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$$- \frac{\sin{\left(x \right)}}{x^{2}} - \frac{2 \cos{\left(x \right)}}{x^{3}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 34.4996123350132$$
$$x_{2} = -5.95939190757933$$
$$x_{3} = 9.21096438740149$$
$$x_{4} = -31.3522215217643$$
$$x_{5} = 56.5132926241755$$
$$x_{6} = -56.5132926241755$$
$$x_{7} = -59.6567478435559$$
$$x_{8} = 94.2265573558031$$
$$x_{9} = -103.653264863525$$
$$x_{10} = -50.2256832197934$$
$$x_{11} = -47.0814357397523$$
$$x_{12} = 91.0842327848165$$
$$x_{13} = -25.053079662454$$
$$x_{14} = -53.3696181339615$$
$$x_{15} = -37.6460352959305$$
$$x_{16} = 65.9431258539286$$
$$x_{17} = -18.7432530945386$$
$$x_{18} = 53.3696181339615$$
$$x_{19} = -12.4065403639626$$
$$x_{20} = 84.7994209518635$$
$$x_{21} = 81.6569211705466$$
$$x_{22} = 47.0814357397523$$
$$x_{23} = -28.2035393053095$$
$$x_{24} = 37.6460352959305$$
$$x_{25} = -15.5802941824244$$
$$x_{26} = -91.0842327848165$$
$$x_{27} = -81.6569211705466$$
$$x_{28} = 75.3716947511882$$
$$x_{29} = 25.053079662454$$
$$x_{30} = -78.5143487963623$$
$$x_{31} = 40.7917141624847$$
$$x_{32} = -97.3688346960149$$
$$x_{33} = -9.21096438740149$$
$$x_{34} = -72.2289483771681$$
$$x_{35} = 135.073678452493$$
$$x_{36} = 78.5143487963623$$
$$x_{37} = -62.8000167068325$$
$$x_{38} = -109.93755273626$$
$$x_{39} = 28.2035393053095$$
$$x_{40} = 50.2256832197934$$
$$x_{41} = -34.4996123350132$$
$$x_{42} = 72.2289483771681$$
$$x_{43} = 18.7432530945386$$
$$x_{44} = 12.4065403639626$$
$$x_{45} = 43.9368086315937$$
$$x_{46} = -87.9418559209576$$
$$x_{47} = 100.511069234565$$
$$x_{48} = 87.9418559209576$$
$$x_{49} = 21.9000773156394$$
$$x_{50} = -21.9000773156394$$
$$x_{51} = -94.2265573558031$$
$$x_{52} = 15.5802941824244$$
$$x_{53} = 59.6567478435559$$
$$x_{54} = -100.511069234565$$
$$x_{55} = -75.3716947511882$$
$$x_{56} = 31.3522215217643$$
$$x_{57} = -65.9431258539286$$
$$x_{58} = -69.0860970774096$$
$$x_{59} = 69.0860970774096$$
$$x_{60} = -43.9368086315937$$
$$x_{61} = 97.3688346960149$$
$$x_{62} = -40.7917141624847$$
$$x_{63} = 5.95939190757933$$
$$x_{64} = 62.8000167068325$$
$$x_{65} = -84.7994209518635$$
$$x_{66} = 2.45871417599962$$
$$x_{67} = -2.45871417599962$$
The values of the extrema at the points:
(34.4996123350132, -0.000838770260526343)
(-5.9593919075793265, 0.0266944281300046)
(9.210964387401486, -0.0115182384102548)
(-31.352221521764292, 0.0010152698990766)
(56.513292624175506, 0.000312915432650295)
(-56.513292624175506, 0.000312915432650295)
(-59.656747843555884, -0.000280825751144458)
(94.22655735580307, 0.000112604447700661)
(-103.65326486352524, -9.30578895300905e-5)
(-50.2256832197934, 0.000396099456126142)
(-47.081435739752315, -0.000450722368032648)
(91.08423278481655, -0.000120506069272649)
(-25.053079662453992, 0.00158817477024791)
(-53.36961813396146, -0.000350838362181669)
(-37.64603529593052, 0.000704611119614408)
(65.94312585392862, -0.000229858880631)
(-18.74325309453857, 0.00283042465312132)
(53.36961813396146, -0.000350838362181669)
(-12.406540363962565, 0.00641398077993427)
(84.79942095186354, -0.000139025181535869)
(81.65692117054658, 0.000149928353545869)
(47.081435739752315, -0.000450722368032648)
(-28.20353930530947, -0.00125401736797822)
(37.64603529593052, 0.000704611119614408)
(-15.580294182424433, -0.00408601227579287)
(-91.08423278481655, -0.000120506069272649)
(-81.65692117054658, 0.000149928353545869)
(75.37169475118824, 0.000175966743144092)
(25.053079662453992, 0.00158817477024791)
(-78.51434879636227, -0.000162166475547147)
(40.79171416248471, -0.00060025351930421)
(-97.36883469601494, -0.000105455311245964)
(-9.210964387401486, -0.0115182384102548)
(-72.22894837716808, -0.00019160683134921)
(135.07367845249348, -5.48038341935653e-5)
(78.51434879636227, -0.000162166475547147)
(-62.80001670683253, 0.000253431359776371)
(-109.93755273625987, -8.27248552367837e-5)
(28.20353930530947, -0.00125401736797822)
(50.2256832197934, 0.000396099456126142)
(-34.4996123350132, -0.000838770260526343)
(72.22894837716808, -0.00019160683134921)
(18.74325309453857, 0.00283042465312132)
(12.406540363962565, 0.00641398077993427)
(43.936808631593706, 0.000517479923876906)
(-87.94185592095755, 0.000129269617694298)
(100.51106923456473, 9.89660537297585e-5)
(87.94185592095755, 0.000129269617694298)
(21.90007731563936, -0.00207637214990232)
(-21.90007731563936, -0.00207637214990232)
(-94.22655735580307, 0.000112604447700661)
(15.580294182424433, -0.00408601227579287)
(59.656747843555884, -0.000280825751144458)
(-100.51106923456473, 9.89660537297585e-5)
(-75.37169475118824, 0.000175966743144092)
(31.352221521764292, 0.0010152698990766)
(-65.94312585392862, -0.000229858880631)
(-69.08609707740959, 0.000209428978902002)
(69.08609707740959, 0.000209428978902002)
(-43.936808631593706, 0.000517479923876906)
(97.36883469601494, -0.000105455311245964)
(-40.79171416248471, -0.00060025351930421)
(5.9593919075793265, 0.0266944281300046)
(62.80001670683253, 0.000253431359776371)
(-84.79942095186354, -0.000139025181535869)
(2.4587141759996247, -0.128324928485094)
(-2.4587141759996247, -0.128324928485094)
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:
$$x_{1} = 34.4996123350132$$
$$x_{2} = 9.21096438740149$$
$$x_{3} = -59.6567478435559$$
$$x_{4} = -103.653264863525$$
$$x_{5} = -47.0814357397523$$
$$x_{6} = 91.0842327848165$$
$$x_{7} = -53.3696181339615$$
$$x_{8} = 65.9431258539286$$
$$x_{9} = 53.3696181339615$$
$$x_{10} = 84.7994209518635$$
$$x_{11} = 47.0814357397523$$
$$x_{12} = -28.2035393053095$$
$$x_{13} = -15.5802941824244$$
$$x_{14} = -91.0842327848165$$
$$x_{15} = -78.5143487963623$$
$$x_{16} = 40.7917141624847$$
$$x_{17} = -97.3688346960149$$
$$x_{18} = -9.21096438740149$$
$$x_{19} = -72.2289483771681$$
$$x_{20} = 135.073678452493$$
$$x_{21} = 78.5143487963623$$
$$x_{22} = -109.93755273626$$
$$x_{23} = 28.2035393053095$$
$$x_{24} = -34.4996123350132$$
$$x_{25} = 72.2289483771681$$
$$x_{26} = 21.9000773156394$$
$$x_{27} = -21.9000773156394$$
$$x_{28} = 15.5802941824244$$
$$x_{29} = 59.6567478435559$$
$$x_{30} = -65.9431258539286$$
$$x_{31} = 97.3688346960149$$
$$x_{32} = -40.7917141624847$$
$$x_{33} = -84.7994209518635$$
$$x_{34} = 2.45871417599962$$
$$x_{35} = -2.45871417599962$$
Maxima of the function at points:
$$x_{35} = -5.95939190757933$$
$$x_{35} = -31.3522215217643$$
$$x_{35} = 56.5132926241755$$
$$x_{35} = -56.5132926241755$$
$$x_{35} = 94.2265573558031$$
$$x_{35} = -50.2256832197934$$
$$x_{35} = -25.053079662454$$
$$x_{35} = -37.6460352959305$$
$$x_{35} = -18.7432530945386$$
$$x_{35} = -12.4065403639626$$
$$x_{35} = 81.6569211705466$$
$$x_{35} = 37.6460352959305$$
$$x_{35} = -81.6569211705466$$
$$x_{35} = 75.3716947511882$$
$$x_{35} = 25.053079662454$$
$$x_{35} = -62.8000167068325$$
$$x_{35} = 50.2256832197934$$
$$x_{35} = 18.7432530945386$$
$$x_{35} = 12.4065403639626$$
$$x_{35} = 43.9368086315937$$
$$x_{35} = -87.9418559209576$$
$$x_{35} = 100.511069234565$$
$$x_{35} = 87.9418559209576$$
$$x_{35} = -94.2265573558031$$
$$x_{35} = -100.511069234565$$
$$x_{35} = -75.3716947511882$$
$$x_{35} = 31.3522215217643$$
$$x_{35} = -69.0860970774096$$
$$x_{35} = 69.0860970774096$$
$$x_{35} = -43.9368086315937$$
$$x_{35} = 5.95939190757933$$
$$x_{35} = 62.8000167068325$$
Decreasing at intervals
$$\left[135.073678452493, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -109.93755273626\right]$$