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 derivadax2cos(2x)−x2sin(2x)=0Resolvermos esta ecuaciónRaíces de esta ecuación
x1=2.24670472895453x2=−10.1856514796438x3=−24.3370721159772x4=98.172223901556x5=−32.1935597952787x6=69.8968599047927x7=−46.3330961388114x8=−98.172223901556x9=−77.7512028363303x10=24.3370721159772x11=−19.6222161805821x12=41.6200962353617x13=40.0490643144726x14=−3.86262591846885x15=46.3330961388114x16=−16.4781945199112x17=11.7597262493445x18=85.6054794697228x19=54.1878598258373x20=66.7550989265392x21=51.0459832324538x22=−60.4715244985757x23=84.0346285545694x24=91.8888644664832x25=−63.6133213216672x26=52.6169257678188x27=−54.1878598258373x28=63.6133213216672x29=99.7430603324317x30=−76.1803402100956x31=−35.3358428558098x32=30.6223651301872x33=−82.4637755597094x34=13.3330271294063x35=18.0503111221878x36=−47.9040693934309x37=−68.3259813506395x38=32.1935597952787x39=−99.7430603324317x40=−41.6200962353617x41=38.4780131551656x42=76.1803402100956x43=8.61037763596538x44=−5.45206082971445x45=55.7587861230655x46=−27.4798391439445x47=−85.6054794697228x48=−58.9006179191122x49=−55.7587861230655x50=22.7655670069956x51=−84.0346285545694x52=−38.4780131551656x53=47.9040693934309x54=74.6094747920599x55=204.987701063789x56=82.4637755597094x57=−71.4677348441946x58=−2.24670472895453x59=33.7647173885721x60=−49.4750314121659x61=−62.0424254948814x62=16.4781945199112x63=−18.0503111221878x64=77.7512028363303x65=−91.8888644664832x66=−33.7647173885721x67=−40.0490643144726x68=−69.8968599047927x69=96.6013861664138x70=60.4715244985757x71=88.7471755026564x72=10.1856514796438x73=68.3259813506395x74=44.7621104652086x75=−13.3330271294063x76=−79.3220628366317x77=−90.3180208221014x78=−57.3297052975115x79=−25.9084912436398x80=−11.7597262493445x81=19.6222161805821x82=25.9084912436398x83=−93.4597065202651x84=3.86262591846885x85=62.0424254948814x86=90.3180208221014Signos de extremos en los puntos:
(2.246704728954532, -0.434467256422443)
(-10.18565147964378, 0.0980592480281483)
(-24.337072115977193, -0.0410809080835075)
(98.172223901556, 0.0101860484638785)
(-32.19355979527871, 0.0310583676149227)
(69.8968599047927, 0.0143064283116353)
(-46.33309613881142, -0.0215815876990685)
(-98.172223901556, 0.0101860484638785)
(-77.75120283633034, -0.0128612714243586)
(24.337072115977193, -0.0410809080835075)
(-19.622216180582097, 0.0509461061857615)
(41.6200962353617, 0.0240251209641055)
(40.04906431447256, -0.024967426643558)
(-3.8626259184688534, 0.256749107051798)
(46.33309613881142, -0.0215815876990685)
(-16.478194519911238, 0.0606583423726206)
(11.759726249344503, -0.0849592339552253)
(85.60547946972281, 0.0116812959808693)
(54.18785982583734, 0.0184535325015639)
(66.75509892653919, 0.0149797089170242)
(51.04598323245382, 0.0195892402934823)
(-60.47152449857575, 0.0165361437007152)
(84.0346285545694, -0.0118996456204748)
(91.88886446648316, 0.0108825503716759)
(-63.613321321667165, 0.015719492253233)
(52.6169257678188, -0.0190044332375671)
(-54.18785982583734, 0.0184535325015639)
(63.613321321667165, 0.015719492253233)
(99.74306033243167, -0.0100256341886906)
(-76.18034021009562, 0.0131264635863328)
(-35.33584285580975, 0.0282970441297328)
(30.6223651301872, -0.0326515186419956)
(-82.46377555970939, 0.0121263137918205)
(13.333027129406338, 0.0749490399878624)
(18.050311122187804, -0.0553794646022984)
(-47.90406939343085, 0.0208739162691316)
(-68.3259813506395, -0.0146353291349374)
(32.19355979527871, 0.0310583676149227)
(-99.74306033243167, -0.0100256341886906)
(-41.6200962353617, 0.0240251209641055)
(38.47801315516559, 0.0259866739740854)
(76.18034021009562, 0.0131264635863328)
(8.610377635965385, -0.115943604692308)
(-5.4520608297144495, -0.182650405646115)
(55.758786123065505, -0.0179336722809866)
(-27.479839143944467, -0.0363842926436063)
(-85.60547946972281, 0.0116812959808693)
(-58.90061791911219, -0.0169771388955304)
(-55.758786123065505, -0.0179336722809866)
(22.76556700699564, 0.0439153964569649)
(-84.0346285545694, -0.0118996456204748)
(-38.47801315516559, 0.0259866739740854)
(47.90406939343085, 0.0208739162691316)
(74.60947479205991, -0.0134028224709878)
(204.98770106378876, 0.00487832694374757)
(82.46377555970939, 0.0121263137918205)
(-71.46773484419464, -0.0139919857530453)
(-2.246704728954532, -0.434467256422443)
(33.76471738857206, -0.0296134678930985)
(-49.47503141216594, -0.0202111834730081)
(-62.04242549488138, -0.0161174796093628)
(16.478194519911238, 0.0606583423726206)
(-18.050311122187804, -0.0553794646022984)
(77.75120283633034, -0.0128612714243586)
(-91.88886446648316, 0.0108825503716759)
(-33.76471738857206, -0.0296134678930985)
(-40.04906431447256, -0.024967426643558)
(-69.8968599047927, 0.0143064283116353)
(96.60138616641379, -0.0103516796697785)
(60.47152449857575, 0.0165361437007152)
(88.7471755026564, 0.0112677854121748)
(10.18565147964378, 0.0980592480281483)
(68.3259813506395, -0.0146353291349374)
(44.76211046520859, 0.0223389292683471)
(-13.333027129406338, 0.0749490399878624)
(-79.32206283663172, 0.0126065825610424)
(-90.31802082210145, -0.01107181786798)
(-57.32970529751154, 0.0174423008954086)
(-25.908491243639833, 0.0385901989751759)
(-11.759726249344503, -0.0849592339552253)
(19.622216180582097, 0.0509461061857615)
(25.908491243639833, 0.0385901989751759)
(-93.45970652026512, -0.0106996450858762)
(3.8626259184688534, 0.256749107051798)
(62.04242549488138, -0.0161174796093628)
(90.31802082210145, -0.01107181786798)
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=2.24670472895453x2=−24.3370721159772x3=−46.3330961388114x4=−77.7512028363303x5=24.3370721159772x6=40.0490643144726x7=46.3330961388114x8=11.7597262493445x9=84.0346285545694x10=52.6169257678188x11=99.7430603324317x12=30.6223651301872x13=18.0503111221878x14=−68.3259813506395x15=−99.7430603324317x16=8.61037763596538x17=−5.45206082971445x18=55.7587861230655x19=−27.4798391439445x20=−58.9006179191122x21=−55.7587861230655x22=−84.0346285545694x23=74.6094747920599x24=−71.4677348441946x25=−2.24670472895453x26=33.7647173885721x27=−49.4750314121659x28=−62.0424254948814x29=−18.0503111221878x30=77.7512028363303x31=−33.7647173885721x32=−40.0490643144726x33=96.6013861664138x34=68.3259813506395x35=−90.3180208221014x36=−11.7597262493445x37=−93.4597065202651x38=62.0424254948814x39=90.3180208221014Puntos máximos de la función:
x39=−10.1856514796438x39=98.172223901556x39=−32.1935597952787x39=69.8968599047927x39=−98.172223901556x39=−19.6222161805821x39=41.6200962353617x39=−3.86262591846885x39=−16.4781945199112x39=85.6054794697228x39=54.1878598258373x39=66.7550989265392x39=51.0459832324538x39=−60.4715244985757x39=91.8888644664832x39=−63.6133213216672x39=−54.1878598258373x39=63.6133213216672x39=−76.1803402100956x39=−35.3358428558098x39=−82.4637755597094x39=13.3330271294063x39=−47.9040693934309x39=32.1935597952787x39=−41.6200962353617x39=38.4780131551656x39=76.1803402100956x39=−85.6054794697228x39=22.7655670069956x39=−38.4780131551656x39=47.9040693934309x39=204.987701063789x39=82.4637755597094x39=16.4781945199112x39=−91.8888644664832x39=−69.8968599047927x39=60.4715244985757x39=88.7471755026564x39=10.1856514796438x39=44.7621104652086x39=−13.3330271294063x39=−79.3220628366317x39=−57.3297052975115x39=−25.9084912436398x39=19.6222161805821x39=25.9084912436398x39=3.86262591846885Decrece en los intervalos
[99.7430603324317,∞)Crece en los intervalos
(−∞,−99.7430603324317]