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$$\left(x + 3\right) \cos{\left(x \right)} + \sin{\left(x \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -95.8293479731014$$
$$x_{2} = 23.599521866075$$
$$x_{3} = 80.1226425025801$$
$$x_{4} = 1.77714670942846$$
$$x_{5} = 67.5584137575819$$
$$x_{6} = -8.0494916184802$$
$$x_{7} = 48.7140208365965$$
$$x_{8} = 61.2766132480572$$
$$x_{9} = 76.9815222493336$$
$$x_{10} = -89.546944536085$$
$$x_{11} = -73.841542432988$$
$$x_{12} = -26.7456257056362$$
$$x_{13} = 98.9699750820314$$
$$x_{14} = -45.5765762525042$$
$$x_{15} = -29.8823122458921$$
$$x_{16} = 26.7371528553266$$
$$x_{17} = 83.2637971494076$$
$$x_{18} = -33.0200216517861$$
$$x_{19} = 36.1538502374869$$
$$x_{20} = -48.716556556407$$
$$x_{21} = 14.1952570831341$$
$$x_{22} = 17.3279134102008$$
$$x_{23} = 54.9951125615717$$
$$x_{24} = -54.9971009087336$$
$$x_{25} = -1.08874792349856$$
$$x_{26} = -58.1375985478334$$
$$x_{27} = 7.9450938109612$$
$$x_{28} = -5.14808338805474$$
$$x_{29} = -14.2260112763823$$
$$x_{30} = -76.9825358933422$$
$$x_{31} = -67.5597303454395$$
$$x_{32} = 33.0144823377454$$
$$x_{33} = 73.8404406052928$$
$$x_{34} = 89.5461956212586$$
$$x_{35} = -51.8567439300492$$
$$x_{36} = 11.0665455476582$$
$$x_{37} = 92.6874335935384$$
$$x_{38} = 4.83926669562784$$
$$x_{39} = 58.1358196557249$$
$$x_{40} = 42.4335074573344$$
$$x_{41} = 39.2935480335985$$
$$x_{42} = -86.4057869750076$$
$$x_{43} = 45.5736778511703$$
$$x_{44} = -98.9705880700397$$
$$x_{45} = -83.2646634582605$$
$$x_{46} = -36.1584645899842$$
$$x_{47} = -80.1235781433617$$
$$x_{48} = 20.4629468621164$$
$$x_{49} = -3.07085543704304$$
$$x_{50} = 70.6994025041995$$
$$x_{51} = -20.4775063343064$$
$$x_{52} = -39.2974513464482$$
$$x_{53} = -17.3483415455941$$
$$x_{54} = -42.4368523841599$$
$$x_{55} = -61.2782141319878$$
$$x_{56} = 64.4174812589959$$
$$x_{57} = 95.8286941079906$$
$$x_{58} = -23.610426015495$$
$$x_{59} = 29.8755385856919$$
$$x_{60} = -92.6881325660443$$
$$x_{61} = 86.4049825662619$$
$$x_{62} = -11.1181378268582$$
$$x_{63} = -70.7006045485144$$
$$x_{64} = 51.8545068078183$$
$$x_{65} = -64.4189295854489$$
Signos de extremos en los puntos:
(-95.82934797310143, 92.8239622142013)
(23.59952186607495, -26.5807444377984)
(80.1226425025801, -83.1166279472508)
(1.7771467094284592, 4.67580048963318)
(67.5584137575819, -70.5513284978385)
(-8.049491618480197, 4.95329236022816)
(48.71402083659647, -51.7043549894224)
(61.276613248057174, -64.2688357808478)
(76.9815222493336, 79.97527153826)
(-89.546944536085, 86.541167903022)
(-73.84154243298796, -70.8344854820511)
(-26.745625705636247, 23.7245971624142)
(98.9699750820314, -101.965072031526)
(-45.5765762525042, 42.5648375625315)
(-29.88231224589206, -26.8637319364326)
(26.737152855326624, 29.7203531187616)
(83.26379714940755, 86.2580015592677)
(-33.020021651786074, 30.0033799491541)
(36.153850237486935, -39.1410863454228)
(-48.716556556407006, -45.7056235228711)
(14.19525708313409, 17.1662528475409)
(17.327913410200786, -20.3033612430302)
(54.99511256157169, -57.9864930674055)
(-54.9971009087336, -51.9874876546271)
(-1.0887479234985609, -1.69345946020889)
(-58.13759854783343, 55.1285325620455)
(7.945093810961197, 10.8996952715985)
(-5.148083388054739, -1.94740340840285)
(-14.226011276382284, 11.181735184982)
(-76.9825358933422, 73.9757784675333)
(-67.55973034543955, -64.5519869728571)
(33.01448233774537, 36.000607056636)
(73.84044060529283, -76.8339344414338)
(89.54619562125856, 92.5407933870874)
(-51.85674393004924, 48.8465131428041)
(11.066545547658214, -14.0311343843941)
(92.6874335935384, -95.6822086749824)
(4.839266695627836, -7.7762532162745)
(58.13581965572485, 61.1276427850604)
(42.4335074573344, -45.4225063607175)
(39.293548033598505, 42.2817308539914)
(-86.40578697500756, -83.3997928333734)
(45.573677851170316, 48.5633874811662)
(-98.97058807003971, -95.9653785647247)
(-83.26466345826046, 80.2584347920204)
(-36.15846458998425, -33.143395762301)
(-80.12357814336166, -77.1170958590207)
(20.462946862116432, 23.4416656545276)
(-3.0708554370430403, 0.00500793753985072)
(70.69940250419954, 73.6926191253227)
(-20.47750633430642, 17.4489681843926)
(-39.297451346448156, 36.2836841112163)
(-17.348341545594117, -14.3136207490662)
(-42.43685238415992, -39.4241799982444)
(-61.278214131987816, -58.2696364907219)
(64.41748125899586, 67.4100660085347)
(95.82869410799064, 98.823635237046)
(-23.610426015495015, -20.5862091973645)
(29.87553858569191, -32.8603402551421)
(-92.6881325660443, -89.6825582122061)
(86.40498256626192, -89.3993905614769)
(-11.118137826858208, -8.05723951047467)
(-70.70060454851438, 67.6932202983926)
(51.85450680781833, 54.8453940577788)
(-64.4189295854489, 61.4107903909679)
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:
$$x_{1} = 23.599521866075$$
$$x_{2} = 80.1226425025801$$
$$x_{3} = 67.5584137575819$$
$$x_{4} = 48.7140208365965$$
$$x_{5} = 61.2766132480572$$
$$x_{6} = -73.841542432988$$
$$x_{7} = 98.9699750820314$$
$$x_{8} = -29.8823122458921$$
$$x_{9} = 36.1538502374869$$
$$x_{10} = -48.716556556407$$
$$x_{11} = 17.3279134102008$$
$$x_{12} = 54.9951125615717$$
$$x_{13} = -54.9971009087336$$
$$x_{14} = -1.08874792349856$$
$$x_{15} = -5.14808338805474$$
$$x_{16} = -67.5597303454395$$
$$x_{17} = 73.8404406052928$$
$$x_{18} = 11.0665455476582$$
$$x_{19} = 92.6874335935384$$
$$x_{20} = 4.83926669562784$$
$$x_{21} = 42.4335074573344$$
$$x_{22} = -86.4057869750076$$
$$x_{23} = -98.9705880700397$$
$$x_{24} = -36.1584645899842$$
$$x_{25} = -80.1235781433617$$
$$x_{26} = -17.3483415455941$$
$$x_{27} = -42.4368523841599$$
$$x_{28} = -61.2782141319878$$
$$x_{29} = -23.610426015495$$
$$x_{30} = 29.8755385856919$$
$$x_{31} = -92.6881325660443$$
$$x_{32} = 86.4049825662619$$
$$x_{33} = -11.1181378268582$$
Puntos máximos de la función:
$$x_{33} = -95.8293479731014$$
$$x_{33} = 1.77714670942846$$
$$x_{33} = -8.0494916184802$$
$$x_{33} = 76.9815222493336$$
$$x_{33} = -89.546944536085$$
$$x_{33} = -26.7456257056362$$
$$x_{33} = -45.5765762525042$$
$$x_{33} = 26.7371528553266$$
$$x_{33} = 83.2637971494076$$
$$x_{33} = -33.0200216517861$$
$$x_{33} = 14.1952570831341$$
$$x_{33} = -58.1375985478334$$
$$x_{33} = 7.9450938109612$$
$$x_{33} = -14.2260112763823$$
$$x_{33} = -76.9825358933422$$
$$x_{33} = 33.0144823377454$$
$$x_{33} = 89.5461956212586$$
$$x_{33} = -51.8567439300492$$
$$x_{33} = 58.1358196557249$$
$$x_{33} = 39.2935480335985$$
$$x_{33} = 45.5736778511703$$
$$x_{33} = -83.2646634582605$$
$$x_{33} = 20.4629468621164$$
$$x_{33} = -3.07085543704304$$
$$x_{33} = 70.6994025041995$$
$$x_{33} = -20.4775063343064$$
$$x_{33} = -39.2974513464482$$
$$x_{33} = 64.4174812589959$$
$$x_{33} = 95.8286941079906$$
$$x_{33} = -70.7006045485144$$
$$x_{33} = 51.8545068078183$$
$$x_{33} = -64.4189295854489$$
Decrece en los intervalos
$$\left[98.9699750820314, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.9705880700397\right]$$