847213$$
$$x_{5} = -69.8633281463648$$
$$x_{6} = -19.5906721136269$$
$$x_{7} = -35.8090209187912$$
$$x_{8} = 86.0759881988554$$
$$x_{9} = 1.83727554500964$$
$$x_{10} = 45.8678544894251$$
$$x_{11} = 14.4475895488415$$
$$x_{12} = -11.7989647636976$$
$$x_{13} = 68.3611274829809$$
$$x_{14} = 96.1343777924765$$
$$x_{15} = 71.8604747646449$$
$$x_{16} = 76.1467434060946$$
$$x_{17} = -25.8762837653952$$
$$x_{18} = -86.0759881988554$$
$$x_{19} = -34.1603645737713$$
$$x_{20} = 99.7779466705706$$
$$x_{21} = -7.00622456665035$$
$$x_{22} = -8.15950719268422$$
$$x_{23} = -82.4301235713939$$
$$x_{24} = 25.8762837653952$$
$$x_{25} = -89.8511260956203$$
$$x_{26} = -96.1343777924765$$
$$x_{27} = -97.7839596791271$$
$$x_{28} = 42.0925855916645$$
$$x_{29} = 58.4346533077773$$
$$x_{30} = -21.5928861245147$$
$$x_{31} = 89.8511260956203$$
$$x_{32} = 7.00622456665035$$
$$x_{33} = -55.7941187699105$$
$$x_{34} = -38.4451217685033$$
$$x_{35} = -13.3027521230436$$
$$x_{36} = -24.3737285159504$$
$$x_{37} = 94.9968064998663$$
$$x_{38} = 79.7927195468083$$
$$x_{39} = -9.81341609268515$$
$$x_{40} = 40.4438455565141$$
$$x_{41} = -18.0877983628054$$
$$x_{42} = 96.9934503588524$$
$$x_{43} = 38.4451217685033$$
$$x_{44} = -71.8604747646449$$
$$x_{45} = -78.1437327982763$$
$$x_{46} = 112.344534664003$$
$$x_{47} = -1.83727554500964$$
$$x_{48} = 22.383498752184$$
$$x_{49} = 30.6585364459467$$
$$x_{50} = -57.2963460877875$$
$$x_{51} = -65.577202792437$$
$$x_{52} = 19.5906721136269$$
$$x_{53} = -62.0776553750778$$
$$x_{54} = 69.8633281463648$$
$$x_{55} = -39.5843530220467$$
$$x_{56} = -5.49740887315255$$
$$x_{57} = -60.0844309903617$$
$$x_{58} = 4.37500995127285$$
$$x_{59} = 63.5798673727277$$
$$x_{60} = 34.1603645737713$$
$$x_{61} = 27.8767502519047$$
$$x_{62} = 82.4301235713939$$
$$x_{63} = 74.6445513627976$$
$$x_{64} = 60.0844309903617$$
$$x_{65} = 66.3677186539667$$
$$x_{66} = 62.0776553750778$$
$$x_{67} = 52.1512797139686$$
$$x_{68} = -27.8767502519047$$
$$x_{69} = 9.81341609268515$$
$$x_{70} = -53.8011194230872$$
$$x_{71} = -68.3611274829809$$
$$x_{72} = 18.0877983628054$$
$$x_{73} = -42.0925855916645$$
$$x_{74} = 55.7941187699105$$
$$x_{75} = 11.7989647636976$$
$$x_{76} = -52.1512797139686$$
$$x_{77} = 35.8090209187912$$
$$x_{78} = 32.1609444948825$$
$$x_{79} = -45.8678544894251$$
$$x_{80} = 24.3737285159504$$
$$x_{81} = -83.567864410722$$
$$x_{82} = -73.5094366544198$$
$$x_{83} = 8.15950719268422$$
$$x_{84} = 78.1437327982763$$
$$x_{85} = -93.4946307809577$$
$$x_{86} = -16.0991815919966$$
$$x_{87} = 53.8011194230872$$
$$x_{88} = 84.4269800057337$$
$$x_{89} = 16.0991815919966$$
$$x_{90} = 91.5007287487714$$
$$x_{91} = 48.3760517934113$$
$$x_{92} = -53.0105972030033$$
$$x_{93} = 92.3592455156957$$
$$x_{94} = -76.1467434060946$$
$$x_{95} = -63.5798673727277$$
$$x_{96} = -32.1609444948825$$
Signos de extremos en los puntos:
(-99.77794667057061, -0.0100221571017624)
(-49.51049315316803, -0.0201969403704625)
(-79.79271954680826, 0.0125323820146768)
(-29.525294984721327, 0.0338674954056953)
(-69.86332814636481, 0.0143133757995405)
(-19.590672113626873, 0.0510317123956476)
(-35.809020918791184, 0.0279249325950273)
(86.0759881988554, 0.0116175703661652)
(1.8372755450096407, -0.536868065984678)
(45.86785448942512, -0.021801287951527)
(14.447589548841549, -0.0692005286453167)
(-11.798964763697647, -0.0846947179126038)
(68.36112748298089, -0.0146278929555536)
(96.13437779247647, -0.0104020547764842)
(71.86047476464489, 0.0139157681661346)
(76.1467434060946, 0.013132317696647)
(-25.876283765395236, 0.0386397964527449)
(-86.0759881988554, 0.0116175703661652)
(-34.160364573771346, 0.0292728769309361)
(99.77794667057061, -0.0100221571017624)
(-7.006224566650347, 0.142443959679329)
(-8.159507192684218, -0.122472030869094)
(-82.43012357139392, 0.0121313137001735)
(25.876283765395236, 0.0386397964527449)
(-89.85112609562027, -0.0111294581650656)
(-96.13437779247647, -0.0104020547764842)
(-97.7839596791271, -0.0102265914140216)
(42.09258559166454, 0.0237565424989448)
(58.43465330777727, -0.0171129043898863)
(-21.592886124514678, 0.0463083200638799)
(89.85112609562027, -0.0111294581650656)
(7.006224566650347, 0.142443959679329)
(-55.79411876991055, -0.0179224780696005)
(-38.44512176850326, 0.0260093883162492)
(-13.30275212304358, 0.0751308317855903)
(-24.373728515950447, -0.0410211080046512)
(94.99680649986631, 0.0105265562331111)
(79.79271954680826, 0.0125323820146768)
(-9.81341609268515, -0.101866996986817)
(40.443845556514106, 0.0247251495816776)
(-18.087798362805444, -0.055269590362083)
(96.99345035885241, 0.010309938890444)
(38.44512176850326, 0.0260093883162492)
(-71.86047476464489, 0.0139157681661346)
(-78.14373279827633, 0.0127968634550925)
(112.34453466400294, -0.00890112114997739)
(-1.8372755450096407, -0.536868065984678)
(22.383498752183968, -0.0446728728919734)
(30.658536445946744, -0.0326139863550781)
(-57.296346087787526, 0.0174526026002064)
(-65.57720279243703, 0.0152490865844274)
(19.590672113626873, 0.0510317123956476)
(-62.07765537507777, -0.0161084506414798)
(69.86332814636481, 0.0143133757995405)
(-39.58435302204672, -0.0252617710620953)
(-5.497408873152548, -0.181334261917197)
(-60.084430990361746, -0.0166430968184287)
(4.375009951272855, 0.228034680889803)
(63.57986737272771, 0.01572787069242)
(34.160364573771346, 0.0292728769309361)
(27.87675025190466, 0.0358706866920223)
(82.43012357139392, 0.0121313137001735)
(74.6445513627976, -0.0133965919791545)
(60.084430990361746, -0.0166430968184287)
(66.36771865396672, -0.0150674551637916)
(62.07765537507777, -0.0161084506414798)
(52.151279713968634, -0.0191746631024101)
(-27.87675025190466, 0.0358706866920223)
(9.81341609268515, -0.101866996986817)
(-53.80111942308716, -0.0185867652739121)
(-68.36112748298089, -0.0146278929555536)
(18.087798362805444, -0.055269590362083)
(-42.09258559166454, 0.0237565424989448)
(55.79411876991055, -0.0179224780696005)
(11.798964763697647, -0.0846947179126038)
(-52.151279713968634, -0.0191746631024101)
(35.809020918791184, 0.0279249325950273)
(32.16094449488251, 0.0310906845424721)
(-45.86785448942512, -0.021801287951527)
(24.373728515950447, -0.0410211080046512)
(-83.56786441072195, -0.0119662441300589)
(-73.50943665441976, 0.0136035808954803)
(8.159507192684218, -0.122472030869094)
(78.14373279827633, 0.0127968634550925)
(-93.49463078095775, -0.0106956826908534)
(-16.099181591996576, -0.062107179778095)
(53.80111942308716, -0.0185867652739121)
(84.42698000573372, 0.0118445009057129)
(16.099181591996576, -0.062107179778095)
(91.50072874877144, -0.0109288323007837)
(48.3760517934113, 0.0206709828755252)
(-53.01059720300328, 0.0188639345385404)
(92.35924551569566, 0.0108272285278506)
(-76.1467434060946, 0.013132317696647)
(-63.57986737272771, 0.01572787069242)
(-32.16094449488251, 0.0310906845424721)
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:
$$x_{1} = -99.7779466705706$$
$$x_{2} = -49.510493153168$$
$$x_{3} = 1.83727554500964$$
$$x_{4} = 45.8678544894251$$
$$x_{5} = 14.4475895488415$$
$$x_{6} = -11.7989647636976$$
$$x_{7} = 68.3611274829809$$
$$x_{8} = 96.1343777924765$$
$$x_{9} = 99.7779466705706$$
$$x_{10} = -8.15950719268422$$
$$x_{11} = -89.8511260956203$$
$$x_{12} = -96.1343777924765$$
$$x_{13} = -97.7839596791271$$
$$x_{14} = 58.4346533077773$$
$$x_{15} = 89.8511260956203$$
$$x_{16} = -55.7941187699105$$
$$x_{17} = -24.3737285159504$$
$$x_{18} = -9.81341609268515$$
$$x_{19} = -18.0877983628054$$
$$x_{20} = 112.344534664003$$
$$x_{21} = -1.83727554500964$$
$$x_{22} = 22.383498752184$$
$$x_{23} = 30.6585364459467$$
$$x_{24} = -62.0776553750778$$
$$x_{25} = -39.5843530220467$$
$$x_{26} = -5.49740887315255$$
$$x_{27} = -60.0844309903617$$
$$x_{28} = 74.6445513627976$$
$$x_{29} = 60.0844309903617$$
$$x_{30} = 66.3677186539667$$
$$x_{31} = 62.0776553750778$$
$$x_{32} = 52.1512797139686$$
$$x_{33} = 9.81341609268515$$
$$x_{34} = -53.8011194230872$$
$$x_{35} = -68.3611274829809$$
$$x_{36} = 18.0877983628054$$
$$x_{37} = 55.7941187699105$$
$$x_{38} = 11.7989647636976$$
$$x_{39} = -52.1512797139686$$
$$x_{40} = -45.8678544894251$$
$$x_{41} = 24.3737285159504$$
$$x_{42} = -83.567864410722$$
$$x_{43} = 8.15950719268422$$
$$x_{44} = -93.4946307809577$$
$$x_{45} = -16.0991815919966$$
$$x_{46} = 53.8011194230872$$
$$x_{47} = 16.0991815919966$$
$$x_{48} = 91.5007287487714$$
Puntos máximos de la función:
$$x_{48} = -79.7927195468083$$
$$x_{48} = -29.5252949847213$$
$$x_{48} = -69.8633281463648$$
$$x_{48} = -19.5906721136269$$
$$x_{48} = -35.8090209187912$$
$$x_{48} = 86.0759881988554$$
$$x_{48} = 71.8604747646449$$
$$x_{48} = 76.1467434060946$$
$$x_{48} = -25.8762837653952$$
$$x_{48} = -86.0759881988554$$
$$x_{48} = -34.1603645737713$$
$$x_{48} = -7.00622456665035$$
$$x_{48} = -82.4301235713939$$
$$x_{48} = 25.8762837653952$$
$$x_{48} = 42.0925855916645$$
$$x_{48} = -21.5928861245147$$
$$x_{48} = 7.00622456665035$$
$$x_{48} = -38.4451217685033$$
$$x_{48} = -13.3027521230436$$
$$x_{48} = 94.9968064998663$$
$$x_{48} = 79.7927195468083$$
$$x_{48} = 40.4438455565141$$
$$x_{48} = 96.9934503588524$$
$$x_{48} = 38.4451217685033$$
$$x_{48} = -71.8604747646449$$
$$x_{48} = -78.1437327982763$$
$$x_{48} = -57.2963460877875$$
$$x_{48} = -65.577202792437$$
$$x_{48} = 19.5906721136269$$
$$x_{48} = 69.8633281463648$$
$$x_{48} = 4.37500995127285$$
$$x_{48} = 63.5798673727277$$
$$x_{48} = 34.1603645737713$$
$$x_{48} = 27.8767502519047$$
$$x_{48} = 82.4301235713939$$
$$x_{48} = -27.8767502519047$$
$$x_{48} = -42.0925855916645$$
$$x_{48} = 35.8090209187912$$
$$x_{48} = 32.1609444948825$$
$$x_{48} = -73.5094366544198$$
$$x_{48} = 78.1437327982763$$
$$x_{48} = 84.4269800057337$$
$$x_{48} = 48.3760517934113$$
$$x_{48} = -53.0105972030033$$
$$x_{48} = 92.3592455156957$$
$$x_{48} = -76.1467434060946$$
$$x_{48} = -63.5798673727277$$
$$x_{48} = -32.1609444948825$$
Decrece en los intervalos
$$\left[112.344534664003, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.7779466705706\right]$$