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$$\frac{1 - \cos{\left(x \right)}}{x^{2}} - \frac{2 \left(x - \sin{\left(x \right)}\right)}{x^{3}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 47.038904997378$$
$$x_{2} = 15.4505036738754$$
$$x_{3} = -97.3893722612836$$
$$x_{4} = 53.4070751110265$$
$$x_{5} = 97.3893722612836$$
$$x_{6} = 78.5398163397448$$
$$x_{7} = -40.7426059185751$$
$$x_{8} = -59.6902604182061$$
$$x_{9} = -65.9734457253857$$
$$x_{10} = 34.4415105438615$$
$$x_{11} = -21.9911485751286$$
$$x_{12} = 97.3482884639088$$
$$x_{13} = -34.5575191894877$$
$$x_{14} = -15.707963267949$$
$$x_{15} = 21.9911485751286$$
$$x_{16} = -34.4415105438615$$
$$x_{17} = 191.637151868977$$
$$x_{18} = 8.98681891581813$$
$$x_{19} = -103.672557568463$$
$$x_{20} = -40.8407044966673$$
$$x_{21} = 9.42477796076938$$
$$x_{22} = 34.5575191894877$$
$$x_{23} = -91.0622680279826$$
$$x_{24} = 65.9734457253857$$
$$x_{25} = 84.7758271362638$$
$$x_{26} = -8.98681891581813$$
$$x_{27} = 21.8082433188578$$
$$x_{28} = -21.8082433188578$$
$$x_{29} = 405.265452313083$$
$$x_{30} = -28.2743338823081$$
$$x_{31} = 91.0622680279826$$
$$x_{32} = 28.1323878256629$$
$$x_{33} = -15.4505036738754$$
$$x_{34} = -59.6231975817859$$
$$x_{35} = -53.4070751110265$$
$$x_{36} = 59.6231975817859$$
$$x_{37} = -128.774239181115$$
$$x_{38} = -9.42477796076938$$
$$x_{39} = 40.8407044966673$$
$$x_{40} = -91.106186954104$$
$$x_{41} = -28.1323878256629$$
$$x_{42} = -84.7758271362638$$
$$x_{43} = 59.6902604182061$$
$$x_{44} = 65.912778079645$$
$$x_{45} = 47.1238898038469$$
$$x_{46} = -141.343371423239$$
$$x_{47} = -235.619449019234$$
$$x_{48} = 91.106186954104$$
$$x_{49} = -97.3482884639088$$
$$x_{50} = 78.4888647223284$$
$$x_{51} = 28.2743338823081$$
$$x_{52} = -323.571681455931$$
$$x_{53} = -78.4888647223284$$
$$x_{54} = -47.1238898038469$$
$$x_{55} = -72.2012444887512$$
$$x_{56} = -53.3321085176254$$
$$x_{57} = 72.2012444887512$$
$$x_{58} = -3.14159265358979$$
$$x_{59} = 40.7426059185751$$
$$x_{60} = 72.2566310325652$$
$$x_{61} = -279.587439619171$$
$$x_{62} = -78.5398163397448$$
$$x_{63} = -72.2566310325652$$
$$x_{64} = -84.8230016469244$$
$$x_{65} = 84.8230016469244$$
$$x_{66} = 53.3321085176254$$
$$x_{67} = -47.038904997378$$
$$x_{68} = -65.912778079645$$
$$x_{69} = 15.707963267949$$
$$x_{70} = 3.14159265358979$$
Signos de extremos en los puntos:
(47.03890499737801, 0.021220636028774)
(15.450503673875414, 0.0636561755375016)
(-97.3893722612836, -0.0102680608446384)
(53.40707511102649, 0.0187241109519877)
(97.3893722612836, 0.0102680608446384)
(78.53981633974483, 0.0127323954473516)
(-40.74260591857512, -0.0244853286498844)
(-59.69026041820607, -0.01675315190441)
(-65.97344572538566, -0.0151576136277996)
(34.44151054386154, 0.0289371532895896)
(-21.991148575128552, -0.0454728408833987)
(97.34828846390877, 0.010268060235269)
(-34.55751918948773, -0.0289372623803446)
(-15.707963267948966, -0.0636619772367581)
(21.991148575128552, 0.0454728408833987)
(-34.44151054386154, -0.0289371532895896)
(191.63715186897738, 0.00521819485547198)
(8.986818915818128, 0.106023005437119)
(-103.67255756846318, -0.00964575412678153)
(-40.840704496667314, -0.0244853758602916)
(9.42477796076938, 0.106103295394597)
(34.55751918948773, 0.0289372623803446)
(-91.06226802798255, -0.0109762021211833)
(65.97344572538566, 0.0151576136277996)
(84.77582713626384, 0.0117892538276292)
(-8.986818915818128, -0.106023005437119)
(21.808243318857798, 0.0454717829781287)
(-21.808243318857798, -0.0454717829781287)
(405.2654523130833, 0.00246751849754877)
(-28.274333882308138, -0.0353677651315323)
(91.06226802798255, 0.0109762021211833)
(28.132387825662946, 0.035367466403301)
(-15.450503673875414, -0.0636561755375016)
(-59.62319758178592, -0.0167531448468799)
(-53.40707511102649, -0.0187241109519877)
(59.62319758178592, 0.0167531448468799)
(-128.77423918111484, -0.00776365561006469)
(-9.42477796076938, -0.106103295394597)
(40.840704496667314, 0.0244853758602916)
(-91.106186954104, -0.0109762029718549)
(-28.132387825662946, -0.035367466403301)
(-84.77582713626384, -0.0117892538276292)
(59.69026041820607, 0.01675315190441)
(65.91277807964495, 0.0151576093510719)
(47.1238898038469, 0.0212206590789194)
(-141.343371423239, -0.00707355293181437)
(-235.61944901923448, -0.00424413181578388)
(91.106186954104, 0.0109762029718549)
(-97.34828846390877, -0.010268060235269)
(78.48886472232839, 0.0127323936599357)
(28.274333882308138, 0.0353677651315323)
(-323.57168145593135, -0.00309038724299932)
(-78.48886472232839, -0.0127323936599357)
(-47.1238898038469, -0.0212206590789194)
(-72.20124448875121, -0.0138395575561134)
(-53.33210851762535, -0.0187240986360253)
(72.20124448875121, 0.0138395575561134)
(-3.141592653589793, -0.318309886183791)
(40.74260591857512, 0.0244853286498844)
(72.25663103256524, 0.0138395602688605)
(-279.5874396191708, -0.00357651557197735)
(-78.53981633974483, -0.0127323954473516)
(-72.25663103256524, -0.0138395602688605)
(-84.82300164692441, -0.0117892550438441)
(84.82300164692441, 0.0117892550438441)
(53.33210851762535, 0.0187240986360253)
(-47.03890499737801, -0.021220636028774)
(-65.91277807964495, -0.0151576093510719)
(15.707963267948966, 0.0636619772367581)
(3.141592653589793, 0.318309886183791)
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:
$$x_{1} = 47.038904997378$$
$$x_{2} = 15.4505036738754$$
$$x_{3} = -59.6902604182061$$
$$x_{4} = -65.9734457253857$$
$$x_{5} = 34.4415105438615$$
$$x_{6} = -21.9911485751286$$
$$x_{7} = -34.5575191894877$$
$$x_{8} = -15.707963267949$$
$$x_{9} = 8.98681891581813$$
$$x_{10} = -40.8407044966673$$
$$x_{11} = 21.8082433188578$$
$$x_{12} = -28.2743338823081$$
$$x_{13} = 28.1323878256629$$
$$x_{14} = -53.4070751110265$$
$$x_{15} = 59.6231975817859$$
$$x_{16} = -9.42477796076938$$
$$x_{17} = -91.106186954104$$
$$x_{18} = 65.912778079645$$
$$x_{19} = -47.1238898038469$$
$$x_{20} = -3.14159265358979$$
$$x_{21} = 40.7426059185751$$
$$x_{22} = -78.5398163397448$$
$$x_{23} = -72.2566310325652$$
$$x_{24} = 53.3321085176254$$
Puntos máximos de la función:
$$x_{24} = 53.4070751110265$$
$$x_{24} = 78.5398163397448$$
$$x_{24} = -40.7426059185751$$
$$x_{24} = 21.9911485751286$$
$$x_{24} = -34.4415105438615$$
$$x_{24} = 9.42477796076938$$
$$x_{24} = 34.5575191894877$$
$$x_{24} = 65.9734457253857$$
$$x_{24} = -8.98681891581813$$
$$x_{24} = -21.8082433188578$$
$$x_{24} = -15.4505036738754$$
$$x_{24} = -59.6231975817859$$
$$x_{24} = 40.8407044966673$$
$$x_{24} = -28.1323878256629$$
$$x_{24} = 59.6902604182061$$
$$x_{24} = 47.1238898038469$$
$$x_{24} = 91.106186954104$$
$$x_{24} = 28.2743338823081$$
$$x_{24} = -53.3321085176254$$
$$x_{24} = 72.2566310325652$$
$$x_{24} = -47.038904997378$$
$$x_{24} = -65.912778079645$$
$$x_{24} = 15.707963267949$$
$$x_{24} = 3.14159265358979$$
Decrece en los intervalos
$$\left[65.912778079645, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -91.106186954104\right]$$