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$$\frac{\left(- 600 \left(\frac{x}{3} + 5\right)^{3} \sin{\left(x \right)} \cos{\left(x \right)} - 300 \left(\frac{x}{3} + 5\right)^{2} \sin^{2}{\left(x \right)}\right) \left(10 \log{\left(\left(x + 5\right)^{2} \right)} + 5\right)}{\left(\left(\frac{x}{3} + 5\right)^{3} \cdot 300 \sin^{2}{\left(x \right)} + 7\right)^{2}} + \frac{10 \left(2 x + 10\right)}{\left(x + 5\right)^{2} \left(\left(\frac{x}{3} + 5\right)^{3} \cdot 300 \sin^{2}{\left(x \right)} + 7\right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -83.2727874973584$$
$$x_{2} = 95.8310897856742$$
$$x_{3} = -39.3276550335433$$
$$x_{4} = 67.560889266173$$
$$x_{5} = 70.7018927392371$$
$$x_{6} = -36.1946474863643$$
$$x_{7} = -33.0646303012334$$
$$x_{8} = -80.1321829512664$$
$$x_{9} = -76.9916785989834$$
$$x_{10} = -13.3747250275116$$
$$x_{11} = -17.7653088173889$$
$$x_{12} = 51.8566623761523$$
$$x_{13} = -26.8229665926194$$
$$x_{14} = -29.9394719831602$$
$$x_{15} = -70.7110384364717$$
$$x_{16} = -45.5989676823298$$
$$x_{17} = -48.7362866272455$$
$$x_{18} = 98.9723464721323$$
$$x_{19} = 33.0145731944214$$
$$x_{20} = 89.5486353126801$$
$$x_{21} = -89.5542464262811$$
$$x_{22} = 20.4570078590476$$
$$x_{23} = 86.4074410623569$$
$$x_{24} = 73.8429364199345$$
$$x_{25} = 61.2790218683648$$
$$x_{26} = -61.2913758541987$$
$$x_{27} = 29.8747766975811$$
$$x_{28} = 1.62571032846112$$
$$x_{29} = 83.2662715234224$$
$$x_{30} = -73.8512905214352$$
$$x_{31} = 23.5959433880961$$
$$x_{32} = -95.8359736534704$$
$$x_{33} = -23.7240583014129$$
$$x_{34} = 14.1804076736695$$
$$x_{35} = 36.1545700235369$$
$$x_{36} = -51.8743358627882$$
$$x_{37} = 76.9840163471242$$
$$x_{38} = -1.55694240961403$$
$$x_{39} = 92.6898521764109$$
$$x_{40} = 4.76797091488681$$
$$x_{41} = 64.4199305552876$$
$$x_{42} = -42.4626302394958$$
$$x_{43} = 42.4350478667949$$
$$x_{44} = -58.1519860167309$$
$$x_{45} = 80.1251290602825$$
$$x_{46} = -92.6950805761372$$
$$x_{47} = 17.3184749964622$$
$$x_{48} = 54.9973800268033$$
$$x_{49} = -86.4134789918403$$
$$x_{50} = -7.49785025429627$$
$$x_{51} = -64.4310462520234$$
$$x_{52} = 45.5754833964017$$
$$x_{53} = -67.5709467881468$$
$$x_{54} = -10.6252638359778$$
$$x_{55} = -55.0129430380975$$
$$x_{56} = 39.2947365767581$$
$$x_{57} = 26.7352181076125$$
$$x_{58} = 48.7160262498578$$
$$x_{59} = -15.6656497396484$$
$$x_{60} = 7.90562211078873$$
$$x_{61} = -20.6690930522039$$
$$x_{62} = 58.1381693149369$$
$$x_{63} = -98.9769190377406$$
$$x_{64} = 11.042834018472$$
Signos de extremos en los puntos:
(-83.27278749735838, -2.60876054959853e-5)
(95.83108978567422, 6.43131332192689e-6)
(-39.32765503354333, -0.000474913916560134)
(67.56088926617302, 1.4507467909696e-5)
(70.70189273923708, 1.30910673431019e-5)
(-36.19464748636427, -0.000700794081610218)
(-33.06463030123335, -0.00110128677978278)
(-80.13218295126643, -2.97806040251224e-5)
(-76.99167859898338, -3.42187254884363e-5)
(-13.374725027511552, 1.48703204380161)
(-17.76530881738893, -0.316751283456898)
(51.85666237615227, 2.58540901594878e-5)
(-26.822966592619352, -0.0036838405646894)
(-29.93947198316022, -0.00188845157155698)
(-70.71103843647168, -4.62004275354222e-5)
(-45.59896768232981, -0.000248934113473829)
(-48.73628662724555, -0.000189168265992073)
(98.97234647213226, 5.95132014594302e-6)
(33.01457319442138, 6.32718558258914e-5)
(89.5486353126801, 7.56056840101053e-6)
(-89.55424642628113, -2.03676922605015e-5)
(20.45700785904762, 0.000140992043746909)
(86.4074410623569, 8.2269016993418e-6)
(73.84293641993447, 1.18552521496999e-5)
(61.27902186836483, 1.80284503322335e-5)
(-61.29137585419873, -7.77450917857276e-5)
(29.87477669758108, 7.57930653886531e-5)
(1.6257103284611183, 0.000840989400699168)
(83.26627152342245, 8.97508828438985e-6)
(-73.85129052143519, -3.96022732594355e-5)
(23.595943388096135, 0.000112938247003005)
(-95.83597365347038, -1.62222548986629e-5)
(-23.72405830141292, -0.00885941658165251)
(14.180407673669468, 0.000232529037135901)
(36.15457002353693, 5.33845281560915e-5)
(-51.87433586278823, -0.000147316137453507)
(76.98401634712424, 1.07719565849402e-5)
(-1.5569424096140303, 0.00110121732983566)
(92.68985217641092, 6.96512010822983e-6)
(4.767970914886807, 0.000591099487611004)
(64.4199305552876, 1.61389889798209e-5)
(-42.46263023949575, -0.00033750581154634)
(42.435047866794875, 3.90620699992672e-5)
(-58.15198601673087, -9.47043749469472e-5)
(80.1251290602825, 9.81821027414399e-6)
(-92.69508057613723, -1.81356163737875e-5)
(17.31847499646221, 0.00017920215327258)
(54.99738002680327, 2.28092784327236e-5)
(-86.41347899184032, -2.29885184762429e-5)
(-7.497850254296271, 0.00564613576665274)
(-64.43104625202345, -6.4654143251366e-5)
(45.57548339640173, 3.38139043165193e-5)
(-67.57094678814683, -5.43807485776194e-5)
(-10.625263835977757, 0.0484960050826985)
(-55.01294303809747, -0.000117095734849419)
(39.29473657675809, 4.5471552816923e-5)
(26.735218107612454, 9.18902101634538e-5)
(48.716026249857805, 2.94743639686436e-5)
(-15.66564973964842, 7.48349266640593)
(7.905622110788726, 0.000421625678784061)
(-20.66909305220391, -0.0316848431855068)
(58.13816931493692, 2.02293790751269e-5)
(-98.97691903774061, -1.45722597118256e-5)
(11.042834018472027, 0.000308976630604364)
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:
$$x_{1} = 95.8310897856742$$
$$x_{2} = 67.560889266173$$
$$x_{3} = 70.7018927392371$$
$$x_{4} = -13.3747250275116$$
$$x_{5} = 51.8566623761523$$
$$x_{6} = 98.9723464721323$$
$$x_{7} = 33.0145731944214$$
$$x_{8} = 89.5486353126801$$
$$x_{9} = 20.4570078590476$$
$$x_{10} = 86.4074410623569$$
$$x_{11} = 73.8429364199345$$
$$x_{12} = 61.2790218683648$$
$$x_{13} = 29.8747766975811$$
$$x_{14} = 1.62571032846112$$
$$x_{15} = 83.2662715234224$$
$$x_{16} = 23.5959433880961$$
$$x_{17} = 14.1804076736695$$
$$x_{18} = 36.1545700235369$$
$$x_{19} = 76.9840163471242$$
$$x_{20} = -1.55694240961403$$
$$x_{21} = 92.6898521764109$$
$$x_{22} = 4.76797091488681$$
$$x_{23} = 64.4199305552876$$
$$x_{24} = 42.4350478667949$$
$$x_{25} = 80.1251290602825$$
$$x_{26} = 17.3184749964622$$
$$x_{27} = 54.9973800268033$$
$$x_{28} = -7.49785025429627$$
$$x_{29} = 45.5754833964017$$
$$x_{30} = -10.6252638359778$$
$$x_{31} = 39.2947365767581$$
$$x_{32} = 26.7352181076125$$
$$x_{33} = 48.7160262498578$$
$$x_{34} = -15.6656497396484$$
$$x_{35} = 7.90562211078873$$
$$x_{36} = 58.1381693149369$$
$$x_{37} = 11.042834018472$$
Puntos máximos de la función:
$$x_{37} = -83.2727874973584$$
$$x_{37} = -39.3276550335433$$
$$x_{37} = -36.1946474863643$$
$$x_{37} = -33.0646303012334$$
$$x_{37} = -80.1321829512664$$
$$x_{37} = -76.9916785989834$$
$$x_{37} = -17.7653088173889$$
$$x_{37} = -26.8229665926194$$
$$x_{37} = -29.9394719831602$$
$$x_{37} = -70.7110384364717$$
$$x_{37} = -45.5989676823298$$
$$x_{37} = -48.7362866272455$$
$$x_{37} = -89.5542464262811$$
$$x_{37} = -61.2913758541987$$
$$x_{37} = -73.8512905214352$$
$$x_{37} = -95.8359736534704$$
$$x_{37} = -23.7240583014129$$
$$x_{37} = -51.8743358627882$$
$$x_{37} = -42.4626302394958$$
$$x_{37} = -58.1519860167309$$
$$x_{37} = -92.6950805761372$$
$$x_{37} = -86.4134789918403$$
$$x_{37} = -64.4310462520234$$
$$x_{37} = -67.5709467881468$$
$$x_{37} = -55.0129430380975$$
$$x_{37} = -20.6690930522039$$
$$x_{37} = -98.9769190377406$$
Decrece en los intervalos
$$\left[98.9723464721323, \infty\right)$$
Crece en los intervalos
$$\left[-17.7653088173889, -15.6656497396484\right]$$