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$$\left(x \operatorname{sign}{\left(x \right)} + \left|{x}\right|\right) \cos{\left(x \left|{x}\right| \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -97.7504683154682$$
$$x_{2} = 40.1647585280723$$
$$x_{3} = 45.483389961489$$
$$x_{4} = 36.042316272118$$
$$x_{5} = -31.6322187044847$$
$$x_{6} = -51.7817470720995$$
$$x_{7} = 34.2088185043301$$
$$x_{8} = -42.4464767305102$$
$$x_{9} = 58.181320750246$$
$$x_{10} = 3.31595752197827$$
$$x_{11} = -7.41470642964517$$
$$x_{12} = -53.6295447898027$$
$$x_{13} = -25.898456161351$$
$$x_{14} = -4.15677273792348$$
$$x_{15} = -38.4463176847146$$
$$x_{16} = -93.7978156802077$$
$$x_{17} = 60.2503967723284$$
$$x_{18} = 32.5137805850529$$
$$x_{19} = -25.1601057171536$$
$$x_{20} = -2.80249560819896$$
$$x_{21} = -5.46306966906912$$
$$x_{22} = 54.1251803643413$$
$$x_{23} = -69.7703218907394$$
$$x_{24} = 1.2533141373155$$
$$x_{25} = 2.1708037636748$$
$$x_{26} = 97.2025714972156$$
$$x_{27} = 56.1199308198413$$
$$x_{28} = 65.4249187276231$$
$$x_{29} = -17.8569777493103$$
$$x_{30} = 41.9626343588803$$
$$x_{31} = -65.8556653468835$$
$$x_{32} = 70.1296178457777$$
$$x_{33} = -42.0000508927152$$
$$x_{34} = -65.7601877100906$$
$$x_{35} = 51.9029450075162$$
$$x_{36} = -51.2328084539026$$
$$x_{37} = -81.792551958106$$
$$x_{38} = -29.8962056111858$$
$$x_{39} = -57.7477324384739$$
$$x_{40} = -23.4138403097702$$
$$x_{41} = 92.1420324625326$$
$$x_{42} = 15.0919007214221$$
$$x_{43} = 6.2665706865775$$
$$x_{44} = 98.5187824897652$$
$$x_{45} = 18.1189522958733$$
$$x_{46} = -90.0207656595681$$
$$x_{47} = -1.2533141373155$$
$$x_{48} = 10.2588183479024$$
$$x_{49} = 96.0483567129934$$
$$x_{50} = -33.7930307841704$$
$$x_{51} = 68.1530977897007$$
$$x_{52} = -9.12426464544565$$
$$x_{53} = -85.7673550856064$$
$$x_{54} = 94.2155509601147$$
$$x_{55} = -79.4150316506989$$
$$x_{56} = 40.3987299743432$$
$$x_{57} = 77.6953684840531$$
$$x_{58} = 0$$
$$x_{59} = -65.5208849232857$$
$$x_{60} = -46.9114661233227$$
$$x_{61} = 20.2479095536667$$
$$x_{62} = -58.1002694166063$$
$$x_{63} = -90.802587624728$$
$$x_{64} = 81.3303346367966$$
$$x_{65} = 75.3553496532499$$
$$x_{66} = -43.9732905100871$$
$$x_{67} = 80.1827248574873$$
$$x_{68} = 62.2255007657586$$
$$x_{69} = -16.0012437412711$$
$$x_{70} = -7.82694427889971$$
$$x_{71} = 27.9969170993996$$
$$x_{72} = 32.1249620498491$$
$$x_{73} = -3.7599424119465$$
$$x_{74} = 88.2941797055903$$
$$x_{75} = -83.5782942231828$$
$$x_{76} = 4.15677273792348$$
$$x_{77} = 22.3146238057912$$
$$x_{78} = 5.74340690380656$$
$$x_{79} = 27.6013807353097$$
$$x_{80} = -13.440296781746$$
$$x_{81} = 46.0326432528734$$
$$x_{82} = -21.7441875995693$$
$$x_{83} = -47.8726627710821$$
$$x_{84} = -80.3979283611276$$
Signos de extremos en los puntos:
(-97.75046831546823, 1)
(40.164758528072326, -1)
(45.483389961489, 1)
(36.042316272117965, -1)
(-31.63221870448466, -1)
(-51.781747072099535, 1)
(34.20881850433011, 1)
(-42.44647673051023, 1)
(58.181320750245966, -1)
(3.3159575219782713, -1)
(-7.414706429645167, 1)
(-53.629544789802715, 1)
(-25.898456161351024, 1)
(-4.15677273792348, 1)
(-38.446317684714586, -1)
(-93.79781568020769, -1)
(60.25039677232844, -1)
(32.51378058505294, 1)
(-25.160105717153563, 1)
(-2.8024956081989645, -1)
(-5.4630696690691245, 1)
(54.12518036434135, 1)
(-69.77032189073937, 1)
(1.2533141373155003, 1)
(2.1708037636748028, -1)
(97.20257149721556, -1)
(56.11993081984125, 1)
(65.42491872762314, 1)
(-17.856977749310325, 1)
(41.96263435888027, 1)
(-65.85566534688348, -1)
(70.1296178457777, -1)
(-42.00005089271522, 1)
(-65.76018771009059, -1)
(51.90294500751617, -1)
(-51.232808453902585, 1)
(-81.79255195810597, 1)
(-29.89620561118578, -1)
(-57.747732438473854, 1)
(-23.41384030977018, -1)
(92.14203246253263, 1)
(15.091900721422071, 1)
(6.266570686577501, 1)
(98.5187824897652, -1)
(18.118952295873328, 1)
(-90.02076565956808, 1)
(-1.2533141373155003, -1)
(10.25881834790236, -1)
(96.04835671299342, 1)
(-33.79303078417042, 1)
(68.15309778970072, 1)
(-9.124264645445654, -1)
(-85.76735508560644, 1)
(94.21555096011465, -1)
(-79.41503165069891, 1)
(40.398729974343226, -1)
(77.69536848405308, -1)
(0, 0)
(-65.52088492328573, -1)
(-46.91146612332267, -1)
(20.24790955366672, 1)
(-58.10026941660626, -1)
(-90.80258762472803, -1)
(81.33033463679655, -1)
(75.35534965324992, -1)
(-43.97329051008712, 1)
(80.1827248574873, 1)
(62.22550076575857, 1)
(-16.001243741271118, 1)
(-7.826944278899714, 1)
(27.996917099399596, -1)
(32.12496204984913, 1)
(-3.7599424119465006, -1)
(88.2941797055903, -1)
(-83.57829422318277, 1)
(4.15677273792348, -1)
(22.314623805791175, 1)
(5.743406903806558, 1)
(27.60138073530969, 1)
(-13.440296781746046, 1)
(46.03264325287345, 1)
(-21.74418759956931, -1)
(-47.87266277108214, 1)
(-80.39792836112757, 1)
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:
$$x_{1} = 40.1647585280723$$
$$x_{2} = 36.042316272118$$
$$x_{3} = -31.6322187044847$$
$$x_{4} = 58.181320750246$$
$$x_{5} = 3.31595752197827$$
$$x_{6} = -38.4463176847146$$
$$x_{7} = -93.7978156802077$$
$$x_{8} = 60.2503967723284$$
$$x_{9} = -2.80249560819896$$
$$x_{10} = 2.1708037636748$$
$$x_{11} = 97.2025714972156$$
$$x_{12} = -65.8556653468835$$
$$x_{13} = 70.1296178457777$$
$$x_{14} = -65.7601877100906$$
$$x_{15} = 51.9029450075162$$
$$x_{16} = -29.8962056111858$$
$$x_{17} = -23.4138403097702$$
$$x_{18} = 98.5187824897652$$
$$x_{19} = -1.2533141373155$$
$$x_{20} = 10.2588183479024$$
$$x_{21} = -9.12426464544565$$
$$x_{22} = 94.2155509601147$$
$$x_{23} = 40.3987299743432$$
$$x_{24} = 77.6953684840531$$
$$x_{25} = -65.5208849232857$$
$$x_{26} = -46.9114661233227$$
$$x_{27} = -58.1002694166063$$
$$x_{28} = -90.802587624728$$
$$x_{29} = 81.3303346367966$$
$$x_{30} = 75.3553496532499$$
$$x_{31} = 27.9969170993996$$
$$x_{32} = -3.7599424119465$$
$$x_{33} = 88.2941797055903$$
$$x_{34} = 4.15677273792348$$
$$x_{35} = -21.7441875995693$$
Puntos máximos de la función:
$$x_{35} = -97.7504683154682$$
$$x_{35} = 45.483389961489$$
$$x_{35} = -51.7817470720995$$
$$x_{35} = 34.2088185043301$$
$$x_{35} = -42.4464767305102$$
$$x_{35} = -7.41470642964517$$
$$x_{35} = -53.6295447898027$$
$$x_{35} = -25.898456161351$$
$$x_{35} = -4.15677273792348$$
$$x_{35} = 32.5137805850529$$
$$x_{35} = -25.1601057171536$$
$$x_{35} = -5.46306966906912$$
$$x_{35} = 54.1251803643413$$
$$x_{35} = -69.7703218907394$$
$$x_{35} = 1.2533141373155$$
$$x_{35} = 56.1199308198413$$
$$x_{35} = 65.4249187276231$$
$$x_{35} = -17.8569777493103$$
$$x_{35} = 41.9626343588803$$
$$x_{35} = -42.0000508927152$$
$$x_{35} = -51.2328084539026$$
$$x_{35} = -81.792551958106$$
$$x_{35} = -57.7477324384739$$
$$x_{35} = 92.1420324625326$$
$$x_{35} = 15.0919007214221$$
$$x_{35} = 6.2665706865775$$
$$x_{35} = 18.1189522958733$$
$$x_{35} = -90.0207656595681$$
$$x_{35} = 96.0483567129934$$
$$x_{35} = -33.7930307841704$$
$$x_{35} = 68.1530977897007$$
$$x_{35} = -85.7673550856064$$
$$x_{35} = -79.4150316506989$$
$$x_{35} = 20.2479095536667$$
$$x_{35} = -43.9732905100871$$
$$x_{35} = 80.1827248574873$$
$$x_{35} = 62.2255007657586$$
$$x_{35} = -16.0012437412711$$
$$x_{35} = -7.82694427889971$$
$$x_{35} = 32.1249620498491$$
$$x_{35} = -83.5782942231828$$
$$x_{35} = 22.3146238057912$$
$$x_{35} = 5.74340690380656$$
$$x_{35} = 27.6013807353097$$
$$x_{35} = -13.440296781746$$
$$x_{35} = 46.0326432528734$$
$$x_{35} = -47.8726627710821$$
$$x_{35} = -80.3979283611276$$
Decrece en los intervalos
$$\left[98.5187824897652, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -93.7978156802077\right]$$