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$$\frac{\left(- 3 x^{2} - 1\right) \cosh{\left(\frac{1}{x} \right)}}{x^{2} \left(x^{2} + 1\right)^{2}} - \frac{\frac{1}{x \left(x^{2} + 1\right)} \sinh{\left(\frac{1}{x} \right)}}{x^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -8924.32241932801$$
$$x_{2} = 5687.22461845844$$
$$x_{3} = 2852.8212852045$$
$$x_{4} = -1947.24501910307$$
$$x_{5} = 7213.60752293321$$
$$x_{6} = 7867.784528521$$
$$x_{7} = -3037.0603364102$$
$$x_{8} = -8488.19878741811$$
$$x_{9} = 9176.15294157982$$
$$x_{10} = -6961.78266980596$$
$$x_{11} = 10702.5991190406$$
$$x_{12} = -10014.6379093033$$
$$x_{13} = -6525.67026127846$$
$$x_{14} = -7179.84015113029$$
$$x_{15} = -9360.44761877016$$
$$x_{16} = 2634.835493011$$
$$x_{17} = 6123.32849463054$$
$$x_{18} = 10266.4702616077$$
$$x_{19} = 10484.5345696987$$
$$x_{20} = -8052.07697757456$$
$$x_{21} = 3506.84143746969$$
$$x_{22} = 4596.99525286914$$
$$x_{23} = 6995.54992011327$$
$$x_{24} = 8740.02837599061$$
$$x_{25} = 3070.81901762428$$
$$x_{26} = -2601.08061964435$$
$$x_{27} = 1327.49545509017$$
$$x_{28} = -2165.16523072303$$
$$x_{29} = -2383.11269559326$$
$$x_{30} = -6089.56186725577$$
$$x_{31} = -9142.38483712015$$
$$x_{32} = -9578.51073944067$$
$$x_{33} = 8958.0904654093$$
$$x_{34} = -6307.61551047145$$
$$x_{35} = 4378.95664464583$$
$$x_{36} = 9830.34243336704$$
$$x_{37} = -10668.8307008529$$
$$x_{38} = 8085.84474093068$$
$$x_{39} = -10886.8954425912$$
$$x_{40} = -5435.40891423752$$
$$x_{41} = -7834.01685095387$$
$$x_{42} = 10048.4062104668$$
$$x_{43} = -1293.78282794355$$
$$x_{44} = -4127.15782640266$$
$$x_{45} = -4345.19256556541$$
$$x_{46} = 6559.43723118159$$
$$x_{47} = 7649.72488690616$$
$$x_{48} = -4781.27173900404$$
$$x_{49} = -7397.89838354129$$
$$x_{50} = 4160.9213441114$$
$$x_{51} = 10920.6638951877$$
$$x_{52} = 6777.49312923367$$
$$x_{53} = 6341.38231795369$$
$$x_{54} = 3942.88989840842$$
$$x_{55} = 5905.27588178744$$
$$x_{56} = -10450.7661880914$$
$$x_{57} = -3909.12703788844$$
$$x_{58} = -5217.36114350749$$
$$x_{59} = 9612.27894894962$$
$$x_{60} = -8270.13763673415$$
$$x_{61} = -3255.06628468658$$
$$x_{62} = -2819.0642901117$$
$$x_{63} = -5871.50945486997$$
$$x_{64} = -5653.45841553203$$
$$x_{65} = 9394.21577758529$$
$$x_{66} = -8706.26039271754$$
$$x_{67} = 5469.17486580756$$
$$x_{68} = 1980.98879681515$$
$$x_{69} = -9796.57417650432$$
$$x_{70} = 3724.86298227001$$
$$x_{71} = -3473.0802785312$$
$$x_{72} = 8521.9667029934$$
$$x_{73} = 7431.66586628837$$
$$x_{74} = 5033.08067653292$$
$$x_{75} = -1729.36222137973$$
$$x_{76} = 5251.12681171199$$
$$x_{77} = 1545.2601310548$$
$$x_{78} = -6743.72601235467$$
$$x_{79} = 8303.90547920117$$
$$x_{80} = -10232.7019189395$$
$$x_{81} = 2416.86484798183$$
$$x_{82} = -4563.23069048471$$
$$x_{83} = -7615.95730257926$$
$$x_{84} = 1763.09931142612$$
$$x_{85} = -1511.5327294226$$
$$x_{86} = -3691.10089784646$$
$$x_{87} = 2198.91381476693$$
$$x_{88} = 4815.03672044901$$
$$x_{89} = -4999.31532937499$$
$$x_{90} = 3288.82632785286$$
Signos de extremos en los puntos:
(-8924.322419328006, -1.40693567122155e-12)
(5687.224618458442, 5.43624285392943e-12)
(2852.821285204498, 4.30701391463048e-11)
(-1947.2450191030653, -1.35437253384406e-10)
(7213.607522933207, 2.6640505856855e-12)
(7867.784528521002, 2.05325374556876e-12)
(-3037.0603364102, -3.56976573380026e-11)
(-8488.198787418114, -1.63513408387853e-12)
(9176.152941579825, 1.29424968224036e-12)
(-6961.782669805956, -2.96372982718315e-12)
(10702.599119040633, 8.15703305563058e-13)
(-10014.63790930326, -9.95621447053096e-13)
(-6525.670261278461, -3.59852578467758e-12)
(-7179.840151130289, -2.70181538007756e-12)
(-9360.447618770157, -1.21929879789127e-12)
(2634.8354930109967, 5.46687889857164e-11)
(6123.328494630542, 4.35549298902863e-12)
(10266.470261607734, 9.24137380628575e-13)
(10484.534569698704, 8.67665900788792e-13)
(-8052.076977574558, -1.91547391632927e-12)
(3506.8414374696895, 2.31873754141503e-11)
(4596.995252869141, 1.02938497251245e-11)
(6995.549920113271, 2.92101922540068e-12)
(8740.02837599061, 1.49782635937671e-12)
(3070.819017624277, 3.45332388429665e-11)
(-2601.0806196443546, -5.68248801266319e-11)
(1327.49545509017, 4.27465183883249e-10)
(-2165.165230723028, -9.85205483957949e-11)
(-2383.11269559326, -7.38866916279439e-11)
(-6089.561867255766, -4.42834912697605e-12)
(-9142.384837120151, -1.30864395132071e-12)
(-9578.510739440673, -1.13790505339796e-12)
(8958.090465409303, 1.3910849928183e-12)
(-6307.615510471454, -3.98478007588904e-12)
(4378.9566446458275, 1.19093498324855e-11)
(9830.342433367043, 1.05267439259783e-12)
(-10668.830700852859, -8.23473309706492e-13)
(8085.844740930678, 1.8915760323767e-12)
(-10886.895442591232, -7.74975266629726e-13)
(-5435.408914237517, -6.22735072266198e-12)
(-7834.01685095387, -2.0799193358159e-12)
(10048.406210466783, 9.85617595596554e-13)
(-1293.782827943549, -4.61759376066616e-10)
(-4127.157826402662, -1.42248192339035e-11)
(-4345.192565565406, -1.21891355053604e-11)
(6559.437231181588, 3.54323741531016e-12)
(7649.724886906163, 2.23389392259955e-12)
(-4781.271739004035, -9.14891745357932e-12)
(-7397.898383541286, -2.46987464143651e-12)
(4160.921344111399, 1.38813424680348e-11)
(10920.663895187696, 7.67808428195332e-13)
(6777.493129233672, 3.2121272173945e-12)
(6341.382317953688, 3.92146355044461e-12)
(3942.889898408417, 1.63138338972446e-11)
(5905.275881787441, 4.85600828005881e-12)
(-10450.76618809145, -8.76103879590501e-13)
(-3909.127037888438, -1.67401996406223e-11)
(-5217.361143507487, -7.04120994498961e-12)
(9612.278948949623, 1.12595465551416e-12)
(-8270.137636734147, -1.76791664444712e-12)
(-3255.0662846865775, -2.8994823485562e-11)
(-2819.0642901116958, -4.46359756345322e-11)
(-5871.50945486997, -4.94027017063802e-12)
(-5653.458415532032, -5.53423233024435e-12)
(9394.215777585292, 1.20619744296409e-12)
(-8706.260392717539, -1.51532239094573e-12)
(5469.174865807563, 6.11272087247919e-12)
(1980.9887968151456, 1.28633440684141e-10)
(-9796.574176504319, -1.06359749116181e-12)
(3724.8629822700063, 1.93494702014268e-11)
(-3473.080278531203, -2.38701702313708e-11)
(8521.966702993403, 1.61577356309481e-12)
(7431.665866288371, 2.43636005508638e-12)
(5033.080676532918, 7.84329077270126e-12)
(-1729.362221379731, -1.93349042265173e-10)
(5251.126811711991, 6.9062528246813e-12)
(1545.2601310547998, 2.7101611178979e-10)
(-6743.726012354672, -3.2606204162023e-12)
(8303.905479201172, 1.74643652474628e-12)
(-10232.701918939507, -9.33316681720004e-13)
(2416.8648479818335, 7.08341794421733e-11)
(-4563.23069048471, -1.05240454587021e-11)
(-7615.957302579259, -2.26373973745589e-12)
(1763.0993114261219, 1.82460811756282e-10)
(-1511.532729422596, -2.89565781091846e-10)
(-3691.100897846461, -1.98853039257663e-11)
(2198.91381476693, 9.40535801920205e-11)
(4815.0367204490085, 8.95779625425101e-12)
(-4999.315329374993, -8.00328715914172e-12)
(3288.826327852859, 2.81110558935193e-11)
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:
La función no tiene puntos mínimos
La función no tiene puntos máximos
Decrece en todo el eje numérico