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$$2 x e^{x} + 2 e^{x} - \frac{\sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\left|{\cos{\left(x \right)}}\right|} + \frac{\cos{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)}}{\left|{\sin{\left(x \right)}}\right|} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -46.3384916404494$$
$$x_{2} = -30.6305283725012$$
$$x_{3} = -98.174770424681$$
$$x_{4} = -16.4933619636257$$
$$x_{5} = -68.329640215578$$
$$x_{6} = -84.037603483527$$
$$x_{7} = -55.7632696012188$$
$$x_{8} = -85.6083998103219$$
$$x_{9} = -19.6349541126022$$
$$x_{10} = -41.6261026600648$$
$$x_{11} = -93.4623814442964$$
$$x_{12} = -71.4712328691678$$
$$x_{13} = -33.7721210260903$$
$$x_{14} = -76.1836218495525$$
$$x_{15} = -54.1924732744239$$
$$x_{16} = -65.1880475619882$$
$$x_{17} = -10.2103455381236$$
$$x_{18} = -96.6039740978861$$
$$x_{19} = -27.488935718926$$
$$x_{20} = 1.52798726886801$$
$$x_{21} = -60.4756585816035$$
$$x_{22} = -43.1968989868597$$
$$x_{23} = -13.3517785974004$$
$$x_{24} = -38.484510006475$$
$$x_{25} = -49.4800842940392$$
$$x_{26} = -5.50693190633047$$
$$x_{27} = -62.0464549083984$$
$$x_{28} = -87.1791961371168$$
$$x_{29} = -57.3340659280137$$
$$x_{30} = -90.3207887907066$$
$$x_{31} = -79.3252145031423$$
$$x_{32} = -2.41901457891826$$
$$x_{33} = -25.9181393921849$$
$$x_{34} = -35.3429173528852$$
$$x_{35} = -63.6172512351933$$
$$x_{36} = -24.3473430656323$$
$$x_{37} = -0.716639609763256$$
$$x_{38} = -52.621676947629$$
$$x_{39} = -69.9004365423729$$
$$x_{40} = -7.07116140316689$$
$$x_{41} = -21.2057504179671$$
$$x_{42} = -32.2013246992955$$
$$x_{43} = -18.0641578800096$$
$$x_{44} = -11.7810136797307$$
$$x_{45} = -77.7544181763474$$
$$x_{46} = -99.7455667514759$$
$$x_{47} = -40.0553063332699$$
$$x_{48} = -82.4668071567321$$
$$x_{49} = -47.9092879672443$$
$$x_{50} = -91.8915851175014$$
$$x_{51} = -3.95526266232386$$
$$x_{52} = -74.6128255227576$$
Signos de extremos en los puntos:
(-46.33849164044945, -0.693147180559945)
(-30.63052837250122, -0.693147180562997)
(-98.17477042468104, -0.693147180559945)
(-16.49336196362572, -0.693149447098994)
(-68.329640215578, -0.693147180559945)
(-84.03760348352696, -0.693147180559945)
(-55.76326960121883, -0.693147180559945)
(-85.60839981032187, -0.693147180559945)
(-19.63495411260219, -0.693147297162393)
(-41.62610266006476, -0.693147180559946)
(-93.46238144429635, -0.693147180559945)
(-71.47123286916779, -0.693147180559945)
(-33.77212102609031, -0.693147180560091)
(-76.18362184955248, -0.693147180559945)
(-54.19247327442393, -0.693147180559945)
(-65.18804756198821, -0.693147180559945)
(-10.210345538123603, -0.693898469323288)
(-96.60397409788614, -0.693147180559945)
(-27.48893571892596, -0.693147180623317)
(1.5279872688680114, 10.9324260034019)
(-60.47565858160352, -0.693147180559945)
(-43.19689898685966, -0.693147180559945)
(-13.351778597400418, -0.693189639321455)
(-38.48451000647497, -0.693147180559947)
(-49.480084294039244, -0.693147180559945)
(-5.506931906330469, -0.738014653199873)
(-62.04645490839842, -0.693147180559945)
(-87.17919613711676, -0.693147180559945)
(-57.33406592801373, -0.693147180559945)
(-90.32078879070656, -0.693147180559945)
(-79.32521450314228, -0.693147180559945)
(-2.4190145789182647, -1.13169028095714)
(-25.91813939218488, -0.693147180847373)
(-35.34291735288518, -0.693147180559977)
(-63.617251235193315, -0.693147180559945)
(-24.347343065632277, -0.693147181858813)
(-0.716639609763256, -1.40263283070348)
(-52.621676947629034, -0.693147180559945)
(-69.9004365423729, -0.693147180559945)
(-7.0711614031668875, -0.705170786262028)
(-21.20575041796708, -0.693147206738355)
(-32.20132469929554, -0.693147180560612)
(-18.064157880009617, -0.693147696600286)
(-11.781013679730744, -0.693327395842177)
(-77.75441817634739, -0.693147180559945)
(-99.74556675147593, -0.693147180559945)
(-40.05530633326986, -0.693147180559946)
(-82.46680715673207, -0.693147180559945)
(-47.909287967244346, -0.693147180559945)
(-91.89158511750145, -0.693147180559945)
(-3.955262662323856, -0.84626195775183)
(-74.61282552275759, -0.693147180559945)
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:
$$x_{52} = -46.3384916404494$$
$$x_{52} = -30.6305283725012$$
$$x_{52} = -98.174770424681$$
$$x_{52} = -16.4933619636257$$
$$x_{52} = -68.329640215578$$
$$x_{52} = -84.037603483527$$
$$x_{52} = -55.7632696012188$$
$$x_{52} = -85.6083998103219$$
$$x_{52} = -19.6349541126022$$
$$x_{52} = -41.6261026600648$$
$$x_{52} = -93.4623814442964$$
$$x_{52} = -71.4712328691678$$
$$x_{52} = -33.7721210260903$$
$$x_{52} = -76.1836218495525$$
$$x_{52} = -54.1924732744239$$
$$x_{52} = -65.1880475619882$$
$$x_{52} = -10.2103455381236$$
$$x_{52} = -96.6039740978861$$
$$x_{52} = -27.488935718926$$
$$x_{52} = 1.52798726886801$$
$$x_{52} = -60.4756585816035$$
$$x_{52} = -43.1968989868597$$
$$x_{52} = -13.3517785974004$$
$$x_{52} = -38.484510006475$$
$$x_{52} = -49.4800842940392$$
$$x_{52} = -5.50693190633047$$
$$x_{52} = -62.0464549083984$$
$$x_{52} = -87.1791961371168$$
$$x_{52} = -57.3340659280137$$
$$x_{52} = -90.3207887907066$$
$$x_{52} = -79.3252145031423$$
$$x_{52} = -2.41901457891826$$
$$x_{52} = -25.9181393921849$$
$$x_{52} = -35.3429173528852$$
$$x_{52} = -63.6172512351933$$
$$x_{52} = -24.3473430656323$$
$$x_{52} = -0.716639609763256$$
$$x_{52} = -52.621676947629$$
$$x_{52} = -69.9004365423729$$
$$x_{52} = -7.07116140316689$$
$$x_{52} = -21.2057504179671$$
$$x_{52} = -32.2013246992955$$
$$x_{52} = -18.0641578800096$$
$$x_{52} = -11.7810136797307$$
$$x_{52} = -77.7544181763474$$
$$x_{52} = -99.7455667514759$$
$$x_{52} = -40.0553063332699$$
$$x_{52} = -82.4668071567321$$
$$x_{52} = -47.9092879672443$$
$$x_{52} = -91.8915851175014$$
$$x_{52} = -3.95526266232386$$
$$x_{52} = -74.6128255227576$$
Decrece en los intervalos
$$\left(-\infty, -99.7455667514759\right]$$
Crece en los intervalos
$$\left[1.52798726886801, \infty\right)$$