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 derivada32x1cos(3x)−2x2sin(3x)=0Resolvermos esta ecuaciónRaíces de esta ecuación
x1=88.4869374039331x2=−65.448149267704x3=−6.79043431976252x4=170.168949124421x5=−49.7396498613733x6=−16.2247147439848x7=−29.8414069768057x8=49.7396498613733x9=66.4953735549544x10=51.8341352242202x11=56.0230857030463x12=36.1252398838916x13=−50.7868934733971x14=−23.5572285705398x15=−1.49780315263635x16=75.9203589492161x17=80.1092256793491x18=34.0306554883025x19=−100.006255101775x20=62.3064710135101x21=26.6993762096484x22=−53.9286135759886x23=−82.2036561197804x24=−71.7314832814509x25=−80.1092256793491x26=−62.3064710135101x27=−56.0230857030463x28=82.2036561197804x29=−84.2980848042281x30=93.722995310418x31=−34.0306554883025x32=86.3925118604052x33=5.74025175731026x34=−9.93719959696432x35=−7.839817499563x36=58.1175522783976x37=−75.9203589492161x38=−67.5425970131389x39=91.6285731099136x40=−27.7467308235745x41=73.8259223276238x42=27.7467308235745x43=−51.8341352242202x44=−3.63470721980963x45=60.212013881401x46=29.8414069768057x47=−73.8259223276238x48=−45.5506542337597x49=95.8174163262761x50=16.2247147439848x51=38.2198035316744x52=45.5506542337597x53=−38.2198035316744x54=2.5750839456459x55=−25.6520087701104x56=−93.722995310418x57=12.0335407481252x58=−14.1293045227106x59=−5.74025175731026x60=−43.4561415684628x61=7.839817499563x62=−69.6370415919254x63=214.151380376346x64=−91.6285731099136x65=71.7314832814509x66=100.006255101775x67=−89.534149641627x68=−58.1175522783976x69=53.9286135759886x70=−78.014793341506x71=84.2980848042281x72=40.3143496657172x73=37.1725240820437x74=−153.413716993446x75=−60.212013881401x76=−21.4623731968525x77=78.014793341506x78=14.1293045227106x79=−36.1252398838916x80=20.4149100867915x81=−12.0335407481252x82=−31.9360462622872x83=18.3198927626296x84=31.9360462622872x85=−95.8174163262761x86=97.9118362335106x87=44.5033992843595x88=−47.6451565627964x89=42.4088808811114x90=−351.334462172035x91=−97.9118362335106x92=69.6370415919254x93=9.93719959696432x94=64.4009241109425x95=22.5098115923814Signos de extremos en los puntos:
(88.4869374039331, 0.00565051144349552)
(-65.448149267704, 0.00763953634790889)
(-6.790434319762521, 0.0735444360211112)
(170.1689491244206, 0.00293825074030404)
(-49.73964986137327, -0.0100521168532409)
(-16.224714743984794, -0.0308106810626306)
(-29.84140697680573, 0.0167541969512603)
(49.73964986137327, -0.0100521168532409)
(66.49537355495444, -0.00751922564151797)
(51.83413522422022, -0.00964595356826895)
(56.02308570304626, -0.00892473421535613)
(36.12523988389156, 0.0138401493761729)
(-50.786893473397086, 0.00984484768974962)
(-23.557228570539834, 0.0212227830972996)
(-1.4978031526363547, -0.325850442316833)
(75.9203589492161, 0.00658578525869541)
(80.10922567934914, 0.00624142434739368)
(34.03065548830255, 0.0146919302201117)
(-100.00625510177518, -0.00499965949213873)
(62.306471013510084, -0.00802473381440714)
(26.69937620964837, -0.0187255699826685)
(-53.92861357598856, -0.00927133882917417)
(-82.20365611978043, 0.00608240451771827)
(-71.73148328145086, 0.00697036473602969)
(-80.10922567934914, 0.00624142434739368)
(-62.306471013510084, -0.00802473381440714)
(-56.02308570304626, -0.00892473421535613)
(82.20365611978043, 0.00608240451771827)
(-84.29808480422805, 0.00593128648461355)
(93.72299531041797, -0.00533483630200461)
(-34.03065548830255, 0.0146919302201117)
(86.39251186040515, 0.00578749555419506)
(5.740251757310256, -0.0869577035192308)
(-9.93719959696432, -0.0502877025320981)
(-7.839817499563003, -0.063719425466419)
(58.11755227839756, -0.00860311139416558)
(-75.9203589492161, 0.00658578525869541)
(-67.54259701313894, 0.00740264563812724)
(91.62857310991362, -0.00545677701319961)
(-27.746730823574467, 0.0180188407230791)
(73.82592232762377, 0.00677261980241303)
(27.746730823574467, 0.0180188407230791)
(-51.83413522422022, -0.00964595356826895)
(-3.6347072198096333, -0.136987804234587)
(60.212013881400964, -0.00830386340098503)
(29.84140697680573, 0.0167541969512603)
(-73.82592232762377, 0.00677261980241303)
(-45.55065423375966, -0.010976496851203)
(95.81741632627612, -0.00521822643124976)
(16.224714743984794, -0.0308106810626306)
(38.21980353167436, 0.0130817256715564)
(45.55065423375966, -0.010976496851203)
(-38.21980353167436, 0.0130817256715564)
(2.5750839456459023, 0.192561830288849)
(-25.652008770110395, 0.0194900054805641)
(-93.72299531041797, -0.00533483630200461)
(12.033540748125203, -0.0415345984517238)
(-14.12930452271064, -0.0353776023435246)
(-5.740251757310256, -0.0869577035192308)
(-43.456141568462805, -0.0115055150594557)
(7.839817499563003, -0.063719425466419)
(-69.63704159192544, 0.00718000449883474)
(214.15138037634594, 0.00233479416954943)
(-91.62857310991362, -0.00545677701319961)
(71.73148328145086, 0.00697036473602969)
(100.00625510177518, -0.00499965949213873)
(-89.53414964162702, -0.00558442266892595)
(-58.11755227839756, -0.00860311139416558)
(53.92861357598856, -0.00927133882917417)
(-78.01479334150599, 0.00640898238229569)
(84.29808480422805, 0.00593128648461355)
(40.31434966571717, 0.0124021077755364)
(37.17252408204367, -0.0134502542107399)
(-153.41371699344606, 0.00325915328541868)
(-60.212013881400964, -0.00830386340098503)
(-21.46237319685247, 0.023293775711192)
(78.01479334150599, 0.00640898238229569)
(14.12930452271064, -0.0353776023435246)
(-36.12523988389156, 0.0138401493761729)
(20.414910086791465, -0.0244886389814967)
(-12.033540748125203, -0.0415345984517238)
(-31.93604626228723, 0.0156554372018487)
(18.319892762629646, -0.0272882194827047)
(31.93604626228723, 0.0156554372018487)
(-95.81741632627612, -0.00521822643124976)
(97.91183623351056, -0.00510660530671998)
(44.50339928435946, 0.0112347816877682)
(-47.64515656279642, -0.010493989314784)
(42.40888088111144, 0.0117896191899248)
(-351.3344621720351, -0.00142314469201501)
(-97.91183623351056, -0.00510660530671998)
(69.63704159192544, 0.00718000449883474)
(9.93719959696432, -0.0502877025320981)
(64.40092411094253, -0.00776375975233391)
(22.50981159238137, -0.0222101009198238)
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=−49.7396498613733x2=−16.2247147439848x3=49.7396498613733x4=66.4953735549544x5=51.8341352242202x6=56.0230857030463x7=−1.49780315263635x8=−100.006255101775x9=62.3064710135101x10=26.6993762096484x11=−53.9286135759886x12=−62.3064710135101x13=−56.0230857030463x14=93.722995310418x15=5.74025175731026x16=−9.93719959696432x17=−7.839817499563x18=58.1175522783976x19=91.6285731099136x20=−51.8341352242202x21=−3.63470721980963x22=60.212013881401x23=−45.5506542337597x24=95.8174163262761x25=16.2247147439848x26=45.5506542337597x27=−93.722995310418x28=12.0335407481252x29=−14.1293045227106x30=−5.74025175731026x31=−43.4561415684628x32=7.839817499563x33=−91.6285731099136x34=100.006255101775x35=−89.534149641627x36=−58.1175522783976x37=53.9286135759886x38=37.1725240820437x39=−60.212013881401x40=14.1293045227106x41=20.4149100867915x42=−12.0335407481252x43=18.3198927626296x44=−95.8174163262761x45=97.9118362335106x46=−47.6451565627964x47=−351.334462172035x48=−97.9118362335106x49=9.93719959696432x50=64.4009241109425x51=22.5098115923814Puntos máximos de la función:
x51=88.4869374039331x51=−65.448149267704x51=−6.79043431976252x51=170.168949124421x51=−29.8414069768057x51=36.1252398838916x51=−50.7868934733971x51=−23.5572285705398x51=75.9203589492161x51=80.1092256793491x51=34.0306554883025x51=−82.2036561197804x51=−71.7314832814509x51=−80.1092256793491x51=82.2036561197804x51=−84.2980848042281x51=−34.0306554883025x51=86.3925118604052x51=−75.9203589492161x51=−67.5425970131389x51=−27.7467308235745x51=73.8259223276238x51=27.7467308235745x51=29.8414069768057x51=−73.8259223276238x51=38.2198035316744x51=−38.2198035316744x51=2.5750839456459x51=−25.6520087701104x51=−69.6370415919254x51=214.151380376346x51=71.7314832814509x51=−78.014793341506x51=84.2980848042281x51=40.3143496657172x51=−153.413716993446x51=−21.4623731968525x51=78.014793341506x51=−36.1252398838916x51=−31.9360462622872x51=31.9360462622872x51=44.5033992843595x51=42.4088808811114x51=69.6370415919254Decrece en los intervalos
[100.006255101775,∞)Crece en los intervalos
(−∞,−351.334462172035]