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$$5 \left(x - 1\right) \cos{\left(x \right)} + 5 \sin{\left(x \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -1.9025816596713$$
$$x_{2} = 54.9963890778611$$
$$x_{3} = 76.9821802337515$$
$$x_{4} = -20.4669019371238$$
$$x_{5} = 45.5755235510474$$
$$x_{6} = -33.0161122316173$$
$$x_{7} = -76.9818428080439$$
$$x_{8} = 20.4716638479466$$
$$x_{9} = -98.9701712382329$$
$$x_{10} = 67.5592651262192$$
$$x_{11} = 11.0943177411687$$
$$x_{12} = 64.418416382217$$
$$x_{13} = -86.4052384364358$$
$$x_{14} = 0.47973100728041$$
$$x_{15} = -58.1363725465042$$
$$x_{16} = -83.2640722170426$$
$$x_{17} = 2.24679137687774$$
$$x_{18} = -92.687656640251$$
$$x_{19} = 83.2643606537498$$
$$x_{20} = 48.7156405519083$$
$$x_{21} = 86.4055062856094$$
$$x_{22} = -23.6025687023826$$
$$x_{23} = 36.1567518873749$$
$$x_{24} = -48.7147981536679$$
$$x_{25} = -14.2028494649391$$
$$x_{26} = -73.8407882621237$$
$$x_{27} = -17.3332512943446$$
$$x_{28} = -61.2771126285488$$
$$x_{29} = 26.74236450316$$
$$x_{30} = 4.95975747525199$$
$$x_{31} = 29.8797427274828$$
$$x_{32} = -70.6997808455739$$
$$x_{33} = 73.8411549997741$$
$$x_{34} = -89.5464342355346$$
$$x_{35} = 70.7001808865412$$
$$x_{36} = -80.1229390117807$$
$$x_{37} = -4.88082214577343$$
$$x_{38} = -54.9957280448611$$
$$x_{39} = -64.417934536029$$
$$x_{40} = -39.2947202239912$$
$$x_{41} = -95.8289030622188$$
$$x_{42} = 39.2960146150878$$
$$x_{43} = 61.2776451215302$$
$$x_{44} = 80.1232505037716$$
$$x_{45} = 23.6061518286588$$
$$x_{46} = -29.8775049299126$$
$$x_{47} = -7.96506651296683$$
$$x_{48} = -26.7395715348192$$
$$x_{49} = 17.3398833066804$$
$$x_{50} = -42.4345199190886$$
$$x_{51} = -45.5745611270113$$
$$x_{52} = 95.8291208275139$$
$$x_{53} = -36.1552231369057$$
$$x_{54} = -11.0781798144108$$
$$x_{55} = 92.6878894142842$$
$$x_{56} = 42.4356299587436$$
$$x_{57} = 33.0179451984154$$
$$x_{58} = -67.5588270317532$$
$$x_{59} = 7.99595954344623$$
$$x_{60} = 89.5466836249472$$
$$x_{61} = -51.8551961430023$$
$$x_{62} = 14.2127076381121$$
$$x_{63} = 58.1369641096559$$
$$x_{64} = 51.8559396371055$$
$$x_{65} = 98.9703754007943$$
Signos de extremos en los puntos:
(-1.9025816596712966, 13.7214078256275)
(54.99638907786112, -269.935657903488)
(76.98218023375154, 379.878002990951)
(-20.46690193712382, 107.218240529184)
(45.57552355104743, 222.821554325083)
(-33.016112231617306, 170.007114177638)
(-76.98184280804387, 389.877159248702)
(20.471663847946616, 97.2301809596448)
(-98.97017123823291, -499.825850608271)
(67.55926512621922, -332.758771477857)
(11.09431774116865, -50.2257327674813)
(64.41841638221703, 317.052668534954)
(-86.4052384364358, -436.99759258515)
(0.4797310072804097, -1.20062622077654)
(-58.13637254650418, 295.639596630236)
(-83.26407221704262, 421.290695583691)
(2.246791376877743, 4.86301476380959)
(-92.687656640251, -468.411601069451)
(83.26436065374978, 411.291416805502)
(48.71564055190829, -238.525826290489)
(86.40550628560945, -426.998262320222)
(-23.602568702382584, -122.911353843023)
(36.15675188737488, -175.712692461281)
(-48.71479815366795, -248.523719184854)
(-14.202849464939055, 75.8503361638903)
(-73.8407882621237, -374.170541538185)
(-17.333251294344578, -91.5301957592509)
(-61.27711262854878, -311.345427745382)
(26.742364503160008, 128.614816116575)
(4.959757475251995, -19.1961144857587)
(29.879742727482828, -144.312225534921)
(-70.69978084557387, 358.46404170382)
(73.84115499977409, -364.171458592563)
(-89.54643423553456, 452.704563560119)
(70.70018088654118, 348.465042056421)
(-80.12293901178074, -405.583881146681)
(-4.880822145773428, -28.9880025069101)
(-54.99572804486114, -279.934004637637)
(-64.41793453602904, 327.05146355648)
(-39.29472022399124, 201.411586896176)
(-95.82890306221881, 484.118698639041)
(39.2960146150878, 191.414825496269)
(61.27764512153021, -301.346759421191)
(80.1232505037716, -395.584660028325)
(23.606151828658817, -112.920331814076)
(-29.877504929912625, -154.306623193335)
(-7.965066512966832, 44.549047901834)
(-26.73957153481918, 138.607821462312)
(17.339883306680367, -81.5468451157945)
(-42.43451991908864, -217.115064561794)
(-45.57456112701128, 232.819146815877)
(95.82912082751392, 474.119243126396)
(-36.15522313690572, -185.70886692607)
(-11.078179814410767, -60.1849723343755)
(92.68788941428421, -458.412183089222)
(42.43562995874359, -207.117841589041)
(33.01794519841537, 160.011701856946)
(-67.55882703175322, -342.757675941634)
(7.995959543446228, 34.6278329451812)
(89.54668362494724, 442.705187130862)
(-51.85519614300229, 264.228694370001)
(14.21270763811213, 65.8751354232112)
(58.13696410965594, 285.641076085333)
(51.85593963710546, 254.230553969829)
(98.97037540079432, -489.826361079821)
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:
$$x_{1} = 54.9963890778611$$
$$x_{2} = -98.9701712382329$$
$$x_{3} = 67.5592651262192$$
$$x_{4} = 11.0943177411687$$
$$x_{5} = -86.4052384364358$$
$$x_{6} = 0.47973100728041$$
$$x_{7} = -92.687656640251$$
$$x_{8} = 48.7156405519083$$
$$x_{9} = 86.4055062856094$$
$$x_{10} = -23.6025687023826$$
$$x_{11} = 36.1567518873749$$
$$x_{12} = -48.7147981536679$$
$$x_{13} = -73.8407882621237$$
$$x_{14} = -17.3332512943446$$
$$x_{15} = -61.2771126285488$$
$$x_{16} = 4.95975747525199$$
$$x_{17} = 29.8797427274828$$
$$x_{18} = 73.8411549997741$$
$$x_{19} = -80.1229390117807$$
$$x_{20} = -4.88082214577343$$
$$x_{21} = -54.9957280448611$$
$$x_{22} = 61.2776451215302$$
$$x_{23} = 80.1232505037716$$
$$x_{24} = 23.6061518286588$$
$$x_{25} = -29.8775049299126$$
$$x_{26} = 17.3398833066804$$
$$x_{27} = -42.4345199190886$$
$$x_{28} = -36.1552231369057$$
$$x_{29} = -11.0781798144108$$
$$x_{30} = 92.6878894142842$$
$$x_{31} = 42.4356299587436$$
$$x_{32} = -67.5588270317532$$
$$x_{33} = 98.9703754007943$$
Puntos máximos de la función:
$$x_{33} = -1.9025816596713$$
$$x_{33} = 76.9821802337515$$
$$x_{33} = -20.4669019371238$$
$$x_{33} = 45.5755235510474$$
$$x_{33} = -33.0161122316173$$
$$x_{33} = -76.9818428080439$$
$$x_{33} = 20.4716638479466$$
$$x_{33} = 64.418416382217$$
$$x_{33} = -58.1363725465042$$
$$x_{33} = -83.2640722170426$$
$$x_{33} = 2.24679137687774$$
$$x_{33} = 83.2643606537498$$
$$x_{33} = -14.2028494649391$$
$$x_{33} = 26.74236450316$$
$$x_{33} = -70.6997808455739$$
$$x_{33} = -89.5464342355346$$
$$x_{33} = 70.7001808865412$$
$$x_{33} = -64.417934536029$$
$$x_{33} = -39.2947202239912$$
$$x_{33} = -95.8289030622188$$
$$x_{33} = 39.2960146150878$$
$$x_{33} = -7.96506651296683$$
$$x_{33} = -26.7395715348192$$
$$x_{33} = -45.5745611270113$$
$$x_{33} = 95.8291208275139$$
$$x_{33} = 33.0179451984154$$
$$x_{33} = 7.99595954344623$$
$$x_{33} = 89.5466836249472$$
$$x_{33} = -51.8551961430023$$
$$x_{33} = 14.2127076381121$$
$$x_{33} = 58.1369641096559$$
$$x_{33} = 51.8559396371055$$
Decrece en los intervalos
$$\left[98.9703754007943, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.9701712382329\right]$$