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$$- \sin{\left(\frac{1}{x} \right)} + \frac{\cos{\left(\frac{1}{x} \right)}}{x} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -33838.8069987955$$
$$x_{2} = 27189.2682985119$$
$$x_{3} = -20277.3091678955$$
$$x_{4} = -37229.2008029351$$
$$x_{5} = 10237.7072403896$$
$$x_{6} = 8542.66546353133$$
$$x_{7} = -12649.1268015143$$
$$x_{8} = -38924.3992166982$$
$$x_{9} = 17018.2064665824$$
$$x_{10} = 38208.0321713516$$
$$x_{11} = 23798.8964040481$$
$$x_{12} = 34817.6371394969$$
$$x_{13} = -10954.0235496446$$
$$x_{14} = 32274.8438867786$$
$$x_{15} = 40750.8308380171$$
$$x_{16} = 36512.8341467568$$
$$x_{17} = 24646.4882156886$$
$$x_{18} = 39903.2310836223$$
$$x_{19} = 22951.3055101033$$
$$x_{20} = 16170.6287133314$$
$$x_{21} = -16886.9773640064$$
$$x_{22} = -10106.4851636254$$
$$x_{23} = -35534.0033488874$$
$$x_{24} = -32991.2092909147$$
$$x_{25} = 17865.7867327802$$
$$x_{26} = 25494.0808534902$$
$$x_{27} = -28753.2265249185$$
$$x_{28} = -24515.2570410387$$
$$x_{29} = -42314.798461984$$
$$x_{30} = 35665.2355068933$$
$$x_{31} = -32143.6119273287$$
$$x_{32} = 21256.1269144408$$
$$x_{33} = 39055.6315237518$$
$$x_{34} = -40619.5984700161$$
$$x_{35} = -21972.4849235058$$
$$x_{36} = 33970.0390649632$$
$$x_{37} = -18582.1393820539$$
$$x_{38} = 19560.9534972764$$
$$x_{39} = -29600.822198113$$
$$x_{40} = 15323.0538900739$$
$$x_{41} = -27905.6313719175$$
$$x_{42} = -30448.418348057$$
$$x_{43} = 28036.8629745587$$
$$x_{44} = -31296.0149360127$$
$$x_{45} = 29732.0539618849$$
$$x_{46} = 18713.369170442$$
$$x_{47} = -21124.8963893094$$
$$x_{48} = -9258.95883733077$$
$$x_{49} = -27058.0367879985$$
$$x_{50} = -23667.6653660369$$
$$x_{51} = -7563.95858775919$$
$$x_{52} = -15191.825714434$$
$$x_{53} = 30579.6501825125$$
$$x_{54} = -26210.4428283739$$
$$x_{55} = 12780.3528304711$$
$$x_{56} = 37360.4330405425$$
$$x_{57} = 13627.9152205591$$
$$x_{58} = -39771.9987451226$$
$$x_{59} = -19429.7234310854$$
$$x_{60} = -14344.2549267547$$
$$x_{61} = -17734.5572628217$$
$$x_{62} = -36381.6019475619$$
$$x_{63} = 28884.4582116784$$
$$x_{64} = 22103.7156394291$$
$$x_{65} = 6847.70624844332$$
$$x_{66} = -34686.4050257305$$
$$x_{67} = -11801.5713963528$$
$$x_{68} = 42446.0308837659$$
$$x_{69} = 33122.4413057753$$
$$x_{70} = -41467.1983793308$$
$$x_{71} = -25362.8495556371$$
$$x_{72} = -38076.7998978638$$
$$x_{73} = -8411.44821783076$$
$$x_{74} = 7695.1721243119$$
$$x_{75} = -6716.49789803616$$
$$x_{76} = 14475.4825115465$$
$$x_{77} = -16039.4000376645$$
$$x_{78} = -22820.0746241929$$
$$x_{79} = 20408.539477891$$
$$x_{80} = 31427.2468354985$$
$$x_{81} = 9390.17882712825$$
$$x_{82} = -13496.6883408414$$
$$x_{83} = 41598.4307750453$$
$$x_{84} = 11085.2472504735$$
$$x_{85} = 11932.7963857739$$
$$x_{86} = 26341.6742377005$$
Signos de extremos en los puntos:
(-33838.80699879548, 1.45552236929802e-10)
(27189.26829851187, 2.25451657343001e-10)
(-20277.309167895473, 4.05347977405768e-10)
(-37229.200802935076, 1.20249032953268e-10)
(10237.707240389647, 1.59016921852384e-9)
(8542.665463531333, 2.28382057709808e-9)
(-12649.126801514269, 1.04166408831929e-9)
(-38924.399216698235, 1.1000311772591e-10)
(17018.206466582393, 5.7546789555829e-10)
(38208.03217135156, 1.14166676112859e-10)
(23798.896404048144, 2.94262614275453e-10)
(34817.63713949693, 1.37483469053734e-10)
(-10954.023549644575, 1.38899736157327e-9)
(32274.8438867786, 1.60000235283064e-10)
(40750.83083801706, 1.00363384269997e-10)
(36512.83414675676, 1.25013666085749e-10)
(24646.48821568855, 2.74371303454757e-10)
(39903.23108362226, 1.04672492895475e-10)
(22951.305510103328, 3.1639824094043e-10)
(16170.628713331358, 6.37374819589809e-10)
(-16886.977364006372, 5.84446713247644e-10)
(-10106.485163625435, 1.63173063949529e-9)
(-35534.00334888743, 1.31995858687617e-10)
(-32991.20929091468, 1.53127177604517e-10)
(17865.786732780187, 5.22161092142426e-10)
(25494.080853490228, 2.56430654488327e-10)
(-28753.226524918457, 2.01593186588411e-10)
(-24515.257041038723, 2.77316725139087e-10)
(-42314.79846198405, 9.30816534960854e-11)
(35665.235506893325, 1.31026189897909e-10)
(-32143.611927328704, 1.613092992514e-10)
(21256.126914440752, 3.68876040823807e-10)
(39055.631523751756, 1.09265041459139e-10)
(-40619.59847001615, 1.01013086784008e-10)
(-21972.484923505832, 3.4521563385681e-10)
(33970.03906496316, 1.44429801451906e-10)
(-18582.139382053876, 4.82677786628471e-10)
(19560.953497276427, 4.35580793656243e-10)
(-29600.82219811303, 1.90213400586003e-10)
(15323.053890073896, 7.09836078804926e-10)
(-27905.63137191755, 2.14025130951256e-10)
(-30448.418348056977, 1.79770753838682e-10)
(28036.86297455866, 2.12026285417721e-10)
(-31296.0149360127, 1.70165215251927e-10)
(29732.053961884885, 1.88537963019542e-10)
(18713.369170442, 4.75931849486244e-10)
(-21124.896389309444, 3.73473252324175e-10)
(-9258.958837330767, 1.94412619336504e-9)
(-27058.03678799853, 2.27643792705123e-10)
(-23667.665366036923, 2.97534774595931e-10)
(-7563.9585877591935, 2.9130670098354e-9)
(-15191.82571443399, 7.22152337928605e-10)
(30579.650182512534, 1.78231207570434e-10)
(-26210.442828373863, 2.42605269207274e-10)
(12780.35283047108, 1.02038266724946e-9)
(37360.433040542535, 1.1940559652146e-10)
(13627.915220559116, 8.97407925748439e-10)
(-39771.99874512261, 1.05364383884421e-10)
(-19429.72343108544, 4.41484404589687e-10)
(-14344.254926754675, 8.10014610941323e-10)
(-17734.557262821672, 5.29917332237062e-10)
(-36381.60194756191, 1.25917165583189e-10)
(28884.45821167836, 1.99765426422971e-10)
(22103.715639429116, 3.41128569836258e-10)
(6847.706248443317, 3.55433615872869e-9)
(-34686.405025730484, 1.38525746429252e-10)
(-11801.571396352814, 1.19665521935985e-9)
(42446.030883765874, 9.2506891036237e-11)
(33122.441305775326, 1.51916146329256e-10)
(-41467.19837933084, 9.69256896965476e-11)
(-25362.84955563705, 2.59091192944538e-10)
(-38076.799897863806, 1.14954934460343e-10)
(-8411.448217830764, 2.35563069050926e-9)
(7695.172124311905, 2.8145700214921e-9)
(-6716.4978980361575, 3.69456221172015e-9)
(14475.482511546521, 7.95394528019244e-10)
(-16039.400037664549, 6.47847109291888e-10)
(-22820.074624192912, 3.20047655044675e-10)
(20408.539477890954, 4.00151911605917e-10)
(31427.246835498532, 1.68747127382574e-10)
(9390.178827128255, 1.89017068663588e-9)
(-13496.688340841416, 9.14943676377789e-10)
(41598.43077504528, 9.63151780553062e-11)
(11085.247250473549, 1.35630695563549e-9)
(11932.796385773858, 1.1704809343982e-9)
(26341.6742377005, 2.40193864797789e-10)
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:
$$x_{1} = 17018.2064665824$$
$$x_{2} = 40750.8308380171$$
$$x_{3} = -29600.822198113$$
$$x_{4} = 28036.8629745587$$
$$x_{5} = -21124.8963893094$$
$$x_{6} = -23667.6653660369$$
$$x_{7} = -39771.9987451226$$
$$x_{8} = 6847.70624844332$$
$$x_{9} = 42446.0308837659$$
$$x_{10} = 41598.4307750453$$
Puntos máximos de la función:
$$x_{10} = -33838.8069987955$$
$$x_{10} = -37229.2008029351$$
$$x_{10} = 8542.66546353133$$
$$x_{10} = 38208.0321713516$$
$$x_{10} = 34817.6371394969$$
$$x_{10} = -10954.0235496446$$
$$x_{10} = -28753.2265249185$$
$$x_{10} = -9258.95883733077$$
$$x_{10} = -7563.95858775919$$
$$x_{10} = -14344.2549267547$$
$$x_{10} = -34686.4050257305$$
$$x_{10} = -22820.0746241929$$
$$x_{10} = -13496.6883408414$$
$$x_{10} = 11932.7963857739$$
Decrece en los intervalos
$$\left[42446.0308837659, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -39771.9987451226\right]$$