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$$- x^{5} \sin{\left(x \right)} + 5 x^{4} \cos{\left(x \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 62.9111635112525$$
$$x_{2} = 37.8305186388889$$
$$x_{3} = 97.4406405869711$$
$$x_{4} = -69.1871806855712$$
$$x_{5} = -75.4643834089706$$
$$x_{6} = 81.7425005689042$$
$$x_{7} = -47.2293632804741$$
$$x_{8} = 50.3644346278355$$
$$x_{9} = -56.6367214297364$$
$$x_{10} = 28.4483142882062$$
$$x_{11} = 6.90959579542153$$
$$x_{12} = 22.2125562085988$$
$$x_{13} = 91.1609800814091$$
$$x_{14} = -97.4406405869711$$
$$x_{15} = -40.9621674789216$$
$$x_{16} = 59.7737149408088$$
$$x_{17} = -66.0490029831692$$
$$x_{18} = 72.3256529801404$$
$$x_{19} = -4.03356779033998$$
$$x_{20} = 84.8818390729824$$
$$x_{21} = -53.5002619192185$$
$$x_{22} = -34.7006238709965$$
$$x_{23} = 31.5729854784403$$
$$x_{24} = 66.0490029831692$$
$$x_{25} = 44.0952059009573$$
$$x_{26} = 1.3138377164929$$
$$x_{27} = 94.30075185089$$
$$x_{28} = 19.1055198623998$$
$$x_{29} = 69.1871806855712$$
$$x_{30} = 75.4643834089706$$
$$x_{31} = 9.89275256512429$$
$$x_{32} = -81.7425005689042$$
$$x_{33} = 0$$
$$x_{34} = -28.4483142882062$$
$$x_{35} = -88.0213377290448$$
$$x_{36} = 47.2293632804741$$
$$x_{37} = -62.9111635112525$$
$$x_{38} = -25.3276477920726$$
$$x_{39} = -12.9352212801115$$
$$x_{40} = 56.6367214297364$$
$$x_{41} = -100.580635384527$$
$$x_{42} = -16.0106585966129$$
$$x_{43} = -22.2125562085988$$
$$x_{44} = 4.03356779033998$$
$$x_{45} = -72.3256529801404$$
$$x_{46} = -9.89275256512429$$
$$x_{47} = -31.5729854784403$$
$$x_{48} = -59.7737149408088$$
$$x_{49} = -6.90959579542153$$
$$x_{50} = 100.580635384527$$
$$x_{51} = 103.720726651558$$
$$x_{52} = -94.30075185089$$
$$x_{53} = -44.0952059009573$$
$$x_{54} = -50.3644346278355$$
$$x_{55} = -91.1609800814091$$
$$x_{56} = 53.5002619192185$$
$$x_{57} = 34.7006238709965$$
$$x_{58} = 78.6033412791698$$
$$x_{59} = 16.0106585966129$$
$$x_{60} = -84.8818390729824$$
$$x_{61} = -37.8305186388889$$
$$x_{62} = -1.3138377164929$$
$$x_{63} = 12.9352212801115$$
$$x_{64} = 40.9621674789216$$
$$x_{65} = 88.0213377290448$$
$$x_{66} = -78.6033412791698$$
$$x_{67} = -19.1055198623998$$
$$x_{68} = 25.3276477920726$$
Signos de extremos en los puntos:
(62.91116351125249, 982361343.674869)
(37.830518638888854, 76815876.763504)
(97.44064058697107, -8772626573.89269)
(-69.18718068557116, -1581237269.79774)
(-75.46438340897063, -2442074835.1523)
(81.74250056890423, 3642744051.26235)
(-47.2293632804741, 233690077.851263)
(50.36443462783552, 322470604.989764)
(-56.63672142973645, -580503316.767324)
(28.44831428820623, -18351703.6596154)
(6.909595795421526, 12759.1487492828)
(22.212556208598823, -5275452.6005768)
(91.16098008140906, -6286264691.69339)
(-97.44064058697107, 8772626573.89269)
(-40.96216747892155, 114473011.24935)
(59.77371494080883, -760391276.15604)
(-66.04900298316916, 1253402274.18613)
(72.32565298014042, -1974360702.22499)
(-4.033567790339982, 670.379412457278)
(84.88183907298242, -4398673622.91735)
(-53.500261919218524, 436406435.530439)
(-34.70062387099647, 49799410.9183366)
(31.572985478440316, 30988431.8368961)
(66.04900298316916, -1253402274.18613)
(44.09520590095732, 165646664.621845)
(1.3138377164928983, 0.994905864288247)
(94.30075185089002, 7446739532.87571)
(19.10551986239984, 2462687.0243185)
(69.18718068557116, 1581237269.79774)
(75.46438340897063, 2442074835.1523)
(9.892752565124287, -84564.1022728481)
(-81.74250056890423, -3642744051.26235)
(0, 0)
(-28.44831428820623, 18351703.6596154)
(-88.02133772904483, -5275216316.74103)
(47.2293632804741, -233690077.851263)
(-62.91116351125249, -982361343.674869)
(-25.327647792072558, -10225214.1985132)
(-12.935221280111474, -337777.725370395)
(56.63672142973645, 580503316.767324)
(-100.58063538452689, -10281013219.2145)
(-16.010658596612945, 1004242.32805887)
(-22.212556208598823, 5275452.6005768)
(4.033567790339982, -670.379412457278)
(-72.32565298014042, 1974360702.22499)
(-9.892752565124287, 84564.1022728481)
(-31.572985478440316, -30988431.8368961)
(-59.77371494080883, 760391276.15604)
(-6.909595795421526, -12759.1487492828)
(100.58063538452689, 10281013219.2145)
(103.72072665155811, -11990125266.5391)
(-94.30075185089002, -7446739532.87571)
(-44.09520590095732, -165646664.621845)
(-50.36443462783552, -322470604.989764)
(-91.16098008140906, 6286264691.69339)
(53.500261919218524, -436406435.530439)
(34.70062387099647, -49799410.9183366)
(78.60334127916975, -2994526546.59522)
(16.010658596612945, -1004242.32805887)
(-84.88183907298242, 4398673622.91735)
(-37.830518638888854, -76815876.763504)
(-1.3138377164928983, -0.994905864288247)
(12.935221280111474, 337777.725370395)
(40.96216747892155, -114473011.24935)
(88.02133772904483, 5275216316.74103)
(-78.60334127916975, 2994526546.59522)
(-19.10551986239984, -2462687.0243185)
(25.327647792072558, 10225214.1985132)
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:
$$x_{1} = 97.4406405869711$$
$$x_{2} = -69.1871806855712$$
$$x_{3} = -75.4643834089706$$
$$x_{4} = -56.6367214297364$$
$$x_{5} = 28.4483142882062$$
$$x_{6} = 22.2125562085988$$
$$x_{7} = 91.1609800814091$$
$$x_{8} = 59.7737149408088$$
$$x_{9} = 72.3256529801404$$
$$x_{10} = 84.8818390729824$$
$$x_{11} = 66.0490029831692$$
$$x_{12} = 9.89275256512429$$
$$x_{13} = -81.7425005689042$$
$$x_{14} = -88.0213377290448$$
$$x_{15} = 47.2293632804741$$
$$x_{16} = -62.9111635112525$$
$$x_{17} = -25.3276477920726$$
$$x_{18} = -12.9352212801115$$
$$x_{19} = -100.580635384527$$
$$x_{20} = 4.03356779033998$$
$$x_{21} = -31.5729854784403$$
$$x_{22} = -6.90959579542153$$
$$x_{23} = 103.720726651558$$
$$x_{24} = -94.30075185089$$
$$x_{25} = -44.0952059009573$$
$$x_{26} = -50.3644346278355$$
$$x_{27} = 53.5002619192185$$
$$x_{28} = 34.7006238709965$$
$$x_{29} = 78.6033412791698$$
$$x_{30} = 16.0106585966129$$
$$x_{31} = -37.8305186388889$$
$$x_{32} = -1.3138377164929$$
$$x_{33} = 40.9621674789216$$
$$x_{34} = -19.1055198623998$$
Puntos máximos de la función:
$$x_{34} = 62.9111635112525$$
$$x_{34} = 37.8305186388889$$
$$x_{34} = 81.7425005689042$$
$$x_{34} = -47.2293632804741$$
$$x_{34} = 50.3644346278355$$
$$x_{34} = 6.90959579542153$$
$$x_{34} = -97.4406405869711$$
$$x_{34} = -40.9621674789216$$
$$x_{34} = -66.0490029831692$$
$$x_{34} = -4.03356779033998$$
$$x_{34} = -53.5002619192185$$
$$x_{34} = -34.7006238709965$$
$$x_{34} = 31.5729854784403$$
$$x_{34} = 44.0952059009573$$
$$x_{34} = 1.3138377164929$$
$$x_{34} = 94.30075185089$$
$$x_{34} = 19.1055198623998$$
$$x_{34} = 69.1871806855712$$
$$x_{34} = 75.4643834089706$$
$$x_{34} = -28.4483142882062$$
$$x_{34} = 56.6367214297364$$
$$x_{34} = -16.0106585966129$$
$$x_{34} = -22.2125562085988$$
$$x_{34} = -72.3256529801404$$
$$x_{34} = -9.89275256512429$$
$$x_{34} = -59.7737149408088$$
$$x_{34} = 100.580635384527$$
$$x_{34} = -91.1609800814091$$
$$x_{34} = -84.8818390729824$$
$$x_{34} = 12.9352212801115$$
$$x_{34} = 88.0213377290448$$
$$x_{34} = -78.6033412791698$$
$$x_{34} = 25.3276477920726$$
Decrece en los intervalos
$$\left[103.720726651558, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100.580635384527\right]$$