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$$- \frac{\sin{\left(x \right)}}{x} - \frac{\cos{\left(x \right)}}{x^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 47.1026627703624$$
$$x_{2} = -84.811211299318$$
$$x_{3} = 25.0929104121121$$
$$x_{4} = 81.6691650818489$$
$$x_{5} = 59.6735041304405$$
$$x_{6} = -28.2389365752603$$
$$x_{7} = -43.9595528888955$$
$$x_{8} = -37.672573565113$$
$$x_{9} = -169.640108529775$$
$$x_{10} = -18.7964043662102$$
$$x_{11} = 91.0952098694071$$
$$x_{12} = 87.9532251106725$$
$$x_{13} = 56.5309801938186$$
$$x_{14} = 100.521017074687$$
$$x_{15} = 69.100567727981$$
$$x_{16} = -91.0952098694071$$
$$x_{17} = 12.4864543952238$$
$$x_{18} = -47.1026627703624$$
$$x_{19} = 50.2455828375744$$
$$x_{20} = -2.79838604578389$$
$$x_{21} = 31.3840740178899$$
$$x_{22} = 62.8159348889734$$
$$x_{23} = 6.12125046689807$$
$$x_{24} = 34.5285657554621$$
$$x_{25} = 78.5270825679419$$
$$x_{26} = -97.3791034786112$$
$$x_{27} = -135.08108127842$$
$$x_{28} = 94.2371684817036$$
$$x_{29} = 40.8162093266346$$
$$x_{30} = 53.3883466217256$$
$$x_{31} = -15.644128370333$$
$$x_{32} = -34.5285657554621$$
$$x_{33} = -25.0929104121121$$
$$x_{34} = -75.3849592185347$$
$$x_{35} = 84.811211299318$$
$$x_{36} = -62.8159348889734$$
$$x_{37} = -65.9582857893902$$
$$x_{38} = -94.2371684817036$$
$$x_{39} = -9.31786646179107$$
$$x_{40} = 18.7964043662102$$
$$x_{41} = -81.6691650818489$$
$$x_{42} = -53.3883466217256$$
$$x_{43} = 2.79838604578389$$
$$x_{44} = -72.2427897046973$$
$$x_{45} = -78.5270825679419$$
$$x_{46} = 97.3791034786112$$
$$x_{47} = -56.5309801938186$$
$$x_{48} = 75.3849592185347$$
$$x_{49} = -31.3840740178899$$
$$x_{50} = -21.945612879981$$
$$x_{51} = -100.521017074687$$
$$x_{52} = -69.100567727981$$
$$x_{53} = -87.9532251106725$$
$$x_{54} = -40.8162093266346$$
$$x_{55} = 65.9582857893902$$
$$x_{56} = -12.4864543952238$$
$$x_{57} = 9.31786646179107$$
$$x_{58} = 15.644128370333$$
$$x_{59} = -109.946647805931$$
$$x_{60} = 43.9595528888955$$
$$x_{61} = -59.6735041304405$$
$$x_{62} = 28.2389365752603$$
$$x_{63} = 21.945612879981$$
$$x_{64} = -50.2455828375744$$
$$x_{65} = 197.91528455229$$
$$x_{66} = -6.12125046689807$$
$$x_{67} = 72.2427897046973$$
$$x_{68} = 37.672573565113$$
Signos de extremos en los puntos:
(47.10266277036235, -0.0212254394164143)
(-84.81121129931802, 0.0117900744410766)
(25.092910412112097, 0.0398202855500511)
(81.66916508184887, 0.0122436055670467)
(59.67350413044053, -0.0167555036571887)
(-28.238936575260272, 0.0353899155541688)
(-43.959552888895495, -0.0227423004725314)
(-37.67257356511297, -0.0265351630103045)
(-169.6401085297751, -0.00589472993500857)
(-18.796404366210158, -0.0531265325613881)
(91.09520986940714, -0.0109768642483425)
(87.95322511067255, 0.0113689449158811)
(56.53098019381864, 0.0176866485521696)
(100.52101707468658, 0.00994767611536293)
(69.10056772798097, 0.0144701459746764)
(-91.09520986940714, 0.0109768642483425)
(12.486454395223781, 0.0798311807800032)
(-47.10266277036235, 0.0212254394164143)
(50.24558283757444, 0.0198983065303553)
(-2.798386045783887, 0.336508416918395)
(31.38407401788986, 0.0318471321112693)
(62.81593488897342, 0.015917510583426)
(6.1212504668980685, 0.161228034325064)
(34.52856575546206, -0.0289493889114503)
(78.52708256794193, -0.0127334276777468)
(-97.3791034786112, 0.0102686022030809)
(-135.0810812784199, 0.00740275832666827)
(94.23716848170359, 0.01061092686295)
(40.81620932663458, -0.0244927205346957)
(53.38834662172563, -0.0187273944640866)
(-15.644128370333028, 0.0637915530395936)
(-34.52856575546206, 0.0289493889114503)
(-25.092910412112097, -0.0398202855500511)
(-75.38495921853475, -0.0132640786518247)
(84.81121129931802, -0.0117900744410766)
(-62.81593488897342, -0.015917510583426)
(-65.95828578939016, 0.0151593553168405)
(-94.23716848170359, -0.01061092686295)
(-9.317866461791066, 0.106707947715237)
(18.796404366210158, 0.0531265325613881)
(-81.66916508184887, -0.0122436055670467)
(-53.38834662172563, 0.0187273944640866)
(2.798386045783887, -0.336508416918395)
(-72.24278970469729, 0.0138408859131547)
(-78.52708256794193, 0.0127334276777468)
(97.3791034786112, -0.0102686022030809)
(-56.53098019381864, -0.0176866485521696)
(75.38495921853475, 0.0132640786518247)
(-31.38407401788986, -0.0318471321112693)
(-21.945612879981045, 0.0455199604051285)
(-100.52101707468658, -0.00994767611536293)
(-69.10056772798097, -0.0144701459746764)
(-87.95322511067255, -0.0113689449158811)
(-40.81620932663458, 0.0244927205346957)
(65.95828578939016, -0.0151593553168405)
(-12.486454395223781, -0.0798311807800032)
(9.317866461791066, -0.106707947715237)
(15.644128370333028, -0.0637915530395936)
(-109.94664780593057, 0.00909494432157336)
(43.959552888895495, 0.0227423004725314)
(-59.67350413044053, 0.0167555036571887)
(28.238936575260272, -0.0353899155541688)
(21.945612879981045, -0.0455199604051285)
(-50.24558283757444, -0.0198983065303553)
(197.91528455229027, -0.00505260236866135)
(-6.1212504668980685, -0.161228034325064)
(72.24278970469729, -0.0138408859131547)
(37.67257356511297, 0.0265351630103045)
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:
$$x_{1} = 47.1026627703624$$
$$x_{2} = 59.6735041304405$$
$$x_{3} = -43.9595528888955$$
$$x_{4} = -37.672573565113$$
$$x_{5} = -169.640108529775$$
$$x_{6} = -18.7964043662102$$
$$x_{7} = 91.0952098694071$$
$$x_{8} = 34.5285657554621$$
$$x_{9} = 78.5270825679419$$
$$x_{10} = 40.8162093266346$$
$$x_{11} = 53.3883466217256$$
$$x_{12} = -25.0929104121121$$
$$x_{13} = -75.3849592185347$$
$$x_{14} = 84.811211299318$$
$$x_{15} = -62.8159348889734$$
$$x_{16} = -94.2371684817036$$
$$x_{17} = -81.6691650818489$$
$$x_{18} = 2.79838604578389$$
$$x_{19} = 97.3791034786112$$
$$x_{20} = -56.5309801938186$$
$$x_{21} = -31.3840740178899$$
$$x_{22} = -100.521017074687$$
$$x_{23} = -69.100567727981$$
$$x_{24} = -87.9532251106725$$
$$x_{25} = 65.9582857893902$$
$$x_{26} = -12.4864543952238$$
$$x_{27} = 9.31786646179107$$
$$x_{28} = 15.644128370333$$
$$x_{29} = 28.2389365752603$$
$$x_{30} = 21.945612879981$$
$$x_{31} = -50.2455828375744$$
$$x_{32} = 197.91528455229$$
$$x_{33} = -6.12125046689807$$
$$x_{34} = 72.2427897046973$$
Puntos máximos de la función:
$$x_{34} = -84.811211299318$$
$$x_{34} = 25.0929104121121$$
$$x_{34} = 81.6691650818489$$
$$x_{34} = -28.2389365752603$$
$$x_{34} = 87.9532251106725$$
$$x_{34} = 56.5309801938186$$
$$x_{34} = 100.521017074687$$
$$x_{34} = 69.100567727981$$
$$x_{34} = -91.0952098694071$$
$$x_{34} = 12.4864543952238$$
$$x_{34} = -47.1026627703624$$
$$x_{34} = 50.2455828375744$$
$$x_{34} = -2.79838604578389$$
$$x_{34} = 31.3840740178899$$
$$x_{34} = 62.8159348889734$$
$$x_{34} = 6.12125046689807$$
$$x_{34} = -97.3791034786112$$
$$x_{34} = -135.08108127842$$
$$x_{34} = 94.2371684817036$$
$$x_{34} = -15.644128370333$$
$$x_{34} = -34.5285657554621$$
$$x_{34} = -65.9582857893902$$
$$x_{34} = -9.31786646179107$$
$$x_{34} = 18.7964043662102$$
$$x_{34} = -53.3883466217256$$
$$x_{34} = -72.2427897046973$$
$$x_{34} = -78.5270825679419$$
$$x_{34} = 75.3849592185347$$
$$x_{34} = -21.945612879981$$
$$x_{34} = -40.8162093266346$$
$$x_{34} = -109.946647805931$$
$$x_{34} = 43.9595528888955$$
$$x_{34} = -59.6735041304405$$
$$x_{34} = 37.672573565113$$
Decrece en los intervalos
$$\left[197.91528455229, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -169.640108529775\right]$$