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$$- 2 \frac{1}{2 x \sin^{2}{\left(3 \right)}} \sin{\left(2 x \right)} - \frac{\cos{\left(2 x \right)} - 1}{2 x^{2} \sin^{2}{\left(3 \right)}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -18.8495559215388$$
$$x_{2} = -15.707963267949$$
$$x_{3} = 26.6848024909251$$
$$x_{4} = 64.3948849627586$$
$$x_{5} = -31.4159265358979$$
$$x_{6} = 21.9911485751286$$
$$x_{7} = -21.9911485751286$$
$$x_{8} = -1350.88484104361$$
$$x_{9} = -58.1108600600615$$
$$x_{10} = 28.2743338823081$$
$$x_{11} = 72.2566310325652$$
$$x_{12} = -14.1017251335659$$
$$x_{13} = -45.5421150692309$$
$$x_{14} = 14.1017251335659$$
$$x_{15} = -94.2477796076938$$
$$x_{16} = -42.3997088362447$$
$$x_{17} = -20.3958423573092$$
$$x_{18} = 87.9645943005142$$
$$x_{19} = -50.2654824574367$$
$$x_{20} = 7.78988375114457$$
$$x_{21} = 73.8206542907788$$
$$x_{22} = -43.9822971502571$$
$$x_{23} = -97.3893722612836$$
$$x_{24} = 50.2654824574367$$
$$x_{25} = 59.6902604182061$$
$$x_{26} = 58.1108600600615$$
$$x_{27} = -80.1043708909521$$
$$x_{28} = -10.9499436485412$$
$$x_{29} = -51.8266315338985$$
$$x_{30} = -53.4070751110265$$
$$x_{31} = 86.3880101981266$$
$$x_{32} = 67.5368388204916$$
$$x_{33} = 12.5663706143592$$
$$x_{34} = -81.6814089933346$$
$$x_{35} = 23.5407082923052$$
$$x_{36} = 94.2477796076938$$
$$x_{37} = -67.5368388204916$$
$$x_{38} = 81.6814089933346$$
$$x_{39} = 95.8133575027966$$
$$x_{40} = -73.8206542907788$$
$$x_{41} = 53.4070751110265$$
$$x_{42} = -28.2743338823081$$
$$x_{43} = -72.2566310325652$$
$$x_{44} = 20.3958423573092$$
$$x_{45} = 37.6991118430775$$
$$x_{46} = 65.9734457253857$$
$$x_{47} = -36.1144715353049$$
$$x_{48} = -95.8133575027966$$
$$x_{49} = 529.357417588754$$
$$x_{50} = 80.1043708909521$$
$$x_{51} = -17.2497818346079$$
$$x_{52} = -89.5298059530594$$
$$x_{53} = 4.60421677720058$$
$$x_{54} = 29.8283692130955$$
$$x_{55} = -7.78988375114457$$
$$x_{56} = -87.9645943005142$$
$$x_{57} = -29.8283692130955$$
$$x_{58} = 36.1144715353049$$
$$x_{59} = -83.2461991121237$$
$$x_{60} = 78.5398163397448$$
$$x_{61} = -6.28318530717959$$
$$x_{62} = 15.707963267949$$
$$x_{63} = 56.5486677646163$$
$$x_{64} = -61.2528940466862$$
$$x_{65} = -62.8318530717959$$
$$x_{66} = -65.9734457253857$$
$$x_{67} = -125.663706143592$$
$$x_{68} = 48.6844162648433$$
$$x_{69} = 43.9822971502571$$
$$x_{70} = -37.6991118430775$$
$$x_{71} = -59.6902604182061$$
$$x_{72} = -1.16556118520721$$
$$x_{73} = 45.5421150692309$$
$$x_{74} = -64.3948849627586$$
$$x_{75} = -114.663771308444$$
$$x_{76} = -75.398223686155$$
$$x_{77} = 92.6715879363332$$
$$x_{78} = 100.530964914873$$
$$x_{79} = -2800.7296721504$$
$$x_{80} = 70.6787605627689$$
$$x_{81} = 34.5575191894877$$
$$x_{82} = 6.28318530717959$$
$$x_{83} = -23.5407082923052$$
$$x_{84} = -39.2571723324086$$
$$x_{85} = 89.5298059530594$$
$$x_{86} = 42.3997088362447$$
$$x_{87} = -9.42477796076938$$
$$x_{88} = 51.8266315338985$$
$$x_{89} = -86.3880101981266$$
$$x_{90} = 197.920337176157$$
Signos de extremos en los puntos:
(-18.84955592153876, 0)
(-15.707963267948966, 0)
-0.0374613617155508
(26.68480249092507, -------------------)
2
sin (3)
-0.0155282475514317
(64.39488496275855, -------------------)
2
sin (3)
(-31.41592653589793, 0)
(21.991148575128552, 0)
(-21.991148575128552, 0)
(-1350.884841043611, 0)
0.0172072134440586
(-58.110860060061505, ------------------)
2
sin (3)
(28.274333882308138, 0)
(72.25663103256524, 0)
0.0708242711210408
(-14.101725133565873, ------------------)
2
sin (3)
0.021955051448177
(-45.5421150692309, -----------------)
2
sin (3)
-0.0708242711210408
(14.101725133565873, -------------------)
2
sin (3)
(-94.2477796076938, 0)
0.0235817882463307
(-42.39970883624466, ------------------)
2
sin (3)
0.0490001524829528
(-20.395842357309167, ------------------)
2
sin (3)
(87.96459430051421, 0)
(-50.26548245743669, 0)
-0.127844922574794
(7.789883751144573, ------------------)
2
sin (3)
-0.0135457228854227
(73.82065429077876, -------------------)
2
sin (3)
(-43.982297150257104, 0)
(-97.3893722612836, 0)
(50.26548245743669, 0)
(59.69026041820607, 0)
-0.0172072134440586
(58.110860060061505, -------------------)
2
sin (3)
0.0124832269403218
(-80.1043708909521, ------------------)
2
sin (3)
0.0911346506917966
(-10.94994364854116, ------------------)
2
sin (3)
0.0192933035363155
(-51.82663153389846, ------------------)
2
sin (3)
(-53.40707511102649, 0)
-0.0115752926793239
(86.38801019812658, -------------------)
2
sin (3)
-0.0148059223769658
(67.53683882049161, -------------------)
2
sin (3)
(12.566370614359172, 0)
(-81.68140899333463, 0)
-0.0424604502887016
(23.54070829230515, -------------------)
2
sin (3)
(94.2477796076938, 0)
0.0148059223769658
(-67.53683882049161, ------------------)
2
sin (3)
(81.68140899333463, 0)
-0.0104366739072752
(95.81335750279658, -------------------)
2
sin (3)
0.0135457228854227
(-73.82065429077876, ------------------)
2
sin (3)
(53.40707511102649, 0)
(-28.274333882308138, 0)
(-72.25663103256524, 0)
-0.0490001524829528
(20.395842357309167, -------------------)
2
sin (3)
(37.69911184307752, 0)
(65.97344572538566, 0)
0.0276844243853039
(-36.11447153530485, ------------------)
2
sin (3)
0.0104366739072752
(-95.81335750279658, ------------------)
2
sin (3)
-0.00188908112858296
(529.3574175887541, --------------------)
2
sin (3)
-0.0124832269403218
(80.1043708909521, -------------------)
2
sin (3)
0.0579230818110724
(-17.249781834607894, ------------------)
2
sin (3)
0.0111691162634939
(-89.52980595305935, ------------------)
2
sin (3)
-0.214660688386019
(4.604216777200577, ------------------)
2
sin (3)
-0.0335157141235985
(29.828369213095506, -------------------)
2
sin (3)
0.127844922574794
(-7.789883751144573, -----------------)
2
sin (3)
(-87.96459430051421, 0)
0.0335157141235985
(-29.828369213095506, ------------------)
2
sin (3)
-0.0276844243853039
(36.11447153530485, -------------------)
2
sin (3)
0.0120121271188891
(-83.24619911212368, ------------------)
2
sin (3)
(78.53981633974483, 0)
(-6.283185307179586, 0)
(15.707963267948966, 0)
(56.548667764616276, 0)
0.0163246714689743
(-61.252894046686194, ------------------)
2
sin (3)
(-62.83185307179586, 0)
(-65.97344572538566, 0)
(-125.66370614359172, 0)
-0.0205382874085413
(48.68441626484328, -------------------)
2
sin (3)
(43.982297150257104, 0)
(-37.69911184307752, 0)
(-59.69026041820607, 0)
0.724611353776708
(-1.1655611852072114, -----------------)
2
sin (3)
-0.021955051448177
(45.5421150692309, ------------------)
2
sin (3)
0.0155282475514317
(-64.39488496275855, ------------------)
2
sin (3)
0.00872098461732392
(-114.66377130844361, -------------------)
2
sin (3)
(-75.39822368615503, 0)
-0.0107904797231539
(92.67158793633321, -------------------)
2
sin (3)
(100.53096491487338, 0)
0.000357049799583499
(-2800.729672150397, --------------------)
2
sin (3)
-0.0141478139878745
(70.67876056276886, -------------------)
2
sin (3)
(34.55751918948773, 0)
(6.283185307179586, 0)
0.0424604502887016
(-23.54070829230515, ------------------)
2
sin (3)
0.0254689206534694
(-39.25717233240859, ------------------)
2
sin (3)
-0.0111691162634939
(89.52980595305935, -------------------)
2
sin (3)
-0.0235817882463307
(42.39970883624466, -------------------)
2
sin (3)
(-9.42477796076938, 0)
-0.0192933035363155
(51.82663153389846, -------------------)
2
sin (3)
0.0115752926793239
(-86.38801019812658, ------------------)
2
sin (3)
(197.92033717615698, 0)
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:
$$x_{1} = -18.8495559215388$$
$$x_{2} = -15.707963267949$$
$$x_{3} = 26.6848024909251$$
$$x_{4} = 64.3948849627586$$
$$x_{5} = -31.4159265358979$$
$$x_{6} = -21.9911485751286$$
$$x_{7} = -1350.88484104361$$
$$x_{8} = 14.1017251335659$$
$$x_{9} = -94.2477796076938$$
$$x_{10} = -50.2654824574367$$
$$x_{11} = 7.78988375114457$$
$$x_{12} = 73.8206542907788$$
$$x_{13} = -43.9822971502571$$
$$x_{14} = -97.3893722612836$$
$$x_{15} = 58.1108600600615$$
$$x_{16} = -53.4070751110265$$
$$x_{17} = 86.3880101981266$$
$$x_{18} = 67.5368388204916$$
$$x_{19} = -81.6814089933346$$
$$x_{20} = 23.5407082923052$$
$$x_{21} = 95.8133575027966$$
$$x_{22} = -28.2743338823081$$
$$x_{23} = -72.2566310325652$$
$$x_{24} = 20.3958423573092$$
$$x_{25} = 529.357417588754$$
$$x_{26} = 80.1043708909521$$
$$x_{27} = 4.60421677720058$$
$$x_{28} = 29.8283692130955$$
$$x_{29} = -87.9645943005142$$
$$x_{30} = 36.1144715353049$$
$$x_{31} = -6.28318530717959$$
$$x_{32} = -62.8318530717959$$
$$x_{33} = -65.9734457253857$$
$$x_{34} = -125.663706143592$$
$$x_{35} = 48.6844162648433$$
$$x_{36} = -37.6991118430775$$
$$x_{37} = -59.6902604182061$$
$$x_{38} = 45.5421150692309$$
$$x_{39} = -75.398223686155$$
$$x_{40} = 92.6715879363332$$
$$x_{41} = 70.6787605627689$$
$$x_{42} = 89.5298059530594$$
$$x_{43} = 42.3997088362447$$
$$x_{44} = -9.42477796076938$$
$$x_{45} = 51.8266315338985$$
Puntos máximos de la función:
$$x_{45} = 21.9911485751286$$
$$x_{45} = -58.1108600600615$$
$$x_{45} = 28.2743338823081$$
$$x_{45} = 72.2566310325652$$
$$x_{45} = -14.1017251335659$$
$$x_{45} = -45.5421150692309$$
$$x_{45} = -42.3997088362447$$
$$x_{45} = -20.3958423573092$$
$$x_{45} = 87.9645943005142$$
$$x_{45} = 50.2654824574367$$
$$x_{45} = 59.6902604182061$$
$$x_{45} = -80.1043708909521$$
$$x_{45} = -10.9499436485412$$
$$x_{45} = -51.8266315338985$$
$$x_{45} = 12.5663706143592$$
$$x_{45} = 94.2477796076938$$
$$x_{45} = -67.5368388204916$$
$$x_{45} = 81.6814089933346$$
$$x_{45} = -73.8206542907788$$
$$x_{45} = 53.4070751110265$$
$$x_{45} = 37.6991118430775$$
$$x_{45} = 65.9734457253857$$
$$x_{45} = -36.1144715353049$$
$$x_{45} = -95.8133575027966$$
$$x_{45} = -17.2497818346079$$
$$x_{45} = -89.5298059530594$$
$$x_{45} = -7.78988375114457$$
$$x_{45} = -29.8283692130955$$
$$x_{45} = -83.2461991121237$$
$$x_{45} = 78.5398163397448$$
$$x_{45} = 15.707963267949$$
$$x_{45} = 56.5486677646163$$
$$x_{45} = -61.2528940466862$$
$$x_{45} = 43.9822971502571$$
$$x_{45} = -1.16556118520721$$
$$x_{45} = -64.3948849627586$$
$$x_{45} = -114.663771308444$$
$$x_{45} = 100.530964914873$$
$$x_{45} = -2800.7296721504$$
$$x_{45} = 34.5575191894877$$
$$x_{45} = 6.28318530717959$$
$$x_{45} = -23.5407082923052$$
$$x_{45} = -39.2571723324086$$
$$x_{45} = -86.3880101981266$$
$$x_{45} = 197.920337176157$$
Decrece en los intervalos
$$\left[529.357417588754, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -1350.88484104361\right]$$