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$$\frac{\sin{\left(x \right)} \operatorname{sign}{\left(x - 7 \right)} + \cos{\left(x \right)} \left|{x - 7}\right|}{- 42 x + \left(x^{3} - x^{2}\right)} + \frac{\left(- 3 x^{2} + 2 x + 42\right) \sin{\left(x \right)} \left|{x - 7}\right|}{\left(- 42 x + \left(x^{3} - x^{2}\right)\right)^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 14.016453093047$$
$$x_{2} = 26.6354574736327$$
$$x_{3} = -42.3604359016656$$
$$x_{4} = -61.2266303830853$$
$$x_{5} = 54.9432759558649$$
$$x_{6} = -76.9419337897232$$
$$x_{7} = 83.2289873280938$$
$$x_{8} = 64.3729136691542$$
$$x_{9} = -64.369993732511$$
$$x_{10} = -13.9420438831868$$
$$x_{11} = 70.6586440153259$$
$$x_{12} = 29.7836865268305$$
$$x_{13} = 36.0768763247778$$
$$x_{14} = -32.919299828625$$
$$x_{15} = -51.7951587828659$$
$$x_{16} = 80.0865146143929$$
$$x_{17} = -95.7970043123775$$
$$x_{18} = -48.6507135445059$$
$$x_{19} = 42.3672514439482$$
$$x_{20} = 98.9405348240556$$
$$x_{21} = -26.6176780767465$$
$$x_{22} = -89.512248812553$$
$$x_{23} = -54.939255228509$$
$$x_{24} = -73.7991351656219$$
$$x_{25} = -67.5131834431642$$
$$x_{26} = 10.8451478646932$$
$$x_{27} = 17.1777448915114$$
$$x_{28} = 95.7983167647195$$
$$x_{29} = 23.485599433969$$
$$x_{30} = -17.1316249007671$$
$$x_{31} = -10.6983350907497$$
$$x_{32} = -39.2143570561418$$
$$x_{33} = 76.9439724516011$$
$$x_{34} = 149.212507171659$$
$$x_{35} = -58.0830633547885$$
$$x_{36} = 73.8013522673499$$
$$x_{37} = 48.6558568457253$$
$$x_{38} = -86.3697819811265$$
$$x_{39} = 89.5137528406582$$
$$x_{40} = 92.6560575082574$$
$$x_{41} = 146.07063730353$$
$$x_{42} = 7.65282388683543$$
$$x_{43} = -20.3017332658149$$
$$x_{44} = -92.6546541656147$$
$$x_{45} = -23.4623877903398$$
$$x_{46} = 67.5158358436257$$
$$x_{47} = 51.799688855665$$
$$x_{48} = 58.0866563372379$$
$$x_{49} = -98.9393046989388$$
$$x_{50} = -7.00360982839007$$
$$x_{51} = -80.0846336468534$$
$$x_{52} = 20.3334172344451$$
$$x_{53} = -70.6562239130531$$
$$x_{54} = 86.3713979521823$$
$$x_{55} = -45.5058407635821$$
$$x_{56} = -83.2272464103056$$
$$x_{57} = -36.0674064816266$$
$$x_{58} = 61.229860625401$$
$$x_{59} = 4.39955639048156$$
$$x_{60} = 39.2223354468251$$
$$x_{61} = -0.53859704600586$$
$$x_{62} = 45.5117316703557$$
$$x_{63} = -29.7696122729537$$
$$x_{64} = 32.9307280335231$$
Signos de extremos en los puntos:
(14.016453093047046, 0.00353836638676052)
(26.635457473632727, 0.00114773837094551)
(-42.36043590166557, -0.000648401503723408)
(-61.2266303830853, -0.000295565464505356)
(54.943275955864934, -0.000298469329531356)
(-76.94193378972315, 0.000183136347207666)
(83.22898732809384, 0.000134617758893156)
(64.3729136691542, 0.000220647630392954)
(-64.369993732511, 0.00026600835720957)
(-13.942043883186841, 0.00885973686894448)
(70.65864401532592, 0.000184549555812555)
(29.783686526830525, -0.000936518043257309)
(36.07687632477782, -0.000657889201238731)
(-32.919299828625, 0.00112589471326007)
(-51.79515878286592, 0.000421234558721923)
(80.08651461439288, -0.000145003794971144)
(-95.79700431237748, 0.000116221150161414)
(-48.65071354450594, -0.000481464727267868)
(42.36725144394817, -0.000487520616923897)
(98.94053482405558, -9.62938965440315e-5)
(-26.61767807674652, 0.00181546269768391)
(-89.51224881255295, 0.000133736838158639)
(-54.939255228508955, -0.000371651579986343)
(-73.79913516562186, -0.000199779414092476)
(-67.51318344316424, -0.000240676511514366)
(10.845147864693171, -0.00541199541314581)
(17.17774489151138, -0.00249886693392249)
(95.79831676471952, 0.000102520896200968)
(23.485599433968986, -0.00143986404452334)
(-17.131624900767097, -0.00518710311846635)
(-10.698335090749685, -0.0190223979798966)
(-39.21435705614177, 0.000766581979435786)
(76.94397245160107, 0.000156640592186579)
(149.21250717165935, -4.31748206644313e-5)
(-58.08306335478854, 0.00033034381495904)
(73.80135226734993, -0.000169737488684419)
(48.655856845725324, -0.000375751471215424)
(-86.36978198112651, -0.000144019112679707)
(89.51375284065819, 0.000116934496654864)
(92.65605750825736, -0.000109372297110004)
(146.07063730352974, 4.50145170384153e-5)
(7.652823886835433, 0.00937797661890569)
(-20.3017332658149, 0.00341991769457062)
(-92.65465416561467, -0.000124518130309472)
(-23.46238779033976, -0.00242866893320253)
(67.51583584362565, -0.000201390142707435)
(51.799688855664975, 0.000333777119038759)
(58.08665633723788, 0.000268486368595447)
(-98.93930469893877, -0.000108726947503291)
(-7.003609828390072, 0.0938560163001481)
(-80.08463364685336, -0.000168490763387854)
(20.33341723444505, 0.00186054082102996)
(-70.65622391305313, 0.000218800755314414)
(86.37139795218235, -0.000125309376682272)
(-45.50584076358215, 0.000555631062996273)
(-83.22724641030557, 0.000155535180840878)
(-36.06740648162657, -0.00092041358758118)
(61.22986062540101, -0.000242808092240104)
(4.399556390481556, 0.0207955020018686)
(39.22233544682508, 0.000563147211410448)
(-0.5385970460058597, -0.174378019814268)
(45.511731670355665, 0.000426185714857683)
(-29.76961227295372, -0.00140917575807083)
(32.930728033523096, 0.000778798198554037)
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:
$$x_{1} = -42.3604359016656$$
$$x_{2} = -61.2266303830853$$
$$x_{3} = 54.9432759558649$$
$$x_{4} = 29.7836865268305$$
$$x_{5} = 36.0768763247778$$
$$x_{6} = 80.0865146143929$$
$$x_{7} = -48.6507135445059$$
$$x_{8} = 42.3672514439482$$
$$x_{9} = 98.9405348240556$$
$$x_{10} = -54.939255228509$$
$$x_{11} = -73.7991351656219$$
$$x_{12} = -67.5131834431642$$
$$x_{13} = 10.8451478646932$$
$$x_{14} = 17.1777448915114$$
$$x_{15} = 23.485599433969$$
$$x_{16} = -17.1316249007671$$
$$x_{17} = -10.6983350907497$$
$$x_{18} = 149.212507171659$$
$$x_{19} = 73.8013522673499$$
$$x_{20} = 48.6558568457253$$
$$x_{21} = -86.3697819811265$$
$$x_{22} = 92.6560575082574$$
$$x_{23} = -92.6546541656147$$
$$x_{24} = -23.4623877903398$$
$$x_{25} = 67.5158358436257$$
$$x_{26} = -98.9393046989388$$
$$x_{27} = -80.0846336468534$$
$$x_{28} = 86.3713979521823$$
$$x_{29} = -36.0674064816266$$
$$x_{30} = 61.229860625401$$
$$x_{31} = -0.53859704600586$$
$$x_{32} = -29.7696122729537$$
Puntos máximos de la función:
$$x_{32} = 14.016453093047$$
$$x_{32} = 26.6354574736327$$
$$x_{32} = -76.9419337897232$$
$$x_{32} = 83.2289873280938$$
$$x_{32} = 64.3729136691542$$
$$x_{32} = -64.369993732511$$
$$x_{32} = -13.9420438831868$$
$$x_{32} = 70.6586440153259$$
$$x_{32} = -32.919299828625$$
$$x_{32} = -51.7951587828659$$
$$x_{32} = -95.7970043123775$$
$$x_{32} = -26.6176780767465$$
$$x_{32} = -89.512248812553$$
$$x_{32} = 95.7983167647195$$
$$x_{32} = -39.2143570561418$$
$$x_{32} = 76.9439724516011$$
$$x_{32} = -58.0830633547885$$
$$x_{32} = 89.5137528406582$$
$$x_{32} = 146.07063730353$$
$$x_{32} = 7.65282388683543$$
$$x_{32} = -20.3017332658149$$
$$x_{32} = 51.799688855665$$
$$x_{32} = 58.0866563372379$$
$$x_{32} = -7.00360982839007$$
$$x_{32} = 20.3334172344451$$
$$x_{32} = -70.6562239130531$$
$$x_{32} = -45.5058407635821$$
$$x_{32} = -83.2272464103056$$
$$x_{32} = 4.39955639048156$$
$$x_{32} = 39.2223354468251$$
$$x_{32} = 45.5117316703557$$
$$x_{32} = 32.9307280335231$$
Decrece en los intervalos
$$\left[149.212507171659, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.9393046989388\right]$$