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$$\left(- \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} - 2\right) e^{x^{1}} - e^{x^{1}} e^{- \frac{x^{2}}{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -60.8720030830002$$
$$x_{2} = -84.8720030830002$$
$$x_{3} = -110.872003083$$
$$x_{4} = -104.872003083$$
$$x_{5} = -50.8720030830002$$
$$x_{6} = -32.8720030830002$$
$$x_{7} = -82.8720030830002$$
$$x_{8} = -116.872003083$$
$$x_{9} = -106.872003083$$
$$x_{10} = -70.8720030830002$$
$$x_{11} = -108.872003083$$
$$x_{12} = -54.8720030830002$$
$$x_{13} = -38.8720030830002$$
$$x_{14} = -98.8720030830002$$
$$x_{15} = -74.8720030830002$$
$$x_{16} = -40.8720030830002$$
$$x_{17} = -48.8720030830002$$
$$x_{18} = -118.872003083$$
$$x_{19} = -94.8720030830002$$
$$x_{20} = -30.8720030830002$$
$$x_{21} = -66.8720030830002$$
$$x_{22} = -44.8720030830002$$
$$x_{23} = -42.8720030830002$$
$$x_{24} = -36.8720030830002$$
$$x_{25} = -64.8720030830002$$
$$x_{26} = -34.8720030830002$$
$$x_{27} = -62.8720030830002$$
$$x_{28} = -80.8720030830002$$
$$x_{29} = -112.872003083$$
$$x_{30} = -102.872003083$$
$$x_{31} = -96.8720030830002$$
$$x_{32} = -88.8720030830002$$
$$x_{33} = -120.872003083$$
$$x_{34} = -28.8720030829998$$
$$x_{35} = -76.8720030830002$$
$$x_{36} = -72.8720030830002$$
$$x_{37} = -56.8720030830002$$
$$x_{38} = -52.8720030830002$$
$$x_{39} = -68.8720030830002$$
$$x_{40} = -58.8720030830002$$
$$x_{41} = -86.8720030830002$$
$$x_{42} = -92.8720030830002$$
$$x_{43} = -46.8720030830002$$
$$x_{44} = -100.872003083$$
$$x_{45} = -114.872003083$$
$$x_{46} = -78.8720030830002$$
$$x_{47} = -90.8720030830002$$
Signos de extremos en los puntos:
___ ____ / ___\
(-60.872003083000195, -7.32242537443395e-27 + 1.83060634360849e-27*\/ 2 *\/ pi *erf\30.4360015415001*\/ 2 /)
___ ____ / ___\
(-84.8720030830002, -2.76431409789314e-37 + 6.91078524473285e-38*\/ 2 *\/ pi *erf\42.4360015415001*\/ 2 /)
___ ____ / ___\
(-110.8720030830002, -1.41231268276666e-48 + 3.53078170691665e-49*\/ 2 *\/ pi *erf\55.4360015415001*\/ 2 /)
___ ____ / ___\
(-104.8720030830002, -5.69767601643042e-46 + 1.42441900410761e-46*\/ 2 *\/ pi *erf\52.4360015415001*\/ 2 /)
___ ____ / ___\
(-50.872003083000195, -1.61287152044994e-22 + 4.03217880112485e-23*\/ 2 *\/ pi *erf\25.4360015415001*\/ 2 /)
___ ____ / ___\
(-32.872003083000195, -1.05901094255222e-14 + 2.64752735638056e-15*\/ 2 *\/ pi *erf\16.4360015415001*\/ 2 /)
___ ____ / ___\
(-82.8720030830002, -2.04256719443973e-36 + 5.10641798609932e-37*\/ 2 *\/ pi *erf\41.4360015415001*\/ 2 /)
___ ____ / ___\
(-116.8720030830002, -3.50077313654137e-51 + 8.75193284135341e-52*\/ 2 *\/ pi *erf\58.4360015415001*\/ 2 /)
___ ____ / ___\
(-106.8720030830002, -7.71096597474066e-47 + 1.92774149368517e-47*\/ 2 *\/ pi *erf\53.4360015415001*\/ 2 /)
___ ____ / ___\
(-70.8720030830002, -3.32437597690338e-31 + 8.31093994225845e-32*\/ 2 *\/ pi *erf\35.4360015415001*\/ 2 /)
___ ____ / ___\
(-108.8720030830002, -1.04356576421941e-47 + 2.60891441054853e-48*\/ 2 *\/ pi *erf\54.4360015415001*\/ 2 /)
___ ____ / ___\
(-54.872003083000195, -2.95407723424848e-24 + 7.38519308562119e-25*\/ 2 *\/ pi *erf\27.4360015415001*\/ 2 /)
___ ____ / ___\
(-38.872003083000195, -2.62502567896482e-17 + 6.56256419741205e-18*\/ 2 *\/ pi *erf\19.4360015415001*\/ 2 /)
___ ____ / ___\
(-98.8720030830002, -2.29860656102102e-43 + 5.74651640255255e-44*\/ 2 *\/ pi *erf\49.4360015415001*\/ 2 /)
___ ____ / ___\
(-74.8720030830002, -6.08880699233452e-33 + 1.52220174808363e-33*\/ 2 *\/ pi *erf\37.4360015415001*\/ 2 /)
___ ____ / ___\
(-40.872003083000195, -3.55258593766085e-18 + 8.88146484415213e-19*\/ 2 *\/ pi *erf\20.4360015415001*\/ 2 /)
___ ____ / ___\
(-48.872003083000195, -1.19175981449722e-21 + 2.97939953624305e-22*\/ 2 *\/ pi *erf\24.4360015415001*\/ 2 /)
___ ____ / ___\
(-118.8720030830002, -4.73778123980951e-52 + 1.18444530995238e-52*\/ 2 *\/ pi *erf\59.4360015415001*\/ 2 /)
___ ____ / ___\
(-94.8720030830002, -1.25499665885795e-41 + 3.13749164714488e-42*\/ 2 *\/ pi *erf\47.4360015415001*\/ 2 /)
___ ____ / ___\
(-30.87200308300019, -7.82509126389984e-14 + 1.95627281597496e-14*\/ 2 *\/ pi *erf\15.4360015415001*\/ 2 /)
___ ____ / ___\
(-66.8720030830002, -1.81504778353551e-29 + 4.53761945883878e-30*\/ 2 *\/ pi *erf\33.4360015415001*\/ 2 /)
___ ____ / ___\
(-44.872003083000195, -6.50678811553913e-20 + 1.62669702888478e-20*\/ 2 *\/ pi *erf\22.4360015415001*\/ 2 /)
___ ____ / ___\
(-42.872003083000195, -4.80790224095739e-19 + 1.20197556023935e-19*\/ 2 *\/ pi *erf\21.4360015415001*\/ 2 /)
___ ____ / ___\
(-36.872003083000195, -1.93964620030046e-16 + 4.84911550075114e-17*\/ 2 *\/ pi *erf\18.4360015415001*\/ 2 /)
___ ____ / ___\
(-64.8720030830002, -1.34114898947836e-28 + 3.35287247369591e-29*\/ 2 *\/ pi *erf\32.4360015415001*\/ 2 /)
___ ____ / ___\
(-34.872003083000195, -1.43321545860978e-15 + 3.58303864652444e-16*\/ 2 *\/ pi *erf\17.4360015415001*\/ 2 /)
___ ____ / ___\
(-62.872003083000195, -9.90982512027978e-28 + 2.47745628006995e-28*\/ 2 *\/ pi *erf\31.4360015415001*\/ 2 /)
___ ____ / ___\
(-80.8720030830002, -1.50926435855505e-35 + 3.77316089638764e-36*\/ 2 *\/ pi *erf\40.4360015415001*\/ 2 /)
___ ____ / ___\
(-112.8720030830002, -1.91135736940886e-49 + 4.77839342352216e-50*\/ 2 *\/ pi *erf\56.4360015415001*\/ 2 /)
___ ____ / ___\
(-102.8720030830002, -4.21004477189361e-45 + 1.0525111929734e-45*\/ 2 *\/ pi *erf\51.4360015415001*\/ 2 /)
___ ____ / ___\
(-96.8720030830002, -1.69845328287544e-42 + 4.24613320718859e-43*\/ 2 *\/ pi *erf\48.4360015415001*\/ 2 /)
___ ____ / ___\
(-88.8720030830002, -5.06301787920477e-39 + 1.26575446980119e-39*\/ 2 *\/ pi *erf\44.4360015415001*\/ 2 /)
___ ____ / ___\
(-120.8720030830002, -6.4118896600273e-53 + 1.60297241500682e-53*\/ 2 *\/ pi *erf\60.4360015415001*\/ 2 /)
___ ____ / ___\
(-28.87200308299975, -5.78200383282335e-13 + 1.44550095820584e-13*\/ 2 *\/ pi *erf\14.4360015414999*\/ 2 /)
___ ____ / ___\
(-76.8720030830002, -8.24030418880661e-34 + 2.06007604720165e-34*\/ 2 *\/ pi *erf\38.4360015415001*\/ 2 /)
___ ____ / ___\
(-72.8720030830002, -4.4990536441921e-32 + 1.12476341104803e-32*\/ 2 *\/ pi *erf\36.4360015415001*\/ 2 /)
___ ____ / ___\
(-56.872003083000195, -3.99790879199847e-25 + 9.99477197999618e-26*\/ 2 *\/ pi *erf\28.4360015415001*\/ 2 /)
___ ____ / ___\
(-52.872003083000195, -2.18278424044359e-23 + 5.45696060110897e-24*\/ 2 *\/ pi *erf\26.4360015415001*\/ 2 /)
___ ____ / ___\
(-68.8720030830002, -2.45640005872765e-30 + 6.14100014681912e-31*\/ 2 *\/ pi *erf\34.4360015415001*\/ 2 /)
___ ____ / ___\
(-58.872003083000195, -5.41058118719257e-26 + 1.35264529679814e-26*\/ 2 *\/ pi *erf\29.4360015415001*\/ 2 /)
___ ____ / ___\
(-86.8720030830002, -3.7410923139333e-38 + 9.35273078483324e-39*\/ 2 *\/ pi *erf\43.4360015415001*\/ 2 /)
___ ____ / ___\
(-92.8720030830002, -9.27324071627194e-41 + 2.31831017906799e-41*\/ 2 *\/ pi *erf\46.4360015415001*\/ 2 /)
___ ____ / ___\
(-46.872003083000195, -8.80598012577114e-21 + 2.20149503144278e-21*\/ 2 *\/ pi *erf\23.4360015415001*\/ 2 /)
___ ____ / ___\
(-100.8720030830002, -3.11082569985316e-44 + 7.77706424963289e-45*\/ 2 *\/ pi *erf\50.4360015415001*\/ 2 /)
___ ____ / ___\
(-114.8720030830002, -2.58674090955336e-50 + 6.46685227388339e-51*\/ 2 *\/ pi *erf\57.4360015415001*\/ 2 /)
___ ____ / ___\
(-78.8720030830002, -1.11520390134799e-34 + 2.78800975336997e-35*\/ 2 *\/ pi *erf\39.4360015415001*\/ 2 /)
___ ____ / ___\
(-90.8720030830002, -6.85204958714212e-40 + 1.71301239678553e-40*\/ 2 *\/ pi *erf\45.4360015415001*\/ 2 /)
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
La función no tiene puntos máximos
Decrece en todo el eje numérico