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 derivada23xcos(3x)+2sin(3x)+1+x32=0Resolvermos esta ecuaciónRaíces de esta ecuación
x1=89.5341496175094x2=73.8319421320736x3=−29.8414070727572x4=7.8396736012599x5=67.5491767500035x6=−78.0147933275031x7=−3.75177195803661x8=−84.2980847924199x9=−16.2520658835412x10=86.3976561110523x11=57.0781065328832x12=35.0779499321882x13=−67.5425969944273x14=3.63183452315228x15=−38.2198035188413x16=−51.842708427932x17=−73.8259223119028x18=12.0335124171734x19=−23.5572290697416x20=−87.4448074455823x21=−69.6370415742658x22=−34.0306555064577x23=97.9118362155217x24=44.5133844234738x25=22.5098087774546x26=−80.1092256661263x27=−91.6334234062544x28=−53.9368539004158x29=−12.0703667254341x30=53.9286134447172x31=−41.3723600294113x32=51.8341350739919x33=−89.5391133882524x34=38.2314296566913x35=40.3253719836236x36=78.020489958096x37=−17.2723300965849x38=−49.7485839674963x39=−58.1251987998863x40=−47.6544832880973x41=80.1147733747798x42=75.9262127024484x43=−27.7467309956061x44=93.7229952896613x45=27.7627423539192x46=36.137539795388x47=42.4193590081437x48=16.2247054307724x49=92.6805798692254x50=−71.7314832647885x51=60.212013791028x52=29.856295359853x53=−5.81650479334164x54=−45.5604096674964x55=−25.6520090653707x56=−1.71221607813122x57=20.414906074639x58=5.7397730916088x59=−14.160695216171x60=82.2090624816772x61=−36.1252398830002x62=88.4919599039137x63=58.1175521765418x64=71.7376788324169x65=14.1292889745525x66=−43.4663670146734x67=−60.2193944752711x68=84.3033568547452x69=−7.89608208530915x70=−82.2036561072884x71=−93.7277372264291x72=66.4953734902289x73=31.9499587197976x74=−0.95989719783345x75=−21.4623740434466x76=100.006255084988x77=−75.9203589343812x78=−97.9163752974962x79=64.4009240388803x80=−56.0310180297196x81=48.7015306215084x82=−95.8220545994953x83=18.3198868045554x84=−31.9360463105638x85=−100.010699111567x86=34.0437120782042x87=56.0230855877076x88=−9.98172626669137x89=−65.448149247889x90=24.60462482455x91=95.8174163069689x92=62.3064709329742x93=9.93714128227479x94=49.7396496884211Signos de extremos en los puntos:
(89.53414961750944, 40.767260320607)
(73.83194213207365, 106.744343711783)
(-29.841407072757196, -18.9227572006474)
(7.839673601259899, -0.0928232644072904)
(67.54917675000348, 97.3198449939322)
(-78.0147933275031, -43.0079170269249)
(-3.7517719580366107, -9.63577788418861)
(-84.2980847924199, -46.1495126395795)
(-16.2520658835412, -28.3665165897607)
(86.39765611105234, 125.593456623441)
(57.07810653288318, 81.6124729402231)
(35.077949932188154, 13.538954386956)
(-67.54259699442734, -37.7719289641448)
(3.631834523152284, -2.25081162630326)
(-38.21980351884126, -23.111313092809)
(-51.842708427932045, -81.7596126747617)
(-73.82592231190284, -40.9135208958062)
(12.033512417173416, 2.01217160130921)
(-23.557229069741553, -15.7815952565633)
(-87.44480744558226, -135.164483041905)
(-69.63704157426584, -38.8191258973999)
(-34.03065550645767, -21.0170074370133)
(97.91183621552172, 44.956097505348)
(44.513384423473845, 62.7639557301852)
(22.509808777454612, 7.2541660356287)
(-80.10922566612635, -44.0551154081801)
(-91.63342340625437, -141.447525978401)
(-53.936853900415834, -84.9009897514051)
(-12.070366725434083, -22.0917451476389)
(53.92861344471723, 22.9644780134213)
(-41.37236002941131, -66.0530821082106)
(51.834135073991895, 21.9172312998043)
(-89.53911338825245, -138.306002776883)
(38.231429656691304, 53.3399212613474)
(40.32537198362361, 56.4812435153022)
(78.02048995809595, 113.027366397324)
(-17.272330096584913, -12.6411234786029)
(-49.74858396749628, -78.6182550138179)
(-58.12519879988633, -91.1837932897095)
(-47.65448328809729, -75.4769195021771)
(80.11477337477979, 116.168883760784)
(75.92621270244842, 109.885852943855)
(-27.746730995606125, -17.8756653242941)
(93.72299528966128, 42.8616801907127)
(27.762742353919162, 37.6338111038007)
(36.13753979538799, 50.1986259870911)
(42.419359008143665, 59.6225893035084)
(16.224705430772445, 4.11027009772762)
(92.68057986922537, 135.018055965606)
(-71.73148326478852, -39.8663232292551)
(60.212013791027964, 26.1061924378668)
(29.856295359853007, 40.774947707108)
(-5.816504793341635, -12.7120199168973)
(-45.56040966749643, -72.3356094094135)
(-25.652009065370727, -16.8286068152483)
(-1.712216078131216, -6.83352581855356)
(20.41490607463904, 6.20641601866254)
(5.739773091608804, -1.15539869871637)
(-14.160695216171039, -25.2283991322338)
(82.20906248167715, 119.310404752608)
(-36.12523988300024, -22.0641550869415)
(88.49195990391365, 128.734987059214)
(58.11755217654183, 25.0589580220582)
(71.73767883241692, 103.602839046315)
(14.129288974552527, 3.06160831731661)
(-43.46636701467342, -69.1943286785897)
(-60.219394475271066, -94.3252161299858)
(84.30335685474525, 122.451929117488)
(-7.896082085309151, -15.8287143054271)
(-82.20365610728835, -45.1023139548754)
(-93.72773722642908, -144.589052405349)
(66.49537349022893, 29.2478783489101)
(31.949958719797575, 43.9161337183775)
(-0.9598971978334502, -5.9209320980479)
(-21.462374043446612, -14.7346515424909)
(100.00625508498813, 46.0033053214858)
(-75.92035893438117, -41.9607188434179)
(-97.91637529749619, -150.872114077811)
(64.40092403888028, 28.2006522622469)
(-56.031018029719625, -88.0423839408958)
(48.70153062150844, 69.0467410714497)
(-95.82205459949529, -147.730581838443)
(18.319886804555388, 5.1584826928003)
(-31.936046310563775, -19.9698733316274)
(-100.01069911156709, -154.013648940895)
(34.0437120782042, 47.0573618170311)
(56.023085587707556, 24.0117200508985)
(-9.981726266691371, -18.9576683040401)
(-65.44814924788905, -36.7247325059037)
(24.604624824550005, 8.30179040932915)
(95.81741630696887, 43.9088891428627)
(62.30647093297416, 27.1534237293258)
(9.937141282274794, 0.961265904515388)
(49.73964968842108, 20.8699791769845)
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=89.5341496175094x2=7.8396736012599x3=−3.75177195803661x4=−16.2520658835412x5=35.0779499321882x6=3.63183452315228x7=−51.842708427932x8=12.0335124171734x9=−87.4448074455823x10=97.9118362155217x11=22.5098087774546x12=−91.6334234062544x13=−53.9368539004158x14=−12.0703667254341x15=53.9286134447172x16=−41.3723600294113x17=51.8341350739919x18=−89.5391133882524x19=−49.7485839674963x20=−58.1251987998863x21=−47.6544832880973x22=93.7229952896613x23=16.2247054307724x24=60.212013791028x25=−5.81650479334164x26=−45.5604096674964x27=−1.71221607813122x28=20.414906074639x29=5.7397730916088x30=−14.160695216171x31=58.1175521765418x32=14.1292889745525x33=−43.4663670146734x34=−60.2193944752711x35=−7.89608208530915x36=−93.7277372264291x37=66.4953734902289x38=100.006255084988x39=−97.9163752974962x40=64.4009240388803x41=−56.0310180297196x42=−95.8220545994953x43=18.3198868045554x44=−100.010699111567x45=56.0230855877076x46=−9.98172626669137x47=24.60462482455x48=95.8174163069689x49=62.3064709329742x50=9.93714128227479x51=49.7396496884211Puntos máximos de la función:
x51=73.8319421320736x51=−29.8414070727572x51=67.5491767500035x51=−78.0147933275031x51=−84.2980847924199x51=86.3976561110523x51=57.0781065328832x51=−67.5425969944273x51=−38.2198035188413x51=−73.8259223119028x51=−23.5572290697416x51=−69.6370415742658x51=−34.0306555064577x51=44.5133844234738x51=−80.1092256661263x51=38.2314296566913x51=40.3253719836236x51=78.020489958096x51=−17.2723300965849x51=80.1147733747798x51=75.9262127024484x51=−27.7467309956061x51=27.7627423539192x51=36.137539795388x51=42.4193590081437x51=92.6805798692254x51=−71.7314832647885x51=29.856295359853x51=−25.6520090653707x51=82.2090624816772x51=−36.1252398830002x51=88.4919599039137x51=71.7376788324169x51=84.3033568547452x51=−82.2036561072884x51=31.9499587197976x51=−0.95989719783345x51=−21.4623740434466x51=−75.9203589343812x51=48.7015306215084x51=−31.9360463105638x51=34.0437120782042x51=−65.448149247889Decrece en los intervalos
[100.006255084988,∞)Crece en los intervalos
(−∞,−100.010699111567]