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$$\frac{2 \cos{\left(2 x \right)}}{x} - \frac{\sin{\left(2 x \right)}}{x^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 2.24670472895453$$
$$x_{2} = -10.1856514796438$$
$$x_{3} = -24.3370721159772$$
$$x_{4} = 98.172223901556$$
$$x_{5} = -32.1935597952787$$
$$x_{6} = 69.8968599047927$$
$$x_{7} = -46.3330961388114$$
$$x_{8} = -98.172223901556$$
$$x_{9} = -77.7512028363303$$
$$x_{10} = 24.3370721159772$$
$$x_{11} = -19.6222161805821$$
$$x_{12} = 41.6200962353617$$
$$x_{13} = 40.0490643144726$$
$$x_{14} = -3.86262591846885$$
$$x_{15} = 46.3330961388114$$
$$x_{16} = -16.4781945199112$$
$$x_{17} = 11.7597262493445$$
$$x_{18} = 85.6054794697228$$
$$x_{19} = 54.1878598258373$$
$$x_{20} = 66.7550989265392$$
$$x_{21} = 51.0459832324538$$
$$x_{22} = -60.4715244985757$$
$$x_{23} = 84.0346285545694$$
$$x_{24} = 91.8888644664832$$
$$x_{25} = -63.6133213216672$$
$$x_{26} = 52.6169257678188$$
$$x_{27} = -54.1878598258373$$
$$x_{28} = 63.6133213216672$$
$$x_{29} = 99.7430603324317$$
$$x_{30} = -76.1803402100956$$
$$x_{31} = -35.3358428558098$$
$$x_{32} = 30.6223651301872$$
$$x_{33} = -82.4637755597094$$
$$x_{34} = 13.3330271294063$$
$$x_{35} = 18.0503111221878$$
$$x_{36} = -47.9040693934309$$
$$x_{37} = -68.3259813506395$$
$$x_{38} = 32.1935597952787$$
$$x_{39} = -99.7430603324317$$
$$x_{40} = -41.6200962353617$$
$$x_{41} = 38.4780131551656$$
$$x_{42} = 76.1803402100956$$
$$x_{43} = 8.61037763596538$$
$$x_{44} = -5.45206082971445$$
$$x_{45} = 55.7587861230655$$
$$x_{46} = -27.4798391439445$$
$$x_{47} = -85.6054794697228$$
$$x_{48} = -58.9006179191122$$
$$x_{49} = -55.7587861230655$$
$$x_{50} = 22.7655670069956$$
$$x_{51} = -84.0346285545694$$
$$x_{52} = -38.4780131551656$$
$$x_{53} = 47.9040693934309$$
$$x_{54} = 74.6094747920599$$
$$x_{55} = 204.987701063789$$
$$x_{56} = 82.4637755597094$$
$$x_{57} = -71.4677348441946$$
$$x_{58} = -2.24670472895453$$
$$x_{59} = 33.7647173885721$$
$$x_{60} = -49.4750314121659$$
$$x_{61} = -62.0424254948814$$
$$x_{62} = 16.4781945199112$$
$$x_{63} = -18.0503111221878$$
$$x_{64} = 77.7512028363303$$
$$x_{65} = -91.8888644664832$$
$$x_{66} = -33.7647173885721$$
$$x_{67} = -40.0490643144726$$
$$x_{68} = -69.8968599047927$$
$$x_{69} = 96.6013861664138$$
$$x_{70} = 60.4715244985757$$
$$x_{71} = 88.7471755026564$$
$$x_{72} = 10.1856514796438$$
$$x_{73} = 68.3259813506395$$
$$x_{74} = 44.7621104652086$$
$$x_{75} = -13.3330271294063$$
$$x_{76} = -79.3220628366317$$
$$x_{77} = -90.3180208221014$$
$$x_{78} = -57.3297052975115$$
$$x_{79} = -25.9084912436398$$
$$x_{80} = -11.7597262493445$$
$$x_{81} = 19.6222161805821$$
$$x_{82} = 25.9084912436398$$
$$x_{83} = -93.4597065202651$$
$$x_{84} = 3.86262591846885$$
$$x_{85} = 62.0424254948814$$
$$x_{86} = 90.3180208221014$$
Signos de extremos en los puntos:
(2.246704728954532, -0.434467256422443)
(-10.18565147964378, 0.0980592480281483)
(-24.337072115977193, -0.0410809080835075)
(98.172223901556, 0.0101860484638785)
(-32.19355979527871, 0.0310583676149227)
(69.8968599047927, 0.0143064283116353)
(-46.33309613881142, -0.0215815876990685)
(-98.172223901556, 0.0101860484638785)
(-77.75120283633034, -0.0128612714243586)
(24.337072115977193, -0.0410809080835075)
(-19.622216180582097, 0.0509461061857615)
(41.6200962353617, 0.0240251209641055)
(40.04906431447256, -0.024967426643558)
(-3.8626259184688534, 0.256749107051798)
(46.33309613881142, -0.0215815876990685)
(-16.478194519911238, 0.0606583423726206)
(11.759726249344503, -0.0849592339552253)
(85.60547946972281, 0.0116812959808693)
(54.18785982583734, 0.0184535325015639)
(66.75509892653919, 0.0149797089170242)
(51.04598323245382, 0.0195892402934823)
(-60.47152449857575, 0.0165361437007152)
(84.0346285545694, -0.0118996456204748)
(91.88886446648316, 0.0108825503716759)
(-63.613321321667165, 0.015719492253233)
(52.6169257678188, -0.0190044332375671)
(-54.18785982583734, 0.0184535325015639)
(63.613321321667165, 0.015719492253233)
(99.74306033243167, -0.0100256341886906)
(-76.18034021009562, 0.0131264635863328)
(-35.33584285580975, 0.0282970441297328)
(30.6223651301872, -0.0326515186419956)
(-82.46377555970939, 0.0121263137918205)
(13.333027129406338, 0.0749490399878624)
(18.050311122187804, -0.0553794646022984)
(-47.90406939343085, 0.0208739162691316)
(-68.3259813506395, -0.0146353291349374)
(32.19355979527871, 0.0310583676149227)
(-99.74306033243167, -0.0100256341886906)
(-41.6200962353617, 0.0240251209641055)
(38.47801315516559, 0.0259866739740854)
(76.18034021009562, 0.0131264635863328)
(8.610377635965385, -0.115943604692308)
(-5.4520608297144495, -0.182650405646115)
(55.758786123065505, -0.0179336722809866)
(-27.479839143944467, -0.0363842926436063)
(-85.60547946972281, 0.0116812959808693)
(-58.90061791911219, -0.0169771388955304)
(-55.758786123065505, -0.0179336722809866)
(22.76556700699564, 0.0439153964569649)
(-84.0346285545694, -0.0118996456204748)
(-38.47801315516559, 0.0259866739740854)
(47.90406939343085, 0.0208739162691316)
(74.60947479205991, -0.0134028224709878)
(204.98770106378876, 0.00487832694374757)
(82.46377555970939, 0.0121263137918205)
(-71.46773484419464, -0.0139919857530453)
(-2.246704728954532, -0.434467256422443)
(33.76471738857206, -0.0296134678930985)
(-49.47503141216594, -0.0202111834730081)
(-62.04242549488138, -0.0161174796093628)
(16.478194519911238, 0.0606583423726206)
(-18.050311122187804, -0.0553794646022984)
(77.75120283633034, -0.0128612714243586)
(-91.88886446648316, 0.0108825503716759)
(-33.76471738857206, -0.0296134678930985)
(-40.04906431447256, -0.024967426643558)
(-69.8968599047927, 0.0143064283116353)
(96.60138616641379, -0.0103516796697785)
(60.47152449857575, 0.0165361437007152)
(88.7471755026564, 0.0112677854121748)
(10.18565147964378, 0.0980592480281483)
(68.3259813506395, -0.0146353291349374)
(44.76211046520859, 0.0223389292683471)
(-13.333027129406338, 0.0749490399878624)
(-79.32206283663172, 0.0126065825610424)
(-90.31802082210145, -0.01107181786798)
(-57.32970529751154, 0.0174423008954086)
(-25.908491243639833, 0.0385901989751759)
(-11.759726249344503, -0.0849592339552253)
(19.622216180582097, 0.0509461061857615)
(25.908491243639833, 0.0385901989751759)
(-93.45970652026512, -0.0106996450858762)
(3.8626259184688534, 0.256749107051798)
(62.04242549488138, -0.0161174796093628)
(90.31802082210145, -0.01107181786798)
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:
$$x_{1} = 2.24670472895453$$
$$x_{2} = -24.3370721159772$$
$$x_{3} = -46.3330961388114$$
$$x_{4} = -77.7512028363303$$
$$x_{5} = 24.3370721159772$$
$$x_{6} = 40.0490643144726$$
$$x_{7} = 46.3330961388114$$
$$x_{8} = 11.7597262493445$$
$$x_{9} = 84.0346285545694$$
$$x_{10} = 52.6169257678188$$
$$x_{11} = 99.7430603324317$$
$$x_{12} = 30.6223651301872$$
$$x_{13} = 18.0503111221878$$
$$x_{14} = -68.3259813506395$$
$$x_{15} = -99.7430603324317$$
$$x_{16} = 8.61037763596538$$
$$x_{17} = -5.45206082971445$$
$$x_{18} = 55.7587861230655$$
$$x_{19} = -27.4798391439445$$
$$x_{20} = -58.9006179191122$$
$$x_{21} = -55.7587861230655$$
$$x_{22} = -84.0346285545694$$
$$x_{23} = 74.6094747920599$$
$$x_{24} = -71.4677348441946$$
$$x_{25} = -2.24670472895453$$
$$x_{26} = 33.7647173885721$$
$$x_{27} = -49.4750314121659$$
$$x_{28} = -62.0424254948814$$
$$x_{29} = -18.0503111221878$$
$$x_{30} = 77.7512028363303$$
$$x_{31} = -33.7647173885721$$
$$x_{32} = -40.0490643144726$$
$$x_{33} = 96.6013861664138$$
$$x_{34} = 68.3259813506395$$
$$x_{35} = -90.3180208221014$$
$$x_{36} = -11.7597262493445$$
$$x_{37} = -93.4597065202651$$
$$x_{38} = 62.0424254948814$$
$$x_{39} = 90.3180208221014$$
Puntos máximos de la función:
$$x_{39} = -10.1856514796438$$
$$x_{39} = 98.172223901556$$
$$x_{39} = -32.1935597952787$$
$$x_{39} = 69.8968599047927$$
$$x_{39} = -98.172223901556$$
$$x_{39} = -19.6222161805821$$
$$x_{39} = 41.6200962353617$$
$$x_{39} = -3.86262591846885$$
$$x_{39} = -16.4781945199112$$
$$x_{39} = 85.6054794697228$$
$$x_{39} = 54.1878598258373$$
$$x_{39} = 66.7550989265392$$
$$x_{39} = 51.0459832324538$$
$$x_{39} = -60.4715244985757$$
$$x_{39} = 91.8888644664832$$
$$x_{39} = -63.6133213216672$$
$$x_{39} = -54.1878598258373$$
$$x_{39} = 63.6133213216672$$
$$x_{39} = -76.1803402100956$$
$$x_{39} = -35.3358428558098$$
$$x_{39} = -82.4637755597094$$
$$x_{39} = 13.3330271294063$$
$$x_{39} = -47.9040693934309$$
$$x_{39} = 32.1935597952787$$
$$x_{39} = -41.6200962353617$$
$$x_{39} = 38.4780131551656$$
$$x_{39} = 76.1803402100956$$
$$x_{39} = -85.6054794697228$$
$$x_{39} = 22.7655670069956$$
$$x_{39} = -38.4780131551656$$
$$x_{39} = 47.9040693934309$$
$$x_{39} = 204.987701063789$$
$$x_{39} = 82.4637755597094$$
$$x_{39} = 16.4781945199112$$
$$x_{39} = -91.8888644664832$$
$$x_{39} = -69.8968599047927$$
$$x_{39} = 60.4715244985757$$
$$x_{39} = 88.7471755026564$$
$$x_{39} = 10.1856514796438$$
$$x_{39} = 44.7621104652086$$
$$x_{39} = -13.3330271294063$$
$$x_{39} = -79.3220628366317$$
$$x_{39} = -57.3297052975115$$
$$x_{39} = -25.9084912436398$$
$$x_{39} = 19.6222161805821$$
$$x_{39} = 25.9084912436398$$
$$x_{39} = 3.86262591846885$$
Decrece en los intervalos
$$\left[99.7430603324317, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.7430603324317\right]$$