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$$\frac{\left(x - 2 \cos{\left(x \right)}\right) \left(5 \sin{\left(x \right)} - 1\right)}{\left(x + 5 \cos{\left(x \right)}\right)^{2}} + \frac{2 \sin{\left(x \right)} + 1}{x + 5 \cos{\left(x \right)}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -69.100567727981$$
$$x_{2} = -12.4864543952238$$
$$x_{3} = 235.615204836452$$
$$x_{4} = -2.79838604578389$$
$$x_{5} = 91.0952098694071$$
$$x_{6} = 78.5270825679419$$
$$x_{7} = -72.2427897046973$$
$$x_{8} = 15.644128370333$$
$$x_{9} = -43.9595528888955$$
$$x_{10} = 81.6691650818489$$
$$x_{11} = 50.2455828375744$$
$$x_{12} = -62.8159348889734$$
$$x_{13} = 6.12125046689807$$
$$x_{14} = -163.356696489782$$
$$x_{15} = -84.811211299318$$
$$x_{16} = 62.8159348889734$$
$$x_{17} = 56.5309801938186$$
$$x_{18} = -28.2389365752603$$
$$x_{19} = -56.5309801938186$$
$$x_{20} = -25.0929104121121$$
$$x_{21} = -59.6735041304405$$
$$x_{22} = 69.100567727981$$
$$x_{23} = -47.1026627703624$$
$$x_{24} = 100.521017074687$$
$$x_{25} = 72.2427897046973$$
$$x_{26} = -40.8162093266346$$
$$x_{27} = 47.1026627703624$$
$$x_{28} = -78.5270825679419$$
$$x_{29} = 97.3791034786112$$
$$x_{30} = -31.3840740178899$$
$$x_{31} = -37.672573565113$$
$$x_{32} = 18.7964043662102$$
$$x_{33} = -75.3849592185347$$
$$x_{34} = 40.8162093266346$$
$$x_{35} = -34.5285657554621$$
$$x_{36} = -53.3883466217256$$
$$x_{37} = 34.5285657554621$$
$$x_{38} = 37.672573565113$$
$$x_{39} = -100.521017074687$$
$$x_{40} = -6.12125046689807$$
$$x_{41} = -65.9582857893902$$
$$x_{42} = 59.6735041304405$$
$$x_{43} = -91.0952098694071$$
$$x_{44} = 75.3849592185347$$
$$x_{45} = 65.9582857893902$$
$$x_{46} = 84.811211299318$$
$$x_{47} = -87.9532251106725$$
$$x_{48} = 150.789815721919$$
$$x_{49} = 25.0929104121121$$
$$x_{50} = 94.2371684817036$$
$$x_{51} = -94.2371684817036$$
$$x_{52} = 2.79838604578389$$
$$x_{53} = 28.2389365752603$$
$$x_{54} = -21.945612879981$$
$$x_{55} = 53.3883466217256$$
$$x_{56} = -50.2455828375744$$
$$x_{57} = 87.9532251106725$$
$$x_{58} = 43.9595528888955$$
$$x_{59} = -81.6691650818489$$
$$x_{60} = -9.31786646179107$$
$$x_{61} = 12.4864543952238$$
$$x_{62} = -97.3791034786112$$
$$x_{63} = 31.3840740178899$$
$$x_{64} = -15.644128370333$$
$$x_{65} = 21.945612879981$$
Signos de extremos en los puntos:
(-69.10056772798097, 1.10919107585662)
(-12.486454395223781, 1.93005534894535)
(235.61520483645214, 1.03035331340083)
(-2.798386045783887, 0.121893023800573)
(91.09520986940714, 1.08130015347894)
(78.52708256794193, 1.09519477249879)
(-72.24278970469729, 0.909384773656961)
(15.644128370333028, 1.65567280332762)
(-43.959552888895495, 1.17962108696221)
(81.66916508184887, 0.919238802280663)
(50.24558283757444, 0.873315854123452)
(-62.81593488897342, 1.12105722215852)
(6.1212504668980685, 0.375133636909343)
(-163.35669648978208, 1.04420314934587)
(-84.81121129931802, 0.922063844299278)
(62.81593488897342, 0.896791535079383)
(56.53098019381864, 0.886252518757009)
(-28.238936575260272, 0.789515699263529)
(-56.53098019381864, 1.1358173045276)
(-25.092910412112097, 1.34803657854872)
(-59.67350413044053, 0.891778041506324)
(69.10056772798097, 0.905543010322104)
(-47.10266277036235, 0.865677223824851)
(100.52101707468658, 0.93366563126969)
(72.24278970469729, 1.10408966745177)
(-40.81620932663458, 0.847256478219416)
(47.10266277036235, 1.1662183651073)
(-78.52708256794193, 0.916201233894233)
(97.3791034786112, 1.07577050109387)
(-31.38407401788986, 1.26515149884497)
(-37.67257356511297, 1.21415999264815)
(18.796404366210158, 0.706166142153014)
(-75.38495921853475, 1.09944369555941)
(40.81620932663458, 1.19537542176621)
(-34.52856575546206, 0.822977712809246)
(-53.38834662172563, 0.88013228834934)
(34.52856575546206, 1.2369424126468)
(37.67257356511297, 0.836011205850915)
(-100.52101707468658, 1.07327848603399)
(-6.1212504668980685, 6.82171277947367)
(-65.95828578939016, 0.901361028841512)
(59.67350413044053, 1.12801314958011)
(-91.09520986940714, 0.927159738569296)
(75.38495921853475, 0.912926216970315)
(65.95828578939016, 1.1148183478799)
(84.81121129931802, 1.0877004980927)
(-87.95322511067255, 1.0843791223833)
(150.78981572191927, 0.955068622538491)
(25.092910412112097, 0.767540933235872)
(94.23716848170359, 0.929465684286188)
(-94.23716848170359, 1.07843798674372)
(2.798386045783887, -2.45115557209475)
(28.238936575260272, 1.30098934668939)
(-21.945612879981045, 0.740436808225191)
(53.38834662172563, 1.14463493903124)
(-50.24558283757444, 1.15467721932994)
(87.95322511067255, 0.924697912039217)
(43.959552888895495, 0.857058041539486)
(-81.66916508184887, 1.09129408266298)
(-9.317866461791066, 0.512920588089769)
(12.486454395223781, 0.600603289524513)
(-97.3791034786112, 0.931630101053182)
(31.38407401788986, 0.807692316269933)
(-15.644128370333028, 0.661444146992734)
(21.945612879981045, 1.41253190207815)
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:
$$x_{1} = -2.79838604578389$$
$$x_{2} = -72.2427897046973$$
$$x_{3} = 81.6691650818489$$
$$x_{4} = 50.2455828375744$$
$$x_{5} = 6.12125046689807$$
$$x_{6} = -84.811211299318$$
$$x_{7} = 62.8159348889734$$
$$x_{8} = 56.5309801938186$$
$$x_{9} = -28.2389365752603$$
$$x_{10} = -59.6735041304405$$
$$x_{11} = 69.100567727981$$
$$x_{12} = -47.1026627703624$$
$$x_{13} = 100.521017074687$$
$$x_{14} = -40.8162093266346$$
$$x_{15} = -78.5270825679419$$
$$x_{16} = 18.7964043662102$$
$$x_{17} = -34.5285657554621$$
$$x_{18} = -53.3883466217256$$
$$x_{19} = 37.672573565113$$
$$x_{20} = -65.9582857893902$$
$$x_{21} = -91.0952098694071$$
$$x_{22} = 75.3849592185347$$
$$x_{23} = 150.789815721919$$
$$x_{24} = 25.0929104121121$$
$$x_{25} = 94.2371684817036$$
$$x_{26} = -21.945612879981$$
$$x_{27} = 87.9532251106725$$
$$x_{28} = 43.9595528888955$$
$$x_{29} = -9.31786646179107$$
$$x_{30} = 12.4864543952238$$
$$x_{31} = -97.3791034786112$$
$$x_{32} = 31.3840740178899$$
$$x_{33} = -15.644128370333$$
Puntos máximos de la función:
$$x_{33} = -69.100567727981$$
$$x_{33} = -12.4864543952238$$
$$x_{33} = 235.615204836452$$
$$x_{33} = 91.0952098694071$$
$$x_{33} = 78.5270825679419$$
$$x_{33} = 15.644128370333$$
$$x_{33} = -43.9595528888955$$
$$x_{33} = -62.8159348889734$$
$$x_{33} = -163.356696489782$$
$$x_{33} = -56.5309801938186$$
$$x_{33} = -25.0929104121121$$
$$x_{33} = 72.2427897046973$$
$$x_{33} = 47.1026627703624$$
$$x_{33} = 97.3791034786112$$
$$x_{33} = -31.3840740178899$$
$$x_{33} = -37.672573565113$$
$$x_{33} = -75.3849592185347$$
$$x_{33} = 40.8162093266346$$
$$x_{33} = 34.5285657554621$$
$$x_{33} = -100.521017074687$$
$$x_{33} = -6.12125046689807$$
$$x_{33} = 59.6735041304405$$
$$x_{33} = 65.9582857893902$$
$$x_{33} = 84.811211299318$$
$$x_{33} = -87.9532251106725$$
$$x_{33} = -94.2371684817036$$
$$x_{33} = 2.79838604578389$$
$$x_{33} = 28.2389365752603$$
$$x_{33} = 53.3883466217256$$
$$x_{33} = -50.2455828375744$$
$$x_{33} = -81.6691650818489$$
$$x_{33} = 21.945612879981$$
Decrece en los intervalos
$$\left[150.789815721919, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -97.3791034786112\right]$$