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$$3 \left(\left(2 x + 1\right) \cos{\left(2 x \right)} + \left(3 x + 1\right) \sin{\left(2 x \right)}\right) e^{3 x} + \left(- 2 \left(2 x + 1\right) \sin{\left(2 x \right)} + 2 \left(3 x + 1\right) \cos{\left(2 x \right)} + 3 \sin{\left(2 x \right)} + 2 \cos{\left(2 x \right)}\right) e^{3 x} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -11.5870823566467$$
$$x_{2} = -6.87732970906955$$
$$x_{3} = -83.840670061203$$
$$x_{4} = -74.4159511177762$$
$$x_{5} = -32.0051544259545$$
$$x_{6} = -24.151581643793$$
$$x_{7} = -46.1419408952529$$
$$x_{8} = -64.9912494546192$$
$$x_{9} = 5.70127131092793$$
$$x_{10} = -68.1328143089569$$
$$x_{11} = -52.4250237498346$$
$$x_{12} = -61.849687452674$$
$$x_{13} = -75.9867365833898$$
$$x_{14} = -97.9777698996318$$
$$x_{15} = -72.8451661245562$$
$$x_{16} = -107.402512995711$$
$$x_{17} = -16.2984182815884$$
$$x_{18} = -14.7278948683244$$
$$x_{19} = -44.5711747579639$$
$$x_{20} = -10.0168714789543$$
$$x_{21} = -96.4069800506118$$
$$x_{22} = -25.7222757391008$$
$$x_{23} = -66.5620315507164$$
$$x_{24} = -36.7173834758069$$
$$x_{25} = -80.6990955366258$$
$$x_{26} = -58.7081287658687$$
$$x_{27} = -17.8689913718302$$
$$x_{28} = -91.6946118441651$$
$$x_{29} = -39.8588908461224$$
$$x_{30} = -86.9822459046496$$
$$x_{31} = -82.2698826245417$$
$$x_{32} = -38.2881353830905$$
$$x_{33} = 7.27085591930572$$
$$x_{34} = -53.9957982318805$$
$$x_{35} = 1.00649021167376$$
$$x_{36} = -3.74189452428571$$
$$x_{37} = -33.5758923617546$$
$$x_{38} = -50.8542506337659$$
$$x_{39} = -90.1238229280977$$
$$x_{40} = -28.8636980341657$$
$$x_{41} = -94.8361904175774$$
$$x_{42} = -60.2789076621501$$
$$x_{43} = -0.409344894484793$$
$$x_{44} = -2.18330949548946$$
$$x_{45} = -43.0004108582336$$
$$x_{46} = -69.7035976841782$$
$$x_{47} = -30.4344226414208$$
$$x_{48} = 4.13248646486902$$
$$x_{49} = -22.5809021677498$$
$$x_{50} = -88.5530342767776$$
$$x_{51} = -47.7127090462317$$
$$x_{52} = -55.75$$
$$x_{53} = -8.44689520600099$$
$$x_{54} = -21.0102406879933$$
Signos de extremos en los puntos:
(-11.587082356646697, -1.83080633292272e-14)
(-6.877329709069555, 1.47382570209898e-8)
(-83.840670061203, -9.7514026023049e-108)
(-74.41595111777625, -1.64621990253583e-95)
(-32.00515442595453, 1.27402832611508e-40)
(-24.15158164379298, -1.63876979567797e-30)
(-46.1419408952529, -7.01629989574314e-59)
(-64.9912494546192, -2.73422609197441e-83)
(5.70127131092793, -313955820.082521)
(-68.13281430895685, -2.31361115133441e-87)
(-52.42502374983461, -5.1951673011731e-67)
(-61.84968745267397, -3.22369555254647e-79)
(-75.98673658338984, 1.51017669816463e-97)
(-97.97776989963178, 4.34371610632048e-126)
(-72.84516612455619, 1.79371307489797e-93)
(-107.40251299571085, 2.50303176417646e-138)
(-16.298418281588436, 1.87995503741104e-20)
(-14.727894868324416, -1.88763088691794e-18)
(-44.57117475796393, 7.54293950390332e-57)
(-10.016871478954263, 1.75511899316108e-12)
(-96.40698005061176, -4.75759370524551e-124)
(-25.72227573910083, 1.56898494381396e-32)
(-66.56203155071637, 2.51584770023241e-85)
(-36.717383475806905, -1.06074972494902e-46)
(-80.69909553662578, -1.16293505469971e-103)
(-58.708128765868715, -3.79089561450872e-75)
(-17.86899137183019, -1.85432892168959e-22)
(-91.69461184416505, 6.24099202691414e-118)
(-39.85889084612238, -9.29804353066796e-51)
(-86.9822459046496, -8.16515738010186e-112)
(-82.26988262454168, 1.06510143861097e-105)
(-38.28813538309046, 9.9396993747243e-49)
(7.270855919305716, 44148887429.5558)
(-53.99579823188055, 4.80752794199676e-69)
(1.0064902116737608, 47.9974995837015)
(-3.741894524285712, 9.57867578664121e-5)
(-33.575892361754555, -1.20114566932e-42)
(-50.854250633765915, 5.60897038638296e-65)
(-90.12382292809771, -6.82796998647828e-116)
(-28.86369803416565, 1.4224469447531e-36)
(-94.83619041757741, 5.20951523076622e-122)
(-60.27890766215009, 3.49700905747677e-77)
(-0.4093448944847935, 0.0850457863773253)
(-2.183309495489456, -0.00583591269994781)
(-43.00041085823357, -8.09891376063053e-55)
(-69.70359768417819, 2.12649150425825e-89)
(-30.434422641420802, -1.34802271838103e-38)
(4.132486464869018, 2078015.63486334)
(-22.58090216774983, 1.70425365611299e-28)
(-88.55303427677755, 7.46787149844157e-114)
(-47.71270904623173, 6.5187770497254e-61)
(-55.75, -3.77681415206845e-71)
(-8.44689520600099, -1.63885984207144e-10)
(-21.010240687993335, -1.76356918563097e-26)
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:
$$x_{1} = -11.5870823566467$$
$$x_{2} = -83.840670061203$$
$$x_{3} = -74.4159511177762$$
$$x_{4} = -24.151581643793$$
$$x_{5} = -46.1419408952529$$
$$x_{6} = -64.9912494546192$$
$$x_{7} = 5.70127131092793$$
$$x_{8} = -68.1328143089569$$
$$x_{9} = -52.4250237498346$$
$$x_{10} = -61.849687452674$$
$$x_{11} = -14.7278948683244$$
$$x_{12} = -96.4069800506118$$
$$x_{13} = -36.7173834758069$$
$$x_{14} = -80.6990955366258$$
$$x_{15} = -58.7081287658687$$
$$x_{16} = -17.8689913718302$$
$$x_{17} = -39.8588908461224$$
$$x_{18} = -86.9822459046496$$
$$x_{19} = -33.5758923617546$$
$$x_{20} = -90.1238229280977$$
$$x_{21} = -0.409344894484793$$
$$x_{22} = -2.18330949548946$$
$$x_{23} = -43.0004108582336$$
$$x_{24} = -30.4344226414208$$
$$x_{25} = -8.44689520600099$$
$$x_{26} = -21.0102406879933$$
Puntos máximos de la función:
$$x_{26} = -6.87732970906955$$
$$x_{26} = -32.0051544259545$$
$$x_{26} = -75.9867365833898$$
$$x_{26} = -97.9777698996318$$
$$x_{26} = -72.8451661245562$$
$$x_{26} = -107.402512995711$$
$$x_{26} = -16.2984182815884$$
$$x_{26} = -44.5711747579639$$
$$x_{26} = -10.0168714789543$$
$$x_{26} = -25.7222757391008$$
$$x_{26} = -66.5620315507164$$
$$x_{26} = -91.6946118441651$$
$$x_{26} = -82.2698826245417$$
$$x_{26} = -38.2881353830905$$
$$x_{26} = 7.27085591930572$$
$$x_{26} = -53.9957982318805$$
$$x_{26} = 1.00649021167376$$
$$x_{26} = -3.74189452428571$$
$$x_{26} = -50.8542506337659$$
$$x_{26} = -28.8636980341657$$
$$x_{26} = -94.8361904175774$$
$$x_{26} = -60.2789076621501$$
$$x_{26} = -69.7035976841782$$
$$x_{26} = 4.13248646486902$$
$$x_{26} = -22.5809021677498$$
$$x_{26} = -88.5530342767776$$
$$x_{26} = -47.7127090462317$$
Decrece en los intervalos
$$\left[5.70127131092793, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -96.4069800506118\right]$$