Para hallar los extremos hay que resolver la ecuación
dxdf(x)=0(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
dxdf(x)=primera derivada23x1cos(2x)−3x2sin(2x)=0Resolvermos esta ecuaciónRaíces de esta ecuación
x1=74.6094747920599x2=96.6013861664138x3=68.3259813506395x4=−85.6054794697228x5=−57.3297052975115x6=−69.8968599047927x7=204.987701063789x8=13.3330271294063x9=−38.4780131551656x10=30.6223651301872x11=−19.6222161805821x12=32.1935597952787x13=38.4780131551656x14=82.4637755597094x15=−2.24670472895453x16=−55.7587861230655x17=−33.7647173885721x18=−91.8888644664832x19=18.0503111221878x20=−41.6200962353617x21=99.7430603324317x22=−40.0490643144726x23=−93.4597065202651x24=98.172223901556x25=2.24670472895453x26=−76.1803402100956x27=22.7655670069956x28=−47.9040693934309x29=3.86262591846885x30=46.3330961388114x31=16.4781945199112x32=88.7471755026564x33=−16.4781945199112x34=−24.3370721159772x35=−63.6133213216672x36=−11.7597262493445x37=−68.3259813506395x38=−630.67432880713x39=−5.45206082971445x40=−58.9006179191122x41=84.0346285545694x42=−82.4637755597094x43=−49.4750314121659x44=40.0490643144726x45=25.9084912436398x46=91.8888644664832x47=−46.3330961388114x48=90.3180208221014x49=19.6222161805821x50=−3.86262591846885x51=−60.4715244985757x52=−99.7430603324317x53=−71.4677348441946x54=−54.1878598258373x55=69.8968599047927x56=−90.3180208221014x57=11.7597262493445x58=62.0424254948814x59=10.1856514796438x60=76.1803402100956x61=−25.9084912436398x62=−84.0346285545694x63=85.6054794697228x64=−77.7512028363303x65=60.4715244985757x66=−98.172223901556x67=33.7647173885721x68=−62.0424254948814x69=−32.1935597952787x70=−18.0503111221878x71=51.0459832324538x72=8.61037763596538x73=44.7621104652086x74=63.6133213216672x75=77.7512028363303x76=−79.3220628366317x77=−10.1856514796438x78=55.7587861230655x79=54.1878598258373x80=66.7550989265392x81=47.9040693934309x82=−35.3358428558098x83=24.3370721159772x84=−27.4798391439445x85=41.6200962353617x86=−13.3330271294063x87=52.6169257678188Signos de extremos en los puntos:
(74.60947479205991, -0.00446760749032927)
(96.60138616641379, -0.00345055988992618)
(68.3259813506395, -0.00487844304497913)
(-85.60547946972281, 0.00389376532695642)
(-57.32970529751154, 0.00581410029846953)
(-69.8968599047927, 0.00476880943721178)
(204.98770106378876, 0.00162610898124919)
(13.333027129406338, 0.0249830133292875)
(-38.47801315516559, 0.00866222465802848)
(30.6223651301872, -0.0108838395473319)
(-19.622216180582097, 0.0169820353952538)
(32.19355979527871, 0.0103527892049742)
(38.47801315516559, 0.00866222465802848)
(82.46377555970939, 0.0040421045972735)
(-2.246704728954532, -0.144822418807481)
(-55.758786123065505, -0.00597789076032886)
(-33.76471738857206, -0.00987115596436615)
(-91.88886446648316, 0.00362751679055862)
(18.050311122187804, -0.0184598215340995)
(-41.6200962353617, 0.00800837365470182)
(99.74306033243167, -0.00334187806289688)
(-40.04906431447256, -0.00832247554785266)
(-93.45970652026512, -0.00356654836195873)
(98.172223901556, 0.00339534948795951)
(2.246704728954532, -0.144822418807481)
(-76.18034021009562, 0.00437548786211094)
(22.76556700699564, 0.014638465485655)
(-47.90406939343085, 0.00695797208971055)
(3.8626259184688534, 0.0855830356839328)
(46.33309613881142, -0.00719386256635616)
(16.478194519911238, 0.0202194474575402)
(88.7471755026564, 0.00375592847072495)
(-16.478194519911238, 0.0202194474575402)
(-24.337072115977193, -0.0136936360278358)
(-63.613321321667165, 0.00523983075107767)
(-11.759726249344503, -0.0283197446517418)
(-68.3259813506395, -0.00487844304497913)
(-630.6743288071303, -0.00052853463880156)
(-5.4520608297144495, -0.0608834685487051)
(-58.90061791911219, -0.00565904629851015)
(84.0346285545694, -0.00396654854015828)
(-82.46377555970939, 0.0040421045972735)
(-49.47503141216594, -0.00673706115766935)
(40.04906431447256, -0.00832247554785266)
(25.908491243639833, 0.012863399658392)
(91.88886446648316, 0.00362751679055862)
(-46.33309613881142, -0.00719386256635616)
(90.31802082210145, -0.00369060595599335)
(19.622216180582097, 0.0169820353952538)
(-3.8626259184688534, 0.0855830356839328)
(-60.47152449857575, 0.00551204790023838)
(-99.74306033243167, -0.00334187806289688)
(-71.46773484419464, -0.0046639952510151)
(-54.18785982583734, 0.00615117750052131)
(69.8968599047927, 0.00476880943721178)
(-90.31802082210145, -0.00369060595599335)
(11.759726249344503, -0.0283197446517418)
(62.04242549488138, -0.00537249320312092)
(10.18565147964378, 0.0326864160093828)
(76.18034021009562, 0.00437548786211094)
(-25.908491243639833, 0.012863399658392)
(-84.0346285545694, -0.00396654854015828)
(85.60547946972281, 0.00389376532695642)
(-77.75120283633034, -0.0042870904747862)
(60.47152449857575, 0.00551204790023838)
(-98.172223901556, 0.00339534948795951)
(33.76471738857206, -0.00987115596436615)
(-62.04242549488138, -0.00537249320312092)
(-32.19355979527871, 0.0103527892049742)
(-18.050311122187804, -0.0184598215340995)
(51.04598323245382, 0.00652974676449409)
(8.610377635965385, -0.0386478682307693)
(44.76211046520859, 0.0074463097561157)
(63.613321321667165, 0.00523983075107767)
(77.75120283633034, -0.0042870904747862)
(-79.32206283663172, 0.00420219418701412)
(-10.18565147964378, 0.0326864160093828)
(55.758786123065505, -0.00597789076032886)
(54.18785982583734, 0.00615117750052131)
(66.75509892653919, 0.00499323630567473)
(47.90406939343085, 0.00695797208971055)
(-35.33584285580975, 0.00943234804324425)
(24.337072115977193, -0.0136936360278358)
(-27.479839143944467, -0.0121280975478688)
(41.6200962353617, 0.00800837365470182)
(-13.333027129406338, 0.0249830133292875)
(52.6169257678188, -0.00633481107918903)
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:
x1=74.6094747920599x2=96.6013861664138x3=68.3259813506395x4=30.6223651301872x5=−2.24670472895453x6=−55.7587861230655x7=−33.7647173885721x8=18.0503111221878x9=99.7430603324317x10=−40.0490643144726x11=−93.4597065202651x12=2.24670472895453x13=46.3330961388114x14=−24.3370721159772x15=−11.7597262493445x16=−68.3259813506395x17=−630.67432880713x18=−5.45206082971445x19=−58.9006179191122x20=84.0346285545694x21=−49.4750314121659x22=40.0490643144726x23=−46.3330961388114x24=90.3180208221014x25=−99.7430603324317x26=−71.4677348441946x27=−90.3180208221014x28=11.7597262493445x29=62.0424254948814x30=−84.0346285545694x31=−77.7512028363303x32=33.7647173885721x33=−62.0424254948814x34=−18.0503111221878x35=8.61037763596538x36=77.7512028363303x37=55.7587861230655x38=24.3370721159772x39=−27.4798391439445x40=52.6169257678188Puntos máximos de la función:
x40=−85.6054794697228x40=−57.3297052975115x40=−69.8968599047927x40=204.987701063789x40=13.3330271294063x40=−38.4780131551656x40=−19.6222161805821x40=32.1935597952787x40=38.4780131551656x40=82.4637755597094x40=−91.8888644664832x40=−41.6200962353617x40=98.172223901556x40=−76.1803402100956x40=22.7655670069956x40=−47.9040693934309x40=3.86262591846885x40=16.4781945199112x40=88.7471755026564x40=−16.4781945199112x40=−63.6133213216672x40=−82.4637755597094x40=25.9084912436398x40=91.8888644664832x40=19.6222161805821x40=−3.86262591846885x40=−60.4715244985757x40=−54.1878598258373x40=69.8968599047927x40=10.1856514796438x40=76.1803402100956x40=−25.9084912436398x40=85.6054794697228x40=60.4715244985757x40=−98.172223901556x40=−32.1935597952787x40=51.0459832324538x40=44.7621104652086x40=63.6133213216672x40=−79.3220628366317x40=−10.1856514796438x40=54.1878598258373x40=66.7550989265392x40=47.9040693934309x40=−35.3358428558098x40=41.6200962353617x40=−13.3330271294063Decrece en los intervalos
[99.7430603324317,∞)Crece en los intervalos
(−∞,−630.67432880713]