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$$\frac{\left(\tan^{2}{\left(\sqrt{e^{x + 2}} \right)} + 1\right) e^{- x - 2} e^{\frac{x}{2} + 1} e^{x + 2}}{2 \sin{\left(\frac{7 x}{4} - \frac{1}{1000} \right)}} - \frac{7 \cos{\left(\frac{7 x}{4} - \frac{1}{1000} \right)} \tan{\left(\sqrt{e^{x + 2}} \right)}}{4 \sin^{2}{\left(\frac{7 x}{4} - \frac{1}{1000} \right)}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -65.6831037228758$$
$$x_{2} = -81.8398659413375$$
$$x_{3} = -0.370825100569674$$
$$x_{4} = -42.3455582962087$$
$$x_{5} = -56.7071247126192$$
$$x_{6} = -8.23703501208389$$
$$x_{7} = -15.4176214152433$$
$$x_{8} = -27.9839918797988$$
$$x_{9} = -22.5984044737583$$
$$x_{10} = -96.201432357748$$
$$x_{11} = -49.526341504414$$
$$x_{12} = -74.6590827331323$$
$$x_{13} = -99.7918239618507$$
$$x_{14} = -35.1647750880035$$
$$x_{15} = -31.5743834839009$$
$$x_{16} = -69.2734953269784$$
$$x_{17} = -72.863886931081$$
$$x_{18} = -11.8272350915717$$
$$x_{19} = -83.6350617433889$$
$$x_{20} = -80.0446701392862$$
$$x_{21} = -85.4302575454402$$
$$x_{22} = -4.65374211427633$$
$$x_{23} = -2.90016061870246$$
$$x_{24} = -94.4062365556967$$
$$x_{25} = -44.14075409826$$
$$x_{26} = -47.7311457023626$$
$$x_{27} = -97.9966281597993$$
$$x_{28} = -13.6224263653095$$
$$x_{29} = -90.8158449515941$$
$$x_{30} = -51.3215373064653$$
$$x_{31} = -62.0927121187731$$
$$x_{32} = -67.4782995249271$$
$$x_{33} = -6.44283173147076$$
$$x_{34} = -17.2128170923721$$
$$x_{35} = -78.2494743372349$$
$$x_{36} = -29.7791876818496$$
$$x_{37} = -24.3936002757145$$
$$x_{38} = -63.8879079208244$$
$$x_{39} = -38.7551666921061$$
$$x_{40} = -10.0320665556463$$
$$x_{41} = -76.4542785351836$$
$$x_{42} = -92.6110407536454$$
$$x_{43} = -60.2975163167218$$
$$x_{44} = -40.5503624941574$$
$$x_{45} = -26.1887960777501$$
$$x_{46} = -87.2254533474915$$
$$x_{47} = -58.5023205146705$$
$$x_{48} = -20.8032086722794$$
$$x_{49} = -45.9359499003113$$
$$x_{50} = -33.3695792859522$$
$$x_{51} = -54.9119289105679$$
Signos de extremos en los puntos:
(-65.68310372287576, -1.5432263358434e-14)
(-81.83986594133755, 4.78666779628583e-18)
(-0.37082510056967405, 2.01266358109232)
(-42.345558296208715, 1.80348787186641e-9)
(-56.7071247126192, 1.37258327613389e-12)
(-8.237035012083892, -0.0460268214591987)
(-15.417621415243326, -0.00126893878679156)
(-27.983991879798754, 2.36966933848724e-6)
(-22.59840447375831, -3.50068630440781e-5)
(-96.20143235774803, 3.64299658904347e-21)
(-49.526341504413956, 4.97537666075055e-11)
(-74.65908273313231, 1.7350841767129e-16)
(-99.79182396185065, 6.05083254103891e-22)
(-35.16477508800347, 6.53733119268715e-8)
(-31.574383483900867, 3.9359005682082e-7)
(-69.27349532697838, -2.56322066268017e-15)
(-72.863886931081, -4.25737949968296e-16)
(-11.82723509157167, -0.00763997433255879)
(-83.63506174338886, -1.95079427453795e-18)
(-80.04467013928624, -1.17450563040157e-17)
(-85.43025754544017, 7.9504124028055e-19)
(-4.653742114276328, -0.283641696748437)
(-2.9001606187024604, 0.792690797093598)
(-94.40623655569672, -8.93882798527462e-21)
(-44.140754098260025, -7.35006890882542e-10)
(-47.731145702362646, -1.22080916205515e-10)
(-97.99662815979934, -1.48469398557002e-21)
(-13.622426365309462, 0.00311360538532328)
(-90.8158449515941, -5.38175856620394e-20)
(-51.32153730646527, -2.02770209183792e-11)
(-62.09271211877313, -9.29123098262817e-14)
(-67.47829952492707, 6.28937964446894e-15)
(-6.442831731470764, 0.113307341311121)
(-17.212817092372088, 0.00051715255810779)
(-78.24947433723493, 2.88188680425113e-17)
(-29.77918768184963, -9.65752706224852e-7)
(-24.393600275714547, 1.42669579090917e-5)
(-63.88790792082444, 3.7866175334718e-14)
(-38.755166692106094, 1.08581755006931e-8)
(-10.032066555646251, 0.0187479133661977)
(-76.45427853518362, -7.07129139063991e-17)
(-92.61104075364541, 2.19332200284352e-20)
(-60.29751631672182, 2.27979119648242e-13)
(-40.550362494157405, -4.42522178270163e-9)
(-26.188796077750066, -5.81446237490517e-6)
(-87.22545334749148, -3.24017033470406e-19)
(-58.50232051467051, -5.59392819883223e-13)
(-20.803208672279414, 8.58964097275147e-5)
(-45.935949900311336, 2.99550187207933e-10)
(-33.36957928595216, -1.6040662566073e-7)
(-54.91192891056789, -3.36791031804043e-12)
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:
$$x_{1} = -81.8398659413375$$
$$x_{2} = -0.370825100569674$$
$$x_{3} = -42.3455582962087$$
$$x_{4} = -56.7071247126192$$
$$x_{5} = -27.9839918797988$$
$$x_{6} = -96.201432357748$$
$$x_{7} = -49.526341504414$$
$$x_{8} = -74.6590827331323$$
$$x_{9} = -99.7918239618507$$
$$x_{10} = -35.1647750880035$$
$$x_{11} = -31.5743834839009$$
$$x_{12} = -85.4302575454402$$
$$x_{13} = -2.90016061870246$$
$$x_{14} = -13.6224263653095$$
$$x_{15} = -67.4782995249271$$
$$x_{16} = -6.44283173147076$$
$$x_{17} = -17.2128170923721$$
$$x_{18} = -78.2494743372349$$
$$x_{19} = -24.3936002757145$$
$$x_{20} = -63.8879079208244$$
$$x_{21} = -38.7551666921061$$
$$x_{22} = -10.0320665556463$$
$$x_{23} = -92.6110407536454$$
$$x_{24} = -60.2975163167218$$
$$x_{25} = -20.8032086722794$$
$$x_{26} = -45.9359499003113$$
Puntos máximos de la función:
$$x_{26} = -65.6831037228758$$
$$x_{26} = -8.23703501208389$$
$$x_{26} = -15.4176214152433$$
$$x_{26} = -22.5984044737583$$
$$x_{26} = -69.2734953269784$$
$$x_{26} = -72.863886931081$$
$$x_{26} = -11.8272350915717$$
$$x_{26} = -83.6350617433889$$
$$x_{26} = -80.0446701392862$$
$$x_{26} = -4.65374211427633$$
$$x_{26} = -94.4062365556967$$
$$x_{26} = -44.14075409826$$
$$x_{26} = -47.7311457023626$$
$$x_{26} = -97.9966281597993$$
$$x_{26} = -90.8158449515941$$
$$x_{26} = -51.3215373064653$$
$$x_{26} = -62.0927121187731$$
$$x_{26} = -29.7791876818496$$
$$x_{26} = -76.4542785351836$$
$$x_{26} = -40.5503624941574$$
$$x_{26} = -26.1887960777501$$
$$x_{26} = -87.2254533474915$$
$$x_{26} = -58.5023205146705$$
$$x_{26} = -33.3695792859522$$
$$x_{26} = -54.9119289105679$$
Decrece en los intervalos
$$\left[-0.370825100569674, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.7918239618507\right]$$