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$$\frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{\sin^{2}{\left(x \right)}}{x^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -50.2654824574367$$
$$x_{2} = 70.6787605627689$$
$$x_{3} = 58.1108600600615$$
$$x_{4} = -20.3958423573092$$
$$x_{5} = -86.3880101981266$$
$$x_{6} = 20.3958423573092$$
$$x_{7} = 51.8266315338985$$
$$x_{8} = 72.2566310325652$$
$$x_{9} = -43.9822971502571$$
$$x_{10} = -37.6991118430775$$
$$x_{11} = 34.5575191894877$$
$$x_{12} = -2678.20755049327$$
$$x_{13} = 4.60421677720058$$
$$x_{14} = -94.2477796076938$$
$$x_{15} = 89.5298059530594$$
$$x_{16} = -21.9911485751286$$
$$x_{17} = 50.2654824574367$$
$$x_{18} = -15.707963267949$$
$$x_{19} = -23.5407082923052$$
$$x_{20} = -87.9645943005142$$
$$x_{21} = 21.9911485751286$$
$$x_{22} = -42.3997088362447$$
$$x_{23} = -72.2566310325652$$
$$x_{24} = 59.6902604182061$$
$$x_{25} = 80.1043708909521$$
$$x_{26} = -29.8283692130955$$
$$x_{27} = -80.1043708909521$$
$$x_{28} = -9.42477796076938$$
$$x_{29} = -51.8266315338985$$
$$x_{30} = 86.3880101981266$$
$$x_{31} = 36.1144715353049$$
$$x_{32} = -65.9734457253857$$
$$x_{33} = 15.707963267949$$
$$x_{34} = 23.5407082923052$$
$$x_{35} = -10.9499436485412$$
$$x_{36} = 28.2743338823081$$
$$x_{37} = 94.2477796076938$$
$$x_{38} = -7.78988375114457$$
$$x_{39} = 37.6991118430775$$
$$x_{40} = -89.5298059530594$$
$$x_{41} = 6.28318530717959$$
$$x_{42} = 26.6848024909251$$
$$x_{43} = -67.5368388204916$$
$$x_{44} = -53.4070751110265$$
$$x_{45} = 29.8283692130955$$
$$x_{46} = -6.28318530717959$$
$$x_{47} = -75.398223686155$$
$$x_{48} = 45.5421150692309$$
$$x_{49} = 42.3997088362447$$
$$x_{50} = 64.3948849627586$$
$$x_{51} = -14.1017251335659$$
$$x_{52} = 14.1017251335659$$
$$x_{53} = -64.3948849627586$$
$$x_{54} = -1740.44233008875$$
$$x_{55} = -62.8318530717959$$
$$x_{56} = -97.3893722612836$$
$$x_{57} = -45.5421150692309$$
$$x_{58} = -125.663706143592$$
$$x_{59} = -83.2461991121237$$
$$x_{60} = -28.2743338823081$$
$$x_{61} = 81.6814089933346$$
$$x_{62} = 100.530964914873$$
$$x_{63} = 7.78988375114457$$
$$x_{64} = -73.8206542907788$$
$$x_{65} = -114.663771308444$$
$$x_{66} = -95.8133575027966$$
$$x_{67} = -61.2528940466862$$
$$x_{68} = -18.8495559215388$$
$$x_{69} = 78.5398163397448$$
$$x_{70} = -17.2497818346079$$
$$x_{71} = -1.16556118520721$$
$$x_{72} = 12.5663706143592$$
$$x_{73} = 87.9645943005142$$
$$x_{74} = 48.6844162648433$$
$$x_{75} = 43.9822971502571$$
$$x_{76} = -36.1144715353049$$
$$x_{77} = -31.4159265358979$$
$$x_{78} = -81.6814089933346$$
$$x_{79} = 56.5486677646163$$
$$x_{80} = 73.8206542907788$$
$$x_{81} = -58.1108600600615$$
$$x_{82} = -39.2571723324086$$
$$x_{83} = 92.6715879363332$$
$$x_{84} = -59.6902604182061$$
$$x_{85} = 65.9734457253857$$
$$x_{86} = 67.5368388204916$$
$$x_{87} = 197.920337176157$$
$$x_{88} = 95.8133575027966$$
$$x_{89} = 53.4070751110265$$
Signos de extremos en los puntos:
(-50.26548245743669, -7.63821385155396e-32)
(70.67876056276886, 0.0141478139878745)
(58.110860060061505, 0.0172072134440586)
(-20.395842357309167, -0.0490001524829528)
(-86.38801019812658, -0.0115752926793239)
(20.395842357309167, 0.0490001524829528)
(51.82663153389846, 0.0192933035363155)
(72.25663103256524, 5.6146090061508e-31)
(-43.982297150257104, -6.68343712010972e-32)
(-37.69911184307752, -5.72866038866547e-32)
(34.55751918948773, 1.40770552330931e-31)
(-2678.2075504932727, -0.000373384043728018)
(4.604216777200577, 0.214660688386019)
(-94.2477796076938, -1.24937720620631e-31)
(89.52980595305935, 0.0111691162634939)
(-21.991148575128552, -3.34171856005486e-32)
(50.26548245743669, 7.63821385155396e-32)
(-15.707963267948966, -2.38694182861061e-32)
(-23.54070829230515, -0.0424604502887016)
(-87.96459430051421, -1.33668742402194e-31)
(21.991148575128552, 3.34171856005486e-32)
(-42.39970883624466, -0.0235817882463307)
(-72.25663103256524, -5.6146090061508e-31)
(59.69026041820607, 2.51765268789636e-32)
(80.1043708909521, 0.0124832269403218)
(-29.828369213095506, -0.0335157141235985)
(-80.1043708909521, -0.0124832269403218)
(-9.42477796076938, -1.43216509716637e-32)
(-51.82663153389846, -0.0192933035363155)
(86.38801019812658, 0.0115752926793239)
(36.11447153530485, 0.0276844243853039)
(-65.97344572538566, -1.45857698861786e-32)
(15.707963267948966, 2.38694182861061e-32)
(23.54070829230515, 0.0424604502887016)
(-10.94994364854116, -0.0911346506917966)
(28.274333882308138, 4.2964952914991e-32)
(94.2477796076938, 1.24937720620631e-31)
(-7.789883751144573, -0.127844922574794)
(37.69911184307752, 5.72866038866547e-32)
(-89.52980595305935, -0.0111691162634939)
(6.283185307179586, 9.54776731444245e-33)
(26.68480249092507, 0.0374613617155508)
(-67.53683882049161, -0.0148059223769658)
(-53.40707511102649, -4.05057601793315e-32)
(29.828369213095506, 0.0335157141235985)
(-6.283185307179586, -9.54776731444245e-33)
(-75.39822368615503, -1.14573207773309e-31)
(45.5421150692309, 0.021955051448177)
(42.39970883624466, 0.0235817882463307)
(64.39488496275855, 0.0155282475514317)
(-14.101725133565873, -0.0708242711210408)
(14.101725133565873, 0.0708242711210408)
(-64.39488496275855, -0.0155282475514317)
(-1740.4423300887454, -2.11977620970517e-30)
(-62.83185307179586, -9.54776731444245e-32)
(-97.3893722612836, -4.83455425149761e-31)
(-45.5421150692309, -0.021955051448177)
(-125.66370614359172, -1.90955346288849e-31)
(-83.24619911212368, -0.0120121271188891)
(-28.274333882308138, -4.2964952914991e-32)
(81.68140899333463, 1.88255223925938e-31)
(100.53096491487338, 1.52764277031079e-31)
(7.789883751144573, 0.127844922574794)
(-73.82065429077876, -0.0135457228854227)
(-114.66377130844361, -0.00872098461732392)
(-95.81335750279658, -0.0104366739072752)
(-61.252894046686194, -0.0163246714689743)
(-18.84955592153876, -2.86433019433273e-32)
(78.53981633974483, 3.07074756807772e-33)
(-17.249781834607894, -0.0579230818110724)
(-1.1655611852072114, -0.724611353776708)
(12.566370614359172, 1.90955346288849e-32)
(87.96459430051421, 1.33668742402194e-31)
(48.68441626484328, 0.0205382874085413)
(43.982297150257104, 6.68343712010972e-32)
(-36.11447153530485, -0.0276844243853039)
(-31.41592653589793, -4.77388365722123e-32)
(-81.68140899333463, -1.88255223925938e-31)
(56.548667764616276, 8.59299058299821e-32)
(73.82065429077876, 0.0135457228854227)
(-58.110860060061505, -0.0172072134440586)
(-39.25717233240859, -0.0254689206534694)
(92.67158793633321, 0.0107904797231539)
(-59.69026041820607, -2.51765268789636e-32)
(65.97344572538566, 1.45857698861786e-32)
(67.53683882049161, 0.0148059223769658)
(197.92033717615698, 4.37573096585357e-32)
(95.81335750279658, 0.0104366739072752)
(53.40707511102649, 4.05057601793315e-32)
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:
$$x_{1} = -20.3958423573092$$
$$x_{2} = -86.3880101981266$$
$$x_{3} = 72.2566310325652$$
$$x_{4} = 34.5575191894877$$
$$x_{5} = -2678.20755049327$$
$$x_{6} = 50.2654824574367$$
$$x_{7} = -23.5407082923052$$
$$x_{8} = 21.9911485751286$$
$$x_{9} = -42.3997088362447$$
$$x_{10} = 59.6902604182061$$
$$x_{11} = -29.8283692130955$$
$$x_{12} = -80.1043708909521$$
$$x_{13} = -51.8266315338985$$
$$x_{14} = 15.707963267949$$
$$x_{15} = -10.9499436485412$$
$$x_{16} = 28.2743338823081$$
$$x_{17} = 94.2477796076938$$
$$x_{18} = -7.78988375114457$$
$$x_{19} = 37.6991118430775$$
$$x_{20} = -89.5298059530594$$
$$x_{21} = 6.28318530717959$$
$$x_{22} = -67.5368388204916$$
$$x_{23} = -14.1017251335659$$
$$x_{24} = -64.3948849627586$$
$$x_{25} = -45.5421150692309$$
$$x_{26} = -83.2461991121237$$
$$x_{27} = 81.6814089933346$$
$$x_{28} = 100.530964914873$$
$$x_{29} = -73.8206542907788$$
$$x_{30} = -114.663771308444$$
$$x_{31} = -95.8133575027966$$
$$x_{32} = -61.2528940466862$$
$$x_{33} = 78.5398163397448$$
$$x_{34} = -17.2497818346079$$
$$x_{35} = -1.16556118520721$$
$$x_{36} = 12.5663706143592$$
$$x_{37} = 87.9645943005142$$
$$x_{38} = 43.9822971502571$$
$$x_{39} = -36.1144715353049$$
$$x_{40} = 56.5486677646163$$
$$x_{41} = -58.1108600600615$$
$$x_{42} = -39.2571723324086$$
$$x_{43} = 65.9734457253857$$
$$x_{44} = 197.920337176157$$
$$x_{45} = 53.4070751110265$$
Puntos máximos de la función:
$$x_{45} = -50.2654824574367$$
$$x_{45} = 70.6787605627689$$
$$x_{45} = 58.1108600600615$$
$$x_{45} = 20.3958423573092$$
$$x_{45} = 51.8266315338985$$
$$x_{45} = -43.9822971502571$$
$$x_{45} = -37.6991118430775$$
$$x_{45} = 4.60421677720058$$
$$x_{45} = -94.2477796076938$$
$$x_{45} = 89.5298059530594$$
$$x_{45} = -21.9911485751286$$
$$x_{45} = -15.707963267949$$
$$x_{45} = -87.9645943005142$$
$$x_{45} = -72.2566310325652$$
$$x_{45} = 80.1043708909521$$
$$x_{45} = -9.42477796076938$$
$$x_{45} = 86.3880101981266$$
$$x_{45} = 36.1144715353049$$
$$x_{45} = -65.9734457253857$$
$$x_{45} = 23.5407082923052$$
$$x_{45} = 26.6848024909251$$
$$x_{45} = -53.4070751110265$$
$$x_{45} = 29.8283692130955$$
$$x_{45} = -6.28318530717959$$
$$x_{45} = -75.398223686155$$
$$x_{45} = 45.5421150692309$$
$$x_{45} = 42.3997088362447$$
$$x_{45} = 64.3948849627586$$
$$x_{45} = 14.1017251335659$$
$$x_{45} = -1740.44233008875$$
$$x_{45} = -62.8318530717959$$
$$x_{45} = -97.3893722612836$$
$$x_{45} = -125.663706143592$$
$$x_{45} = -28.2743338823081$$
$$x_{45} = 7.78988375114457$$
$$x_{45} = -18.8495559215388$$
$$x_{45} = 48.6844162648433$$
$$x_{45} = -31.4159265358979$$
$$x_{45} = -81.6814089933346$$
$$x_{45} = 73.8206542907788$$
$$x_{45} = 92.6715879363332$$
$$x_{45} = -59.6902604182061$$
$$x_{45} = 67.5368388204916$$
$$x_{45} = 95.8133575027966$$
Decrece en los intervalos
$$\left[197.920337176157, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -2678.20755049327\right]$$