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$$x e^{\sin{\left(x \right)}} \cos{\left(x \right)} + e^{\sin{\left(x \right)}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 29.8115799030901$$
$$x_{2} = -39.2444240846477$$
$$x_{3} = -83.240191603726$$
$$x_{4} = -80.1230937867295$$
$$x_{5} = 92.6661916492115$$
$$x_{6} = -7.72415319239641$$
$$x_{7} = -36.1559769880743$$
$$x_{8} = -64.3871177170664$$
$$x_{9} = 70.699979453112$$
$$x_{10} = 89.5465582344838$$
$$x_{11} = -73.8409703906111$$
$$x_{12} = 23.5194140147849$$
$$x_{13} = -17.336473487102$$
$$x_{14} = 20.4692255293053$$
$$x_{15} = 48.6741398947227$$
$$x_{16} = 86.3822212589452$$
$$x_{17} = -89.5242202334874$$
$$x_{18} = 42.3879070002498$$
$$x_{19} = 7.97963107097301$$
$$x_{20} = 36.1006116108761$$
$$x_{21} = -61.2773767058956$$
$$x_{22} = -95.8081382182729$$
$$x_{23} = 73.8138793572668$$
$$x_{24} = 10.9037335277384$$
$$x_{25} = -42.4350684201498$$
$$x_{26} = 4.48766960334109$$
$$x_{27} = 95.829011377113$$
$$x_{28} = -14.0660135689384$$
$$x_{29} = 80.0981276558536$$
$$x_{30} = -45.5311287148944$$
$$x_{31} = -32.9563750616135$$
$$x_{32} = -76.9560252131026$$
$$x_{33} = 76.9820104261667$$
$$x_{34} = 61.2447280834131$$
$$x_{35} = 2.07393280909122$$
$$x_{36} = -86.405371586641$$
$$x_{37} = 98.9500623082067$$
$$x_{38} = 64.4181735917203$$
$$x_{39} = 114.676852122197$$
$$x_{40} = 26.740942117298$$
$$x_{41} = 17.2206571155732$$
$$x_{42} = 45.5750370742992$$
$$x_{43} = 67.5294331532335$$
$$x_{44} = 54.959675275262$$
$$x_{45} = -70.671684294851$$
$$x_{46} = -23.6043227065406$$
$$x_{47} = 58.1366657885594$$
$$x_{48} = -51.8169788924771$$
$$x_{49} = -29.8786052250774$$
$$x_{50} = 51.8555643132686$$
$$x_{51} = -58.1022522048044$$
$$x_{52} = -67.5590444598741$$
$$x_{53} = -20.3712437074438$$
$$x_{54} = -26.6660278619112$$
$$x_{55} = 14.2076100006438$$
Signos de extremos en los puntos:
(29.811579903090074, 10.9732409049855)
(-39.2444240846477, -14.4419053517253)
(-83.240191603726, -30.6245650761363)
(-80.12309378672954, -217.780186719688)
(92.66619164921153, 34.0919718861739)
(-7.724153192396411, -2.86557227944294)
(-36.15597698807427, -98.2445441921587)
(-64.38711771706639, -23.6895540072882)
(70.69997945311201, 192.16324535601)
(89.54655823448383, 243.397604013506)
(-73.84097039061113, -200.702161683353)
(23.519414014784864, 8.66013672223406)
(-17.336473487101994, -47.0470230402469)
(20.46922552930527, 55.574724548253)
(48.67413989472275, 17.9099951847023)
(86.3822212589452, 31.7803727966998)
(-89.52422023348744, -32.9361748753839)
(42.3879070002498, 15.5979801862797)
(7.979631070973006, 21.5205580854683)
(36.100611610876136, 13.2857699778249)
(-61.27737670589561, -166.546999453251)
(-95.80813821827292, -35.2477643277421)
(73.81387935726681, 27.1571008568398)
(10.903733527738439, 4.02820011709472)
(-42.43506842014976, -115.318446655742)
(4.487669603341088, 1.69295728126075)
(95.82901137711305, 260.476077285986)
(-14.066013568938363, -5.18770724663809)
(80.09812765585362, 29.4687510496163)
(-45.53112871489442, -16.7540070295327)
(-32.95637506161347, -12.1295567206951)
(-76.95602521310259, -28.3129299445716)
(76.9820104261667, 209.241144754145)
(61.24472808341312, 22.533680098419)
(2.073932809091215, 4.98045499108776)
(-86.40537158664104, -234.858421645529)
(98.9500623082067, 36.4035526354082)
(64.41817359172032, 175.085651978291)
(114.67685212219666, 311.712151348327)
(26.740942117297966, 72.6385908126525)
(17.220657115573236, 6.34582510262999)
(45.57503707429922, 123.855973062763)
(67.52943315323353, 24.8454142749004)
(54.959675275261986, 20.2218819936998)
(-70.671684294851, -26.0012627213949)
(-23.60432270654059, -64.1056213085331)
(58.13666578855936, 158.008463802741)
(-51.816978892477124, -19.0659516956551)
(-29.87860522507741, -81.1729808735538)
(51.855564313268616, 140.931828050329)
(-58.10225220480441, -21.3777903341693)
(-67.55904445987407, -183.624405080059)
(-20.371243707443842, -7.5032020264622)
(-26.666027861911218, -9.81678618716885)
(14.20761000064383, 38.5246251107899)
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:
$$x_{1} = 29.8115799030901$$
$$x_{2} = -80.1230937867295$$
$$x_{3} = 92.6661916492115$$
$$x_{4} = -36.1559769880743$$
$$x_{5} = -73.8409703906111$$
$$x_{6} = 23.5194140147849$$
$$x_{7} = -17.336473487102$$
$$x_{8} = 48.6741398947227$$
$$x_{9} = 86.3822212589452$$
$$x_{10} = 42.3879070002498$$
$$x_{11} = 36.1006116108761$$
$$x_{12} = -61.2773767058956$$
$$x_{13} = 73.8138793572668$$
$$x_{14} = 10.9037335277384$$
$$x_{15} = -42.4350684201498$$
$$x_{16} = 4.48766960334109$$
$$x_{17} = 80.0981276558536$$
$$x_{18} = 61.2447280834131$$
$$x_{19} = -86.405371586641$$
$$x_{20} = 98.9500623082067$$
$$x_{21} = 17.2206571155732$$
$$x_{22} = 67.5294331532335$$
$$x_{23} = 54.959675275262$$
$$x_{24} = -23.6043227065406$$
$$x_{25} = -29.8786052250774$$
$$x_{26} = -67.5590444598741$$
Puntos máximos de la función:
$$x_{26} = -39.2444240846477$$
$$x_{26} = -83.240191603726$$
$$x_{26} = -7.72415319239641$$
$$x_{26} = -64.3871177170664$$
$$x_{26} = 70.699979453112$$
$$x_{26} = 89.5465582344838$$
$$x_{26} = 20.4692255293053$$
$$x_{26} = -89.5242202334874$$
$$x_{26} = 7.97963107097301$$
$$x_{26} = -95.8081382182729$$
$$x_{26} = 95.829011377113$$
$$x_{26} = -14.0660135689384$$
$$x_{26} = -45.5311287148944$$
$$x_{26} = -32.9563750616135$$
$$x_{26} = -76.9560252131026$$
$$x_{26} = 76.9820104261667$$
$$x_{26} = 2.07393280909122$$
$$x_{26} = 64.4181735917203$$
$$x_{26} = 114.676852122197$$
$$x_{26} = 26.740942117298$$
$$x_{26} = 45.5750370742992$$
$$x_{26} = -70.671684294851$$
$$x_{26} = 58.1366657885594$$
$$x_{26} = -51.8169788924771$$
$$x_{26} = 51.8555643132686$$
$$x_{26} = -58.1022522048044$$
$$x_{26} = -20.3712437074438$$
$$x_{26} = -26.6660278619112$$
$$x_{26} = 14.2076100006438$$
Decrece en los intervalos
$$\left[98.9500623082067, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -86.405371586641\right]$$