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$$\frac{2 x - 2 \sin{\left(x \right)}}{\sin{\left(x \right)}} - \frac{\left(x^{2} + 2 \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -92.6551683094389$$
$$x_{2} = -26.6313909606767$$
$$x_{3} = 98.9397527512558$$
$$x_{4} = 32.9278998310383$$
$$x_{5} = 1.2849319665371$$
$$x_{6} = 80.0853327571602$$
$$x_{7} = 17.155956342572$$
$$x_{8} = -7.63088453995118$$
$$x_{9} = 39.2202567558535$$
$$x_{10} = -61.2278702799341$$
$$x_{11} = 26.6313909606767$$
$$x_{12} = -14.0054180069591$$
$$x_{13} = -32.9278998310383$$
$$x_{14} = 42.363211889599$$
$$x_{15} = 10.7955143777089$$
$$x_{16} = -23.4733304235899$$
$$x_{17} = 89.5133008395388$$
$$x_{18} = 36.0713918773106$$
$$x_{19} = -39.2202567558535$$
$$x_{20} = 45.5101401905028$$
$$x_{21} = -45.5101401905028$$
$$x_{22} = -1.2849319665371$$
$$x_{23} = -98.9397527512558$$
$$x_{24} = -48.6527574169176$$
$$x_{25} = 23.4733304235899$$
$$x_{26} = -89.5133008395388$$
$$x_{27} = -29.775811593805$$
$$x_{28} = 61.2278702799341$$
$$x_{29} = 92.6551683094389$$
$$x_{30} = -36.0713918773106$$
$$x_{31} = 95.7979195697262$$
$$x_{32} = -95.7979195697262$$
$$x_{33} = 20.3270942073626$$
$$x_{34} = -83.2284683476736$$
$$x_{35} = -80.0853327571602$$
$$x_{36} = -70.6579373151886$$
$$x_{37} = 29.775811593805$$
$$x_{38} = -58.0856381970501$$
$$x_{39} = -86.3703779936973$$
$$x_{40} = 51.7984316046612$$
$$x_{41} = -73.7999666296462$$
$$x_{42} = -54.9408225573151$$
$$x_{43} = -76.9433704296783$$
$$x_{44} = 54.9408225573151$$
$$x_{45} = -64.3720724360942$$
$$x_{46} = -51.7984316046612$$
$$x_{47} = 70.6579373151886$$
$$x_{48} = 64.3720724360942$$
$$x_{49} = 14.0054180069591$$
$$x_{50} = 67.5141887296183$$
$$x_{51} = -20.3270942073626$$
$$x_{52} = -10.7955143777089$$
$$x_{53} = 48.6527574169176$$
$$x_{54} = -17.155956342572$$
$$x_{55} = -67.5141887296183$$
$$x_{56} = -4.15988080716962$$
$$x_{57} = 7.63088453995118$$
$$x_{58} = -42.363211889599$$
$$x_{59} = 58.0856381970501$$
$$x_{60} = 73.7999666296462$$
$$x_{61} = 86.3703779936973$$
$$x_{62} = 83.2284683476736$$
$$x_{63} = 4.15988080716962$$
$$x_{64} = 76.9433704296783$$
Signos de extremos en los puntos:
(-92.65516830943886, 8586.97974862893)
(-26.631390960676704, -711.225360426234)
(98.93975275125575, -9791.07426594427)
(32.92789983103829, 1086.24290487152)
(1.2849319665370957, 2.3087155001914)
(80.08533275716019, -6415.65989935066)
(17.155956342571987, -296.313339112184)
(-7.630884539951176, -60.1639505608096)
(39.220256755853505, 1540.22594297569)
(-61.22787027993408, 3750.85103259194)
(26.631390960676704, 711.225360426234)
(-14.00541800695914, -198.131545970685)
(-32.92789983103829, -1086.24290487152)
(42.36321188959896, -1796.63949522568)
(10.795514377708916, -118.509382882767)
(-23.473330423589946, 552.990007818584)
(89.51330083953881, 8014.63052810254)
(36.07139187731056, -1303.14224246637)
(-39.220256755853505, -1540.22594297569)
(45.510140190502824, 2073.17093074854)
(-45.510140190502824, -2073.17093074854)
(-1.2849319665370957, -2.3087155001914)
(-98.93975275125575, 9791.07426594427)
(-48.652757416917595, 2369.08911585739)
(23.473330423589946, -552.990007818584)
(-89.51330083953881, -8014.63052810254)
(-29.77581159380503, 888.594454590108)
(61.22787027993408, -3750.85103259194)
(92.65516830943886, -8586.97974862893)
(-36.07139187731056, 1303.14224246637)
(95.79791956972616, 9179.24095812185)
(-95.79791956972616, -9179.24095812185)
(20.327094207362574, 415.181124677421)
(-83.2284683476736, -6928.97736621397)
(-80.08533275716019, 6415.65989935066)
(-70.65793731518859, -4994.54330476316)
(29.77581159380503, -888.594454590108)
(-58.08563819705012, -3375.9401799037)
(-86.3703779936973, 7461.84165871348)
(51.79843160466122, 2685.07602698745)
(-73.79996662964622, 5448.43434038109)
(-54.9408225573151, 3020.4926589874)
(-76.94337042967828, -5922.28157766336)
(54.9408225573151, -3020.4926589874)
(-64.37207243609416, -4145.76274487732)
(-51.79843160466122, -2685.07602698745)
(70.65793731518859, 4994.54330476316)
(64.37207243609416, 4145.76274487732)
(14.00541800695914, 198.131545970685)
(67.51418872961827, -4560.16480265732)
(-20.327094207362574, -415.181124677421)
(-10.795514377708916, 118.509382882767)
(48.652757416917595, -2369.08911585739)
(-17.155956342571987, 296.313339112184)
(-67.51418872961827, 4560.16480265732)
(-4.1598808071696185, 19.0962798149071)
(7.630884539951176, 60.1639505608096)
(-42.36321188959896, 1796.63949522568)
(58.08563819705012, 3375.9401799037)
(73.79996662964622, -5448.43434038109)
(86.3703779936973, -7461.84165871348)
(83.2284683476736, 6928.97736621397)
(4.1598808071696185, -19.0962798149071)
(76.94337042967828, 5922.28157766336)
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:
$$x_{1} = -92.6551683094389$$
$$x_{2} = 32.9278998310383$$
$$x_{3} = 1.2849319665371$$
$$x_{4} = 39.2202567558535$$
$$x_{5} = -61.2278702799341$$
$$x_{6} = 26.6313909606767$$
$$x_{7} = -23.4733304235899$$
$$x_{8} = 89.5133008395388$$
$$x_{9} = 45.5101401905028$$
$$x_{10} = -98.9397527512558$$
$$x_{11} = -48.6527574169176$$
$$x_{12} = -29.775811593805$$
$$x_{13} = -36.0713918773106$$
$$x_{14} = 95.7979195697262$$
$$x_{15} = 20.3270942073626$$
$$x_{16} = -80.0853327571602$$
$$x_{17} = -86.3703779936973$$
$$x_{18} = 51.7984316046612$$
$$x_{19} = -73.7999666296462$$
$$x_{20} = -54.9408225573151$$
$$x_{21} = 70.6579373151886$$
$$x_{22} = 64.3720724360942$$
$$x_{23} = 14.0054180069591$$
$$x_{24} = -10.7955143777089$$
$$x_{25} = -17.155956342572$$
$$x_{26} = -67.5141887296183$$
$$x_{27} = -4.15988080716962$$
$$x_{28} = 7.63088453995118$$
$$x_{29} = -42.363211889599$$
$$x_{30} = 58.0856381970501$$
$$x_{31} = 83.2284683476736$$
$$x_{32} = 76.9433704296783$$
Puntos máximos de la función:
$$x_{32} = -26.6313909606767$$
$$x_{32} = 98.9397527512558$$
$$x_{32} = 80.0853327571602$$
$$x_{32} = 17.155956342572$$
$$x_{32} = -7.63088453995118$$
$$x_{32} = -14.0054180069591$$
$$x_{32} = -32.9278998310383$$
$$x_{32} = 42.363211889599$$
$$x_{32} = 10.7955143777089$$
$$x_{32} = 36.0713918773106$$
$$x_{32} = -39.2202567558535$$
$$x_{32} = -45.5101401905028$$
$$x_{32} = -1.2849319665371$$
$$x_{32} = 23.4733304235899$$
$$x_{32} = -89.5133008395388$$
$$x_{32} = 61.2278702799341$$
$$x_{32} = 92.6551683094389$$
$$x_{32} = -95.7979195697262$$
$$x_{32} = -83.2284683476736$$
$$x_{32} = -70.6579373151886$$
$$x_{32} = 29.775811593805$$
$$x_{32} = -58.0856381970501$$
$$x_{32} = -76.9433704296783$$
$$x_{32} = 54.9408225573151$$
$$x_{32} = -64.3720724360942$$
$$x_{32} = -51.7984316046612$$
$$x_{32} = 67.5141887296183$$
$$x_{32} = -20.3270942073626$$
$$x_{32} = 48.6527574169176$$
$$x_{32} = 73.7999666296462$$
$$x_{32} = 86.3703779936973$$
$$x_{32} = 4.15988080716962$$
Decrece en los intervalos
$$\left[95.7979195697262, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.9397527512558\right]$$