In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative$$\frac{- e^{\frac{1}{x}} \cos{\left(x - 1 \right)} + \frac{e^{\frac{1}{x}} \sin{\left(x - 1 \right)}}{x^{2}}}{x} + \frac{e^{\frac{1}{x}} \sin{\left(x - 1 \right)}}{x^{2}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 90.5242222935239$$
$$x_{2} = -82.2401942469512$$
$$x_{3} = -97.9500638776862$$
$$x_{4} = -129.368425163711$$
$$x_{5} = -101.09196742337$$
$$x_{6} = -79.0981306244151$$
$$x_{7} = 882.215604537818$$
$$x_{8} = 65.3871232231487$$
$$x_{9} = -44.5311450895032$$
$$x_{10} = 11.9047687533544$$
$$x_{11} = -35.1006447274204$$
$$x_{12} = -69.6716886266418$$
$$x_{13} = -94.8081399480599$$
$$x_{14} = 125.084851504839$$
$$x_{15} = 52.8169894052803$$
$$x_{16} = -88.5242223558121$$
$$x_{17} = 96.8081399005761$$
$$x_{18} = 49.6741525580375$$
$$x_{19} = 74.8138830211387$$
$$x_{20} = 1.89086316139645$$
$$x_{21} = -31.956418752663$$
$$x_{22} = -38.2444497826469$$
$$x_{23} = 68.5294379303516$$
$$x_{24} = -50.8169899605382$$
$$x_{25} = -47.6741532712731$$
$$x_{26} = -72.8138831559317$$
$$x_{27} = -16.2209759676129$$
$$x_{28} = -41.3879273405629$$
$$x_{29} = -28.8116391999575$$
$$x_{30} = -60.244734759137$$
$$x_{31} = -63.3871234559972$$
$$x_{32} = 99.9500638359527$$
$$x_{33} = -25.6661111861798$$
$$x_{34} = -22.5195363391972$$
$$x_{35} = 33.9564153556302$$
$$x_{36} = -91.6661935619045$$
$$x_{37} = 87.3822235508343$$
$$x_{38} = 21.3714104361107$$
$$x_{39} = 71.6716884662242$$
$$x_{40} = 18.2209301787382$$
$$x_{41} = -57.1022600328727$$
$$x_{42} = 5.50076423875644$$
$$x_{43} = 93.666193507645$$
$$x_{44} = 55.9596840982793$$
$$x_{45} = 30.8116341242798$$
$$x_{46} = -6.7281114008228$$
$$x_{47} = -9.90505640900686$$
$$x_{48} = -85.3822236226929$$
$$x_{49} = 46.5311441578705$$
$$x_{50} = 43.3879261001259$$
$$x_{51} = -75.9560285626656$$
$$x_{52} = 62.2447344746801$$
$$x_{53} = 84.2401941636113$$
$$x_{54} = -3.51145625729743$$
$$x_{55} = 77.9560284485786$$
$$x_{56} = 40.2444480941082$$
$$x_{57} = 15.0665070958637$$
$$x_{58} = -19.3714337743213$$
$$x_{59} = 8.72695149596223$$
$$x_{60} = 27.6661032530785$$
$$x_{61} = 24.519523219747$$
$$x_{62} = 59.1022596816803$$
$$x_{63} = 37.100642368753$$
$$x_{64} = -13.0666103001526$$
$$x_{65} = -66.5294381227839$$
$$x_{66} = 81.0981305272062$$
$$x_{67} = -53.9596845369823$$
The values of the extrema at the points:
(90.52422229352386, -0.0111687780834938)
(-82.24019424695123, -0.0120116795983125)
(-97.95006387768619, 0.0101050686570301)
(-129.36842516371073, 0.00767011539602727)
(-101.09196742337016, -0.00979414408698753)
(-79.09813062441506, 0.0124827240334023)
(882.2156045378176, -0.00113479460425381)
(65.38712322314875, -0.0155273491079369)
(-44.531145089503156, -0.0219522439971534)
(11.904768753354384, 0.0909843646151252)
(-35.10064472742037, 0.0276787062476084)
(-69.67168862664177, -0.0141470785698663)
(-94.80813994805987, -0.0104363816333954)
(125.08485150483934, 0.00805848089806915)
(52.81698940528028, -0.0192915971672041)
(-88.52422235581211, -0.0111687573254055)
(96.80813990057612, -0.0104363974582078)
(49.6741525580375, 0.0205362355942815)
(74.81388302113875, 0.0135451233894866)
(1.890863161396451, -0.697886712906169)
(-31.956418752663, -0.0303146277702685)
(-38.24444978264689, -0.025464495490115)
(68.52943793035158, 0.0148051420955315)
(-50.816989960538216, -0.0192914122052374)
(-47.674153271273084, 0.0205359980296013)
(-72.8138831559317, 0.0135450784733407)
(-16.22097596761286, 0.057866063268526)
(-41.387927340562875, 0.0235782938191481)
(-28.8116391999575, 0.0335054030394208)
(-60.244734759137046, 0.016323534973335)
(-63.38712345599723, -0.0155272715254626)
(99.95006383595273, 0.0101050825656262)
(-25.666111186179823, -0.0374468043728171)
(-22.519536339197245, 0.0424389538935363)
(33.9564153556302, -0.030315758240525)
(-91.66619356190452, 0.010790156396892)
(87.3822235508343, 0.0115749166485247)
(21.371410436110686, -0.0489742458829439)
(71.67168846622415, -0.0141471320231358)
(18.22093017873822, 0.0578812340575476)
(-57.10226003287275, -0.0172058792800939)
(5.500764238756438, 0.213172028282039)
(93.666193507645, 0.0107901744796039)
(55.95968409827928, 0.0181892067692534)
(30.81163412427976, 0.0335070915110918)
(-6.728111400822799, -0.127089049158806)
(-9.905056409006862, 0.0908899121326124)
(-85.38222362269292, 0.0115748927014279)
(46.53114415787051, -0.0219525542719284)
(43.38792610012586, 0.0235787068842675)
(-75.95602856266557, -0.0129922207498032)
(62.24473447468011, 0.0163236297468303)
(84.24019416361132, -0.0120117073710662)
(-3.511456257297425, 0.209897225586145)
(77.95602844857859, -0.0129922587672397)
(40.24444809410818, -0.0254650576791727)
(15.066507095863699, -0.0707501721140639)
(-19.371433774321314, -0.0489665000996869)
(8.726951495962233, -0.127464334832884)
(27.66610325307851, -0.0374494420601855)
(24.519523219747033, 0.0424433128520271)
(59.10225968168029, -0.0172059962818578)
(37.10064236875305, 0.0276794913853983)
(-13.066610300152584, -0.0707160812401601)
(-66.52943812278386, 0.0148050779767495)
(81.0981305272062, 0.0124827564272671)
(-53.959684536982316, 0.018189060622011)
Intervals of increase and decrease of the function:Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
$$x_{1} = 90.5242222935239$$
$$x_{2} = -82.2401942469512$$
$$x_{3} = -101.09196742337$$
$$x_{4} = 882.215604537818$$
$$x_{5} = 65.3871232231487$$
$$x_{6} = -44.5311450895032$$
$$x_{7} = -69.6716886266418$$
$$x_{8} = -94.8081399480599$$
$$x_{9} = 52.8169894052803$$
$$x_{10} = -88.5242223558121$$
$$x_{11} = 96.8081399005761$$
$$x_{12} = 1.89086316139645$$
$$x_{13} = -31.956418752663$$
$$x_{14} = -38.2444497826469$$
$$x_{15} = -50.8169899605382$$
$$x_{16} = -63.3871234559972$$
$$x_{17} = -25.6661111861798$$
$$x_{18} = 33.9564153556302$$
$$x_{19} = 21.3714104361107$$
$$x_{20} = 71.6716884662242$$
$$x_{21} = -57.1022600328727$$
$$x_{22} = -6.7281114008228$$
$$x_{23} = 46.5311441578705$$
$$x_{24} = -75.9560285626656$$
$$x_{25} = 84.2401941636113$$
$$x_{26} = 77.9560284485786$$
$$x_{27} = 40.2444480941082$$
$$x_{28} = 15.0665070958637$$
$$x_{29} = -19.3714337743213$$
$$x_{30} = 8.72695149596223$$
$$x_{31} = 27.6661032530785$$
$$x_{32} = 59.1022596816803$$
$$x_{33} = -13.0666103001526$$
Maxima of the function at points:
$$x_{33} = -97.9500638776862$$
$$x_{33} = -129.368425163711$$
$$x_{33} = -79.0981306244151$$
$$x_{33} = 11.9047687533544$$
$$x_{33} = -35.1006447274204$$
$$x_{33} = 125.084851504839$$
$$x_{33} = 49.6741525580375$$
$$x_{33} = 74.8138830211387$$
$$x_{33} = 68.5294379303516$$
$$x_{33} = -47.6741532712731$$
$$x_{33} = -72.8138831559317$$
$$x_{33} = -16.2209759676129$$
$$x_{33} = -41.3879273405629$$
$$x_{33} = -28.8116391999575$$
$$x_{33} = -60.244734759137$$
$$x_{33} = 99.9500638359527$$
$$x_{33} = -22.5195363391972$$
$$x_{33} = -91.6661935619045$$
$$x_{33} = 87.3822235508343$$
$$x_{33} = 18.2209301787382$$
$$x_{33} = 5.50076423875644$$
$$x_{33} = 93.666193507645$$
$$x_{33} = 55.9596840982793$$
$$x_{33} = 30.8116341242798$$
$$x_{33} = -9.90505640900686$$
$$x_{33} = -85.3822236226929$$
$$x_{33} = 43.3879261001259$$
$$x_{33} = 62.2447344746801$$
$$x_{33} = -3.51145625729743$$
$$x_{33} = 24.519523219747$$
$$x_{33} = 37.100642368753$$
$$x_{33} = -66.5294381227839$$
$$x_{33} = 81.0981305272062$$
$$x_{33} = -53.9596845369823$$
Decreasing at intervals
$$\left[882.215604537818, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -101.09196742337\right]$$