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$$2 x e^{x} \operatorname{sign}{\left(x^{2} - 1 \right)} + e^{x} \left|{x^{2} - 1}\right| = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -95.3933840527324$$
$$x_{2} = -38.7941628623993$$
$$x_{3} = -93.4066913476332$$
$$x_{4} = -83.4849610815635$$
$$x_{5} = -67.672930203154$$
$$x_{6} = -111.306857078541$$
$$x_{7} = -113.298023778539$$
$$x_{8} = -87.4510448712257$$
$$x_{9} = -89.4354616849822$$
$$x_{10} = -107.325672932625$$
$$x_{11} = -40.6259139055514$$
$$x_{12} = -79.5231196989186$$
$$x_{13} = -59.8185637723201$$
$$x_{14} = -73.5902464456082$$
$$x_{15} = -99.3686680130171$$
$$x_{16} = -115.289542922953$$
$$x_{17} = -85.4675183643145$$
$$x_{18} = -69.6433164421341$$
$$x_{19} = 0.414213562373095$$
$$x_{20} = -37.7300917842729$$
$$x_{21} = -81.5034616017267$$
$$x_{22} = -121.266015624442$$
$$x_{23} = -105.33570656639$$
$$x_{24} = -109.316065313131$$
$$x_{25} = -119.273557249601$$
$$x_{26} = -97.3807250722054$$
$$x_{27} = -65.704929844768$$
$$x_{28} = -57.8637540821341$$
$$x_{29} = -52.03009245017$$
$$x_{30} = -117.281393788779$$
$$x_{31} = -77.544048180243$$
$$x_{32} = -101.357170818505$$
$$x_{33} = -44.3672208724359$$
$$x_{34} = -61.7773551203297$$
$$x_{35} = -53.9686782648593$$
$$x_{36} = -35.26226422642$$
$$x_{37} = -50.0989460197047$$
$$x_{38} = -71.6158294137195$$
$$x_{39} = -103.346195271843$$
$$x_{40} = -55.9135416592293$$
$$x_{41} = -48.1767141989494$$
$$x_{42} = -75.5663751001662$$
$$x_{43} = -91.4206982622687$$
$$x_{44} = -37.0008140512653$$
$$x_{45} = -63.7396188614987$$
$$x_{46} = -46.2653023401513$$
$$x_{47} = -42.4858568772826$$
$$x_{48} = -2.41421356237309$$
Signos de extremos en los puntos:
(-95.39338405273239, 3.38975631949078e-38)
(-38.79416286239926, 2.13379769677655e-14)
(-93.40669134763324, 2.36971202639343e-37)
(-83.48496108156348, 3.85563809456968e-33)
(-67.67293020315404, 1.86532692701039e-26)
(-111.3068570785408, 5.66310947340503e-45)
(-113.29802377853865, 8.01132402523455e-46)
(-87.45104487122568, 8.01617131470696e-35)
(-89.43546168498222, 1.15249144751629e-35)
(-107.32567293262527, 2.82112295315694e-43)
(-40.62591390555136, 3.74742406857647e-15)
(-79.52311969891859, 1.83851030746967e-31)
(-59.818563772320104, 3.75559207646351e-23)
(-73.59024644560816, 5.93912354719363e-29)
(-99.36866801301709, 6.90543051051201e-40)
(-115.28954292295346, 1.1322304923496e-46)
(-85.46751836431446, 5.56507628743757e-34)
(-69.64331644213406, 2.75398331789066e-27)
(0.41421356237309503, 1.25355956434731)
(-37.730091784272915, 5.8492458577976e-14)
(-81.50346160172673, 2.66552706251889e-32)
(-121.26601562444176, 3.17897604139287e-49)
(-105.33570656639, 1.98790659016081e-42)
(-109.31606531313103, 3.99914996256706e-44)
(-119.27355724960071, 2.25532983233445e-48)
(-97.38072507220538, 4.84159731605541e-39)
(-65.70492984476797, 1.25836559866923e-25)
(-57.8637540821341, 2.48184345247097e-22)
(-52.030092450170045, 6.85435660589161e-20)
(-117.28139378877921, 1.59869092545485e-47)
(-77.54404818024295, 1.2649474358699e-30)
(-101.35717081850541, 9.83573218957909e-41)
(-44.36722087243588, 1.06038741672969e-16)
(-61.77735512032974, 5.64911447979659e-24)
(-53.96867826485926, 1.06129291359449e-20)
(-35.26226422641998, 6.02648673714973e-13)
(-50.098946019704684, 4.38316991239769e-19)
(-71.61582941371945, 4.0511081306763e-28)
(-103.3461952718431, 1.39915551390878e-41)
(-55.91354165922927, 1.62911769281967e-21)
(-48.176714198949384, 2.77080564749139e-18)
(-75.5663751001662, 8.68003336311423e-30)
(-91.42069826226873, 1.65398784961654e-36)
(-37.000814051265266, 1.1664224044489e-13)
(-63.739618861498705, 8.45176210251136e-25)
(-46.265302340151294, 1.72799450007234e-17)
(-42.48585687728257, 6.3807965305151e-16)
(-2.414213562373095, 0.431843167522967)
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:
La función no tiene puntos mínimos
Puntos máximos de la función:
$$x_{48} = 0.414213562373095$$
$$x_{48} = -2.41421356237309$$
Decrece en los intervalos
$$\left(-\infty, -2.41421356237309\right]$$
Crece en los intervalos
$$\left[0.414213562373095, \infty\right)$$