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$$\frac{- 3 \sin{\left(3 x \right)} + 7 \sin{\left(7 x \right)}}{x^{2}} - \frac{2 \left(\cos{\left(3 x \right)} - \cos{\left(7 x \right)}\right)}{x^{3}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 73.9967329633305$$
$$x_{2} = -73.9967329633305$$
$$x_{3} = -89.7047470121367$$
$$x_{4} = -5.88334097907696$$
$$x_{5} = 10.3386869452249$$
$$x_{6} = -33.1556710687763$$
$$x_{7} = 39.9185887685627$$
$$x_{8} = -26.0505741737963$$
$$x_{9} = -43.9822971502571$$
$$x_{10} = 28.2743338823081$$
$$x_{11} = 12.1711783051505$$
$$x_{12} = 13.4817945658253$$
$$x_{13} = 24.2095258592079$$
$$x_{14} = -23.7306357344751$$
$$x_{15} = -31.8050183397407$$
$$x_{16} = -52.0054617321196$$
$$x_{17} = -15.707963267949$$
$$x_{18} = -37.6991118430775$$
$$x_{19} = -39.9185887685627$$
$$x_{20} = -85.7426369606128$$
$$x_{21} = 20.2496992007472$$
$$x_{22} = -70.0344971468579$$
$$x_{23} = -102.751477358685$$
$$x_{24} = -4.04546347923779$$
$$x_{25} = 17.9253602624254$$
$$x_{26} = 65.9734457253857$$
$$x_{27} = -11.1632443424398$$
$$x_{28} = 76.3177617903111$$
$$x_{29} = 21.9911485751286$$
$$x_{30} = -93.85641703355$$
$$x_{31} = -65.9734457253857$$
$$x_{32} = 30.0140105999544$$
$$x_{33} = -75.7883011951918$$
$$x_{34} = 70.0344971468579$$
$$x_{35} = 60.0801517336434$$
$$x_{36} = 68.1936248680213$$
$$x_{37} = 61.910338966879$$
$$x_{38} = -57.9494986979589$$
$$x_{39} = 64.2327201708699$$
$$x_{40} = -60.0801517336434$$
$$x_{41} = -41.7595109966085$$
$$x_{42} = -79.9407489893177$$
$$x_{43} = 37.6991118430775$$
$$x_{44} = -21.9911485751286$$
$$x_{45} = -53.7968614690947$$
$$x_{46} = -48.0429075650649$$
$$x_{47} = -45.7222196961036$$
$$x_{48} = 34.1651562272963$$
$$x_{49} = -1.72785268115637$$
$$x_{50} = -27.8816169932295$$
$$x_{51} = -35.9581240877896$$
$$x_{52} = -61.0911103639023$$
$$x_{53} = 26.0505741737963$$
$$x_{54} = 90.18501510683$$
$$x_{55} = 100.139638310928$$
$$x_{56} = -61.910338966879$$
$$x_{57} = 8.02089510578949$$
$$x_{58} = -87.9645943005142$$
$$x_{59} = 54.326255289094$$
$$x_{60} = 95.9879479767112$$
$$x_{61} = -9.81004212050563$$
$$x_{62} = -8.02089510578949$$
$$x_{63} = 16.0953925366953$$
$$x_{64} = 6.28318530717959$$
$$x_{65} = -63.7512168713818$$
$$x_{66} = -97.7796098754723$$
$$x_{67} = -71.8650930729889$$
$$x_{68} = 52.0054617321196$$
$$x_{69} = 86.2239538314439$$
$$x_{70} = -49.8736146343732$$
$$x_{71} = 80.2799409683107$$
$$x_{72} = -13.9660390823958$$
$$x_{73} = 87.9645943005142$$
$$x_{74} = 43.9822971502571$$
$$x_{75} = 32.3342623220753$$
$$x_{76} = -67.7135207443646$$
$$x_{77} = 38.0884840641708$$
$$x_{78} = -19.7665667687004$$
$$x_{79} = 92.0258760154351$$
$$x_{80} = 78.1483385546479$$
$$x_{81} = 98.3091082017841$$
$$x_{82} = -95.9879479767112$$
$$x_{83} = 57.9494986979589$$
$$x_{84} = -83.9017736838728$$
$$x_{85} = 72.2566310325652$$
$$x_{86} = 46.2020044431786$$
$$x_{87} = 94.2477796076938$$
$$x_{88} = -30.0140105999544$$
$$x_{89} = 48.0429075650649$$
$$x_{90} = 82.0715410084561$$
$$x_{91} = -17.9253602624254$$
$$x_{92} = 42.2413979761308$$
$$x_{93} = 2.19017794396668$$
$$x_{94} = 56.1569205219136$$
$$x_{95} = 50.2654824574367$$
$$x_{96} = 4.04546347923779$$
$$x_{97} = -92.0258760154351$$
Signos de extremos en los puntos:
(73.99673296333052, 8.03930288521132e-5)
(-73.99673296333052, 8.03930288521132e-5)
(-89.70474701213674, -5.4703255753612e-5)
(-5.883340979076965, 0.037691714404883)
(10.338686945224943, 0.0179121518053217)
(-33.15567106877626, -0.00040042488372479)
(39.91858876856266, 0.0012022176368869)
(-26.05057417379634, -0.00282276105607107)
(-43.982297150257104, 0)
(28.274333882308138, 0)
(12.171178305150516, 0.0088173424068626)
(13.481794565825336, -0.0105364802725626)
(24.20952585920788, -0.00326835479328241)
(-23.730635734475104, 0.000781645578399185)
(-31.80501833974069, 0.00129165137305129)
(-52.005461732119564, -0.000162758554818959)
(-15.707963267948966, 0)
(-37.69911184307752, 0)
(-39.91858876856266, 0.0012022176368869)
(-85.74263696061283, 0.00026058788237464)
(20.249699200747163, 0.00107345875201028)
(-70.03449714685794, -0.000390590674268843)
(-102.75147735868458, 0.00018145667453231)
(-4.045463479237794, 0.116585382063563)
(17.925360262425393, -0.00596107574909238)
(65.97344572538566, 0)
(-11.163244342439842, 0.00353174515763499)
(76.31776179031108, -0.000328924022378928)
(21.991148575128552, 0)
(-93.85641703354997, 0.000148330070689398)
(-65.97344572538566, 0)
(30.01401059995444, 0.000488637473970616)
(-75.78830119519185, 0.000227484272732193)
(70.03449714685794, -0.000390590674268843)
(60.08015173364343, -0.000361985577253256)
(68.1936248680213, -0.000411962811520212)
(61.910338966879, -0.000499824863277712)
(-57.949498697958894, 0.000131081996001023)
(64.23272017086985, 0.000106691617603996)
(-60.08015173364343, -0.000361985577253256)
(-41.75951099660849, 0.00109856121576167)
(-79.94074898931768, -6.88822463246151e-5)
(37.69911184307752, 0)
(-21.991148575128552, 0)
(-53.79686146909471, -0.000451479279972487)
(-48.04290756506492, 0.000830004815147386)
(-45.72221969610364, -0.000210564965458014)
(34.1651562272963, -0.00111936824241238)
(-1.7278526811563655, -0.146376799874948)
(-27.88161699322951, -0.00168071423145777)
(-35.95812408778957, -0.000340442628605085)
(-61.09111036390234, -0.000117946933113421)
(26.05057417379634, -0.00282276105607107)
(90.18501510683002, 0.00023554804493505)
(100.13963831092772, 0.000130300304685561)
(-61.910338966879, -0.000499824863277712)
(8.020895105789487, -0.00683996916068079)
(-87.96459430051421, 0)
(54.32625528909404, 0.000649116134917602)
(95.98794797671118, -4.77761069264751e-5)
(-9.810042120505633, -0.0135697858716332)
(-8.020895105789487, -0.00683996916068079)
(16.095392536695318, -0.00504273203450722)
(6.283185307179586, 0)
(-63.751216871381786, -0.000471376259278609)
(-97.77960987547232, -0.000136666083829747)
(-71.86509307298887, -0.000252999282516496)
(52.005461732119564, -0.000162758554818959)
(86.22395383144388, -5.92090461286597e-5)
(-49.87361463437316, 0.000525301593330343)
(80.27994096831073, 6.83014068895433e-5)
(-13.96603908239582, 0.00225658169391437)
(87.96459430051421, 0)
(43.982297150257104, 0)
(32.33426232207532, -0.00183230572664874)
(-67.71352074436457, 9.60046704253271e-5)
(38.08848406417079, 0.00090065043125887)
(-19.766566768700443, -0.00490246455191702)
(92.02587601543506, 0.000226218695483382)
(78.14833855464792, -0.000213952060694765)
(98.30910820178411, 0.000198226325057273)
(-95.98794797671118, -4.77761069264751e-5)
(57.949498697958894, 0.000131081996001023)
(-83.90177368387278, 0.000272148173617132)
(72.25663103256524, 0)
(46.20200444317864, 0.000897462870855063)
(94.2477796076938, 0)
(-30.01401059995444, 0.000488637473970616)
(48.04290756506492, 0.000830004815147386)
(82.07154100845611, 0.000193986338777878)
(-17.925360262425393, -0.00596107574909238)
(42.24139797613083, -0.000246696736221661)
(2.190177943966682, 0.393771794426958)
(56.15692052191362, 0.000414329468010238)
(50.26548245743669, 0)
(4.045463479237794, 0.116585382063563)
(-92.02587601543506, 0.000226218695483382)
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:
$$x_{1} = -89.7047470121367$$
$$x_{2} = -33.1556710687763$$
$$x_{3} = -26.0505741737963$$
$$x_{4} = -43.9822971502571$$
$$x_{5} = 13.4817945658253$$
$$x_{6} = 24.2095258592079$$
$$x_{7} = -52.0054617321196$$
$$x_{8} = -37.6991118430775$$
$$x_{9} = -70.0344971468579$$
$$x_{10} = 17.9253602624254$$
$$x_{11} = 76.3177617903111$$
$$x_{12} = 70.0344971468579$$
$$x_{13} = 60.0801517336434$$
$$x_{14} = 68.1936248680213$$
$$x_{15} = 61.910338966879$$
$$x_{16} = -60.0801517336434$$
$$x_{17} = -79.9407489893177$$
$$x_{18} = 37.6991118430775$$
$$x_{19} = -53.7968614690947$$
$$x_{20} = -45.7222196961036$$
$$x_{21} = 34.1651562272963$$
$$x_{22} = -1.72785268115637$$
$$x_{23} = -27.8816169932295$$
$$x_{24} = -35.9581240877896$$
$$x_{25} = -61.0911103639023$$
$$x_{26} = 26.0505741737963$$
$$x_{27} = -61.910338966879$$
$$x_{28} = 8.02089510578949$$
$$x_{29} = -87.9645943005142$$
$$x_{30} = 95.9879479767112$$
$$x_{31} = -9.81004212050563$$
$$x_{32} = -8.02089510578949$$
$$x_{33} = 16.0953925366953$$
$$x_{34} = 6.28318530717959$$
$$x_{35} = -63.7512168713818$$
$$x_{36} = -97.7796098754723$$
$$x_{37} = -71.8650930729889$$
$$x_{38} = 52.0054617321196$$
$$x_{39} = 86.2239538314439$$
$$x_{40} = 87.9645943005142$$
$$x_{41} = 43.9822971502571$$
$$x_{42} = 32.3342623220753$$
$$x_{43} = -19.7665667687004$$
$$x_{44} = 78.1483385546479$$
$$x_{45} = -95.9879479767112$$
$$x_{46} = 94.2477796076938$$
$$x_{47} = -17.9253602624254$$
$$x_{48} = 42.2413979761308$$
$$x_{49} = 50.2654824574367$$
Puntos máximos de la función:
$$x_{49} = 73.9967329633305$$
$$x_{49} = -73.9967329633305$$
$$x_{49} = -5.88334097907696$$
$$x_{49} = 10.3386869452249$$
$$x_{49} = 39.9185887685627$$
$$x_{49} = 28.2743338823081$$
$$x_{49} = 12.1711783051505$$
$$x_{49} = -23.7306357344751$$
$$x_{49} = -31.8050183397407$$
$$x_{49} = -15.707963267949$$
$$x_{49} = -39.9185887685627$$
$$x_{49} = -85.7426369606128$$
$$x_{49} = 20.2496992007472$$
$$x_{49} = -102.751477358685$$
$$x_{49} = -4.04546347923779$$
$$x_{49} = 65.9734457253857$$
$$x_{49} = -11.1632443424398$$
$$x_{49} = 21.9911485751286$$
$$x_{49} = -93.85641703355$$
$$x_{49} = -65.9734457253857$$
$$x_{49} = 30.0140105999544$$
$$x_{49} = -75.7883011951918$$
$$x_{49} = -57.9494986979589$$
$$x_{49} = 64.2327201708699$$
$$x_{49} = -41.7595109966085$$
$$x_{49} = -21.9911485751286$$
$$x_{49} = -48.0429075650649$$
$$x_{49} = 90.18501510683$$
$$x_{49} = 100.139638310928$$
$$x_{49} = 54.326255289094$$
$$x_{49} = -49.8736146343732$$
$$x_{49} = 80.2799409683107$$
$$x_{49} = -13.9660390823958$$
$$x_{49} = -67.7135207443646$$
$$x_{49} = 38.0884840641708$$
$$x_{49} = 92.0258760154351$$
$$x_{49} = 98.3091082017841$$
$$x_{49} = 57.9494986979589$$
$$x_{49} = -83.9017736838728$$
$$x_{49} = 72.2566310325652$$
$$x_{49} = 46.2020044431786$$
$$x_{49} = -30.0140105999544$$
$$x_{49} = 48.0429075650649$$
$$x_{49} = 82.0715410084561$$
$$x_{49} = 2.19017794396668$$
$$x_{49} = 56.1569205219136$$
$$x_{49} = 4.04546347923779$$
$$x_{49} = -92.0258760154351$$
Decrece en los intervalos
$$\left[95.9879479767112, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -97.7796098754723\right]$$