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$$\left(\left(2 x + \left(x^{2} + 1\right)\right) + \left(- 2 \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}\right) e^{x} + \left(2 x + \left(- 2 \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} + \left(- \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) e^{x} + 2\right) e^{x} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -93.4135149755732$$
$$x_{2} = -52.0625314136456$$
$$x_{3} = -65.7215707857911$$
$$x_{4} = -38.8866033852926$$
$$x_{5} = -61.7970469090044$$
$$x_{6} = -40.7021606138828$$
$$x_{7} = -105.34083441056$$
$$x_{8} = -109.320766981846$$
$$x_{9} = -67.6883070820612$$
$$x_{10} = -69.6575690448132$$
$$x_{11} = 12.5665178864624$$
$$x_{12} = -59.8401098081453$$
$$x_{13} = 2.88302862300745$$
$$x_{14} = -113.302350381263$$
$$x_{15} = -81.5129914542486$$
$$x_{16} = -87.4590521969804$$
$$x_{17} = -101.362785703358$$
$$x_{18} = 6.30797146170616$$
$$x_{19} = -119.277400270678$$
$$x_{20} = 0.677247254969318$$
$$x_{21} = -48.218087356325$$
$$x_{22} = -57.8874330100957$$
$$x_{23} = -103.351558330607$$
$$x_{24} = -63.7576880804052$$
$$x_{25} = -79.5332552098772$$
$$x_{26} = -35.4100272197758$$
$$x_{27} = -99.3745529423273$$
$$x_{28} = -1.69742009717348$$
$$x_{29} = -91.4278853676496$$
$$x_{30} = -117.285388562094$$
$$x_{31} = -83.4939382270426$$
$$x_{32} = -42.5499668447831$$
$$x_{33} = -44.4219620932777$$
$$x_{34} = -2.97924417035108$$
$$x_{35} = -111.311365356229$$
$$x_{36} = -75.577910111863$$
$$x_{37} = -95.3998711831007$$
$$x_{38} = -121.269715394308$$
$$x_{39} = -55.93969182993$$
$$x_{40} = -107.330580740598$$
$$x_{41} = -115.293698637214$$
$$x_{42} = -85.4759897300653$$
$$x_{43} = -71.6290775076557$$
$$x_{44} = 9.4226838207314$$
$$x_{45} = -77.5548493298981$$
$$x_{46} = -73.6025932625681$$
$$x_{47} = -53.9977149340079$$
$$x_{48} = -97.3869000709892$$
$$x_{49} = -37.1157134040609$$
$$x_{50} = -46.3126397955165$$
$$x_{51} = -50.1354367467416$$
$$x_{52} = 15.7079538468433$$
$$x_{53} = -89.4430422232197$$
Signos de extremos en los puntos:
(-93.41351497557318, 2.30407647152512e-37)
(-52.0625314136456, 6.39343972624134e-20)
(-65.72157078579109, 1.20110939427611e-25)
(-38.88660338529258, 1.85666723330323e-14)
(-61.79704690900439, 5.3659731984164e-24)
(-40.70216061388277, 3.31821504233813e-15)
(-105.34083441056026, 1.94073173390061e-42)
(-109.32076698184598, 3.90856730846474e-44)
(-67.68830708206119, 1.78419012120052e-26)
(-69.65756904481317, 2.63924061365338e-27)
(12.566517886462412, 82279095595.3478)
(-59.84010980814534, 3.55727518601997e-23)
(2.8830286230074504, -202.519543113505)
(-113.30235038126258, 7.83776312909701e-46)
(-81.51299145424862, 2.57685234663256e-32)
(-87.45905219698041, 7.77386830240892e-35)
(-101.36278570335809, 9.59062506726681e-41)
(6.307971461706159, 315612.507291013)
(-119.27740027067756, 2.20946327407096e-48)
(0.6772472549693177, 3.70091126944644)
(-48.21808735632502, 2.55486240594015e-18)
(-57.88743301009572, 2.34336503915629e-22)
(-103.35155833060679, 1.36514077767275e-41)
(-63.75768808040524, 8.04862624344384e-25)
(-79.53325520987718, 1.77522470555565e-31)
(-35.41002721977578, 4.95439474937152e-13)
(-99.37455294232734, 6.72892300611602e-40)
(-1.6974200971734845, 0.151404169344921)
(-91.42788536764964, 1.60686223129953e-36)
(-117.28538856209434, 1.56550059270147e-47)
(-83.49393822704256, 3.73153423063277e-33)
(-42.549966844783086, 5.72697856040409e-16)
(-44.421962093277685, 9.62068234384569e-17)
(-2.9792441703510804, 0.197412114144383)
(-111.31136535622898, 5.53769168851023e-45)
(-75.57791011186302, 8.35893797444063e-30)
(-95.3998711831007, 3.29841406899876e-38)
(-121.26971539430798, 3.11561903698671e-49)
(-55.93969182993003, 1.53275576033734e-21)
(-107.33058074059751, 2.75573477437007e-43)
(-115.29369863721371, 1.10822376178835e-46)
(-85.47598973006525, 5.39157812314283e-34)
(-71.62907750765571, 3.88914345170604e-28)
(9.422683820731399, -152207938.972468)
(-77.55484932989813, 1.21983840758805e-30)
(-73.60259326256813, 5.71089478551578e-29)
(-53.997714934007874, 9.94499704535209e-21)
(-97.38690007098918, 4.71457680860106e-39)
(-37.11571340406087, 9.9138620796487e-14)
(-46.31263979551651, 1.58166615656916e-17)
(-50.13543674674165, 4.0667372566538e-19)
(15.707953846843301, -44029653495906.4)
(-89.44304222321975, 1.11868461008945e-35)
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:
$$x_{1} = 2.88302862300745$$
$$x_{2} = -1.69742009717348$$
$$x_{3} = 9.4226838207314$$
$$x_{4} = 15.7079538468433$$
Puntos máximos de la función:
$$x_{4} = 12.5665178864624$$
$$x_{4} = 6.30797146170616$$
$$x_{4} = 0.677247254969318$$
$$x_{4} = -2.97924417035108$$
Decrece en los intervalos
$$\left[15.7079538468433, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -1.69742009717348\right]$$