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$$\frac{3 x \cos{\left(3 x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{2} + 1 + \frac{2}{x^{3}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 89.5341496175094$$
$$x_{2} = 73.8319421320736$$
$$x_{3} = -29.8414070727572$$
$$x_{4} = 7.8396736012599$$
$$x_{5} = 67.5491767500035$$
$$x_{6} = -78.0147933275031$$
$$x_{7} = -3.75177195803661$$
$$x_{8} = -84.2980847924199$$
$$x_{9} = -16.2520658835412$$
$$x_{10} = 86.3976561110523$$
$$x_{11} = 57.0781065328832$$
$$x_{12} = 35.0779499321882$$
$$x_{13} = -67.5425969944273$$
$$x_{14} = 3.63183452315228$$
$$x_{15} = -38.2198035188413$$
$$x_{16} = -51.842708427932$$
$$x_{17} = -73.8259223119028$$
$$x_{18} = 12.0335124171734$$
$$x_{19} = -23.5572290697416$$
$$x_{20} = -87.4448074455823$$
$$x_{21} = -69.6370415742658$$
$$x_{22} = -34.0306555064577$$
$$x_{23} = 97.9118362155217$$
$$x_{24} = 44.5133844234738$$
$$x_{25} = 22.5098087774546$$
$$x_{26} = -80.1092256661263$$
$$x_{27} = -91.6334234062544$$
$$x_{28} = -53.9368539004158$$
$$x_{29} = -12.0703667254341$$
$$x_{30} = 53.9286134447172$$
$$x_{31} = -41.3723600294113$$
$$x_{32} = 51.8341350739919$$
$$x_{33} = -89.5391133882524$$
$$x_{34} = 38.2314296566913$$
$$x_{35} = 40.3253719836236$$
$$x_{36} = 78.020489958096$$
$$x_{37} = -17.2723300965849$$
$$x_{38} = -49.7485839674963$$
$$x_{39} = -58.1251987998863$$
$$x_{40} = -47.6544832880973$$
$$x_{41} = 80.1147733747798$$
$$x_{42} = 75.9262127024484$$
$$x_{43} = -27.7467309956061$$
$$x_{44} = 93.7229952896613$$
$$x_{45} = 27.7627423539192$$
$$x_{46} = 36.137539795388$$
$$x_{47} = 42.4193590081437$$
$$x_{48} = 16.2247054307724$$
$$x_{49} = 92.6805798692254$$
$$x_{50} = -71.7314832647885$$
$$x_{51} = 60.212013791028$$
$$x_{52} = 29.856295359853$$
$$x_{53} = -5.81650479334164$$
$$x_{54} = -45.5604096674964$$
$$x_{55} = -25.6520090653707$$
$$x_{56} = -1.71221607813122$$
$$x_{57} = 20.414906074639$$
$$x_{58} = 5.7397730916088$$
$$x_{59} = -14.160695216171$$
$$x_{60} = 82.2090624816772$$
$$x_{61} = -36.1252398830002$$
$$x_{62} = 88.4919599039137$$
$$x_{63} = 58.1175521765418$$
$$x_{64} = 71.7376788324169$$
$$x_{65} = 14.1292889745525$$
$$x_{66} = -43.4663670146734$$
$$x_{67} = -60.2193944752711$$
$$x_{68} = 84.3033568547452$$
$$x_{69} = -7.89608208530915$$
$$x_{70} = -82.2036561072884$$
$$x_{71} = -93.7277372264291$$
$$x_{72} = 66.4953734902289$$
$$x_{73} = 31.9499587197976$$
$$x_{74} = -0.95989719783345$$
$$x_{75} = -21.4623740434466$$
$$x_{76} = 100.006255084988$$
$$x_{77} = -75.9203589343812$$
$$x_{78} = -97.9163752974962$$
$$x_{79} = 64.4009240388803$$
$$x_{80} = -56.0310180297196$$
$$x_{81} = 48.7015306215084$$
$$x_{82} = -95.8220545994953$$
$$x_{83} = 18.3198868045554$$
$$x_{84} = -31.9360463105638$$
$$x_{85} = -100.010699111567$$
$$x_{86} = 34.0437120782042$$
$$x_{87} = 56.0230855877076$$
$$x_{88} = -9.98172626669137$$
$$x_{89} = -65.448149247889$$
$$x_{90} = 24.60462482455$$
$$x_{91} = 95.8174163069689$$
$$x_{92} = 62.3064709329742$$
$$x_{93} = 9.93714128227479$$
$$x_{94} = 49.7396496884211$$
Signos de extremos en los puntos:
(89.53414961750944, 40.767260320607)
(73.83194213207365, 106.744343711783)
(-29.841407072757196, -18.9227572006474)
(7.839673601259899, -0.0928232644072904)
(67.54917675000348, 97.3198449939322)
(-78.0147933275031, -43.0079170269249)
(-3.7517719580366107, -9.63577788418861)
(-84.2980847924199, -46.1495126395795)
(-16.2520658835412, -28.3665165897607)
(86.39765611105234, 125.593456623441)
(57.07810653288318, 81.6124729402231)
(35.077949932188154, 13.538954386956)
(-67.54259699442734, -37.7719289641448)
(3.631834523152284, -2.25081162630326)
(-38.21980351884126, -23.111313092809)
(-51.842708427932045, -81.7596126747617)
(-73.82592231190284, -40.9135208958062)
(12.033512417173416, 2.01217160130921)
(-23.557229069741553, -15.7815952565633)
(-87.44480744558226, -135.164483041905)
(-69.63704157426584, -38.8191258973999)
(-34.03065550645767, -21.0170074370133)
(97.91183621552172, 44.956097505348)
(44.513384423473845, 62.7639557301852)
(22.509808777454612, 7.2541660356287)
(-80.10922566612635, -44.0551154081801)
(-91.63342340625437, -141.447525978401)
(-53.936853900415834, -84.9009897514051)
(-12.070366725434083, -22.0917451476389)
(53.92861344471723, 22.9644780134213)
(-41.37236002941131, -66.0530821082106)
(51.834135073991895, 21.9172312998043)
(-89.53911338825245, -138.306002776883)
(38.231429656691304, 53.3399212613474)
(40.32537198362361, 56.4812435153022)
(78.02048995809595, 113.027366397324)
(-17.272330096584913, -12.6411234786029)
(-49.74858396749628, -78.6182550138179)
(-58.12519879988633, -91.1837932897095)
(-47.65448328809729, -75.4769195021771)
(80.11477337477979, 116.168883760784)
(75.92621270244842, 109.885852943855)
(-27.746730995606125, -17.8756653242941)
(93.72299528966128, 42.8616801907127)
(27.762742353919162, 37.6338111038007)
(36.13753979538799, 50.1986259870911)
(42.419359008143665, 59.6225893035084)
(16.224705430772445, 4.11027009772762)
(92.68057986922537, 135.018055965606)
(-71.73148326478852, -39.8663232292551)
(60.212013791027964, 26.1061924378668)
(29.856295359853007, 40.774947707108)
(-5.816504793341635, -12.7120199168973)
(-45.56040966749643, -72.3356094094135)
(-25.652009065370727, -16.8286068152483)
(-1.712216078131216, -6.83352581855356)
(20.41490607463904, 6.20641601866254)
(5.739773091608804, -1.15539869871637)
(-14.160695216171039, -25.2283991322338)
(82.20906248167715, 119.310404752608)
(-36.12523988300024, -22.0641550869415)
(88.49195990391365, 128.734987059214)
(58.11755217654183, 25.0589580220582)
(71.73767883241692, 103.602839046315)
(14.129288974552527, 3.06160831731661)
(-43.46636701467342, -69.1943286785897)
(-60.219394475271066, -94.3252161299858)
(84.30335685474525, 122.451929117488)
(-7.896082085309151, -15.8287143054271)
(-82.20365610728835, -45.1023139548754)
(-93.72773722642908, -144.589052405349)
(66.49537349022893, 29.2478783489101)
(31.949958719797575, 43.9161337183775)
(-0.9598971978334502, -5.9209320980479)
(-21.462374043446612, -14.7346515424909)
(100.00625508498813, 46.0033053214858)
(-75.92035893438117, -41.9607188434179)
(-97.91637529749619, -150.872114077811)
(64.40092403888028, 28.2006522622469)
(-56.031018029719625, -88.0423839408958)
(48.70153062150844, 69.0467410714497)
(-95.82205459949529, -147.730581838443)
(18.319886804555388, 5.1584826928003)
(-31.936046310563775, -19.9698733316274)
(-100.01069911156709, -154.013648940895)
(34.0437120782042, 47.0573618170311)
(56.023085587707556, 24.0117200508985)
(-9.981726266691371, -18.9576683040401)
(-65.44814924788905, -36.7247325059037)
(24.604624824550005, 8.30179040932915)
(95.81741630696887, 43.9088891428627)
(62.30647093297416, 27.1534237293258)
(9.937141282274794, 0.961265904515388)
(49.73964968842108, 20.8699791769845)
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:
$$x_{1} = 89.5341496175094$$
$$x_{2} = 7.8396736012599$$
$$x_{3} = -3.75177195803661$$
$$x_{4} = -16.2520658835412$$
$$x_{5} = 35.0779499321882$$
$$x_{6} = 3.63183452315228$$
$$x_{7} = -51.842708427932$$
$$x_{8} = 12.0335124171734$$
$$x_{9} = -87.4448074455823$$
$$x_{10} = 97.9118362155217$$
$$x_{11} = 22.5098087774546$$
$$x_{12} = -91.6334234062544$$
$$x_{13} = -53.9368539004158$$
$$x_{14} = -12.0703667254341$$
$$x_{15} = 53.9286134447172$$
$$x_{16} = -41.3723600294113$$
$$x_{17} = 51.8341350739919$$
$$x_{18} = -89.5391133882524$$
$$x_{19} = -49.7485839674963$$
$$x_{20} = -58.1251987998863$$
$$x_{21} = -47.6544832880973$$
$$x_{22} = 93.7229952896613$$
$$x_{23} = 16.2247054307724$$
$$x_{24} = 60.212013791028$$
$$x_{25} = -5.81650479334164$$
$$x_{26} = -45.5604096674964$$
$$x_{27} = -1.71221607813122$$
$$x_{28} = 20.414906074639$$
$$x_{29} = 5.7397730916088$$
$$x_{30} = -14.160695216171$$
$$x_{31} = 58.1175521765418$$
$$x_{32} = 14.1292889745525$$
$$x_{33} = -43.4663670146734$$
$$x_{34} = -60.2193944752711$$
$$x_{35} = -7.89608208530915$$
$$x_{36} = -93.7277372264291$$
$$x_{37} = 66.4953734902289$$
$$x_{38} = 100.006255084988$$
$$x_{39} = -97.9163752974962$$
$$x_{40} = 64.4009240388803$$
$$x_{41} = -56.0310180297196$$
$$x_{42} = -95.8220545994953$$
$$x_{43} = 18.3198868045554$$
$$x_{44} = -100.010699111567$$
$$x_{45} = 56.0230855877076$$
$$x_{46} = -9.98172626669137$$
$$x_{47} = 24.60462482455$$
$$x_{48} = 95.8174163069689$$
$$x_{49} = 62.3064709329742$$
$$x_{50} = 9.93714128227479$$
$$x_{51} = 49.7396496884211$$
Puntos máximos de la función:
$$x_{51} = 73.8319421320736$$
$$x_{51} = -29.8414070727572$$
$$x_{51} = 67.5491767500035$$
$$x_{51} = -78.0147933275031$$
$$x_{51} = -84.2980847924199$$
$$x_{51} = 86.3976561110523$$
$$x_{51} = 57.0781065328832$$
$$x_{51} = -67.5425969944273$$
$$x_{51} = -38.2198035188413$$
$$x_{51} = -73.8259223119028$$
$$x_{51} = -23.5572290697416$$
$$x_{51} = -69.6370415742658$$
$$x_{51} = -34.0306555064577$$
$$x_{51} = 44.5133844234738$$
$$x_{51} = -80.1092256661263$$
$$x_{51} = 38.2314296566913$$
$$x_{51} = 40.3253719836236$$
$$x_{51} = 78.020489958096$$
$$x_{51} = -17.2723300965849$$
$$x_{51} = 80.1147733747798$$
$$x_{51} = 75.9262127024484$$
$$x_{51} = -27.7467309956061$$
$$x_{51} = 27.7627423539192$$
$$x_{51} = 36.137539795388$$
$$x_{51} = 42.4193590081437$$
$$x_{51} = 92.6805798692254$$
$$x_{51} = -71.7314832647885$$
$$x_{51} = 29.856295359853$$
$$x_{51} = -25.6520090653707$$
$$x_{51} = 82.2090624816772$$
$$x_{51} = -36.1252398830002$$
$$x_{51} = 88.4919599039137$$
$$x_{51} = 71.7376788324169$$
$$x_{51} = 84.3033568547452$$
$$x_{51} = -82.2036561072884$$
$$x_{51} = 31.9499587197976$$
$$x_{51} = -0.95989719783345$$
$$x_{51} = -21.4623740434466$$
$$x_{51} = -75.9203589343812$$
$$x_{51} = 48.7015306215084$$
$$x_{51} = -31.9360463105638$$
$$x_{51} = 34.0437120782042$$
$$x_{51} = -65.448149247889$$
Decrece en los intervalos
$$\left[100.006255084988, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100.010699111567\right]$$