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$$3 \sin{\left(3 x \right)} - 3 \cos{\left(3 x \right)} + 2 e^{- 2 x} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 68.329640215578$$
$$x_{2} = 84.037603483527$$
$$x_{3} = 62.0464549083984$$
$$x_{4} = 50.5272818452358$$
$$x_{5} = 64.1408500107916$$
$$x_{6} = 94.5095789954929$$
$$x_{7} = 13.8753675533551$$
$$x_{8} = 10.7337748996905$$
$$x_{9} = 6.54498437036171$$
$$x_{10} = 101.839961853869$$
$$x_{11} = 92.4151838930998$$
$$x_{12} = 26.4417381677141$$
$$x_{13} = 9.68657734917473$$
$$x_{14} = -0.355847991830463$$
$$x_{15} = 18.0641577581413$$
$$x_{16} = 86.1319985859202$$
$$x_{17} = 20.1585528605345$$
$$x_{18} = 22.2529479629277$$
$$x_{19} = 55.7632696012188$$
$$x_{20} = 2.35477889580279$$
$$x_{21} = 66.2352451131848$$
$$x_{22} = 70.4240353179712$$
$$x_{23} = 46.3384916404494$$
$$x_{24} = 15.9697626557481$$
$$x_{25} = 40.0553063332699$$
$$x_{26} = 57.857664703612$$
$$x_{27} = 0.140498114252764$$
$$x_{28} = 33.7721210260903$$
$$x_{29} = 48.4328867428426$$
$$x_{30} = 49.4800842940392$$
$$x_{31} = 71.4712328691678$$
$$x_{32} = 11.7809724509709$$
$$x_{33} = 59.9520598060052$$
$$x_{34} = 75.6600230739542$$
$$x_{35} = 215.984494934298$$
$$x_{36} = 52.621676947629$$
$$x_{37} = 44.2440965380563$$
$$x_{38} = 30.6305283725005$$
$$x_{39} = 4.4505681854736$$
$$x_{40} = 88.2263936883134$$
$$x_{41} = 99.7455667514759$$
$$x_{42} = 42.1497014356631$$
$$x_{43} = 35.8665161284835$$
$$x_{44} = 37.9609112308767$$
$$x_{45} = 79.8488132787406$$
$$x_{46} = 90.3207887907066$$
$$x_{47} = 72.5184304203644$$
$$x_{48} = 81.9432083811338$$
$$x_{49} = 77.7544181763474$$
$$x_{50} = 28.5361332701073$$
$$x_{51} = 24.3473430653209$$
Signos de extremos en los puntos:
(68.329640215578, 1.41421356237309)
(84.03760348352696, -1.41421356237309)
(62.04645490839842, 1.41421356237309)
(50.52728184523584, -1.41421356237309)
(64.14085001079161, 1.41421356237309)
(94.50957899549294, -1.4142135623731)
(13.87536755335506, 1.41421356237221)
(10.733774899690475, -1.41421356284817)
(6.544984370361709, -1.41421562822253)
(101.83996185386913, 1.41421356237309)
(92.41518389309975, -1.41421356237309)
(26.441738167714092, 1.41421356237309)
(9.686577349174733, 1.41421355851524)
(-0.35584799183046345, -1.64370234329464)
(18.06415775814131, 1.41421356237309)
(86.13199858592016, -1.41421356237309)
(20.158552860534506, 1.41421356237309)
(22.252947962927703, 1.4142135623731)
(55.76326960121883, 1.41421356237309)
(2.3547788958027898, -1.42320957005441)
(66.2352451131848, 1.41421356237309)
(70.4240353179712, 1.4142135623731)
(46.33849164044945, -1.4142135623731)
(15.969762655748118, 1.41421356237308)
(40.05530633326986, -1.41421356237309)
(57.85766470361202, 1.41421356237309)
(0.1404981142527637, -2.07663423535921)
(33.772121026090275, -1.41421356237309)
(48.43288674284265, -1.41421356237309)
(49.480084294039244, 1.41421356237309)
(71.47123286916779, -1.41421356237309)
(11.780972450970918, 1.41421356231459)
(59.952059806005224, 1.41421356237309)
(75.66002307395419, -1.41421356237309)
(215.98449493429828, -1.4142135623731)
(52.621676947629034, -1.4142135623731)
(44.244096538056255, -1.41421356237309)
(30.630528372500486, 1.41421356237309)
(4.450568185473596, -1.41434979348282)
(88.22639368831337, -1.41421356237309)
(99.74556675147593, 1.41421356237309)
(42.14970143566306, -1.41421356237309)
(35.866516128483475, -1.41421356237309)
(37.96091123087667, -1.41421356237309)
(79.84881327874058, -1.41421356237309)
(90.32078879070656, -1.41421356237309)
(72.5184304203644, 1.41421356237309)
(81.94320838113377, -1.41421356237309)
(77.75441817634739, -1.41421356237309)
(28.53613327010729, 1.41421356237309)
(24.3473430653209, 1.41421356237309)
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:
$$x_{1} = 84.037603483527$$
$$x_{2} = 50.5272818452358$$
$$x_{3} = 94.5095789954929$$
$$x_{4} = 10.7337748996905$$
$$x_{5} = 6.54498437036171$$
$$x_{6} = 92.4151838930998$$
$$x_{7} = 86.1319985859202$$
$$x_{8} = 2.35477889580279$$
$$x_{9} = 46.3384916404494$$
$$x_{10} = 40.0553063332699$$
$$x_{11} = 0.140498114252764$$
$$x_{12} = 33.7721210260903$$
$$x_{13} = 48.4328867428426$$
$$x_{14} = 71.4712328691678$$
$$x_{15} = 75.6600230739542$$
$$x_{16} = 215.984494934298$$
$$x_{17} = 52.621676947629$$
$$x_{18} = 44.2440965380563$$
$$x_{19} = 4.4505681854736$$
$$x_{20} = 88.2263936883134$$
$$x_{21} = 42.1497014356631$$
$$x_{22} = 35.8665161284835$$
$$x_{23} = 37.9609112308767$$
$$x_{24} = 79.8488132787406$$
$$x_{25} = 90.3207887907066$$
$$x_{26} = 81.9432083811338$$
$$x_{27} = 77.7544181763474$$
Puntos máximos de la función:
$$x_{27} = 68.329640215578$$
$$x_{27} = 62.0464549083984$$
$$x_{27} = 64.1408500107916$$
$$x_{27} = 13.8753675533551$$
$$x_{27} = 101.839961853869$$
$$x_{27} = 26.4417381677141$$
$$x_{27} = 9.68657734917473$$
$$x_{27} = -0.355847991830463$$
$$x_{27} = 18.0641577581413$$
$$x_{27} = 20.1585528605345$$
$$x_{27} = 22.2529479629277$$
$$x_{27} = 55.7632696012188$$
$$x_{27} = 66.2352451131848$$
$$x_{27} = 70.4240353179712$$
$$x_{27} = 15.9697626557481$$
$$x_{27} = 57.857664703612$$
$$x_{27} = 49.4800842940392$$
$$x_{27} = 11.7809724509709$$
$$x_{27} = 59.9520598060052$$
$$x_{27} = 30.6305283725005$$
$$x_{27} = 99.7455667514759$$
$$x_{27} = 72.5184304203644$$
$$x_{27} = 28.5361332701073$$
$$x_{27} = 24.3473430653209$$
Decrece en los intervalos
$$\left[215.984494934298, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, 0.140498114252764\right]$$