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Trazado el gráfico de la función/ (sin3x)/2x+x-1/x^2-4

Gráfico de la función y = (sin3x)/2x+x-1/x^2-4

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       sin(3*x)         1     
f(x) = --------*x + x - -- - 4
          2              2    
                        x
f(x)=((xsin⁡(3x)2+x)−1x2)−4f{\left(x \right)} = \left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4 
f = x*(sin(3*x)/2) + x - 1/x^2 - 4
Gráfico de la función
02468-8-6-4-2-1010-500500
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
((xsin⁡(3x)2+x)−1x2)−4=0\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4 = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=5.41431902897203x_{1} = 5.41431902897203
x2=7.76635597363129x_{2} = 7.76635597363129
x3=7.91207856548297x_{3} = 7.91207856548297
x4=5.41431902897203x_{4} = 5.41431902897203
x5=6.04004815600269x_{5} = 6.04004815600269
x6=4.17068707471415x_{6} = 4.17068707471415
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en (sin(3*x)/2)*x + x - 1/x^2 - 4.
−4+(0sin⁡(0⋅3)2−102)-4 + \left(0 \frac{\sin{\left(0 \cdot 3 \right)}}{2} - \frac{1}{0^{2}}\right)
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
3xcos⁡(3x)2+sin⁡(3x)2+1+2x3=0\frac{3 x \cos{\left(3 x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{2} + 1 + \frac{2}{x^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=89.5341496175094x_{1} = 89.5341496175094
x2=73.8319421320736x_{2} = 73.8319421320736
x3=−29.8414070727572x_{3} = -29.8414070727572
x4=7.8396736012599x_{4} = 7.8396736012599
x5=67.5491767500035x_{5} = 67.5491767500035
x6=−78.0147933275031x_{6} = -78.0147933275031
x7=−3.75177195803661x_{7} = -3.75177195803661
x8=−84.2980847924199x_{8} = -84.2980847924199
x9=−16.2520658835412x_{9} = -16.2520658835412
x10=86.3976561110523x_{10} = 86.3976561110523
x11=57.0781065328832x_{11} = 57.0781065328832
x12=35.0779499321882x_{12} = 35.0779499321882
x13=−67.5425969944273x_{13} = -67.5425969944273
x14=3.63183452315228x_{14} = 3.63183452315228
x15=−38.2198035188413x_{15} = -38.2198035188413
x16=−51.842708427932x_{16} = -51.842708427932
x17=−73.8259223119028x_{17} = -73.8259223119028
x18=12.0335124171734x_{18} = 12.0335124171734
x19=−23.5572290697416x_{19} = -23.5572290697416
x20=−87.4448074455823x_{20} = -87.4448074455823
x21=−69.6370415742658x_{21} = -69.6370415742658
x22=−34.0306555064577x_{22} = -34.0306555064577
x23=97.9118362155217x_{23} = 97.9118362155217
x24=44.5133844234738x_{24} = 44.5133844234738
x25=22.5098087774546x_{25} = 22.5098087774546
x26=−80.1092256661263x_{26} = -80.1092256661263
x27=−91.6334234062544x_{27} = -91.6334234062544
x28=−53.9368539004158x_{28} = -53.9368539004158
x29=−12.0703667254341x_{29} = -12.0703667254341
x30=53.9286134447172x_{30} = 53.9286134447172
x31=−41.3723600294113x_{31} = -41.3723600294113
x32=51.8341350739919x_{32} = 51.8341350739919
x33=−89.5391133882524x_{33} = -89.5391133882524
x34=38.2314296566913x_{34} = 38.2314296566913
x35=40.3253719836236x_{35} = 40.3253719836236
x36=78.020489958096x_{36} = 78.020489958096
x37=−17.2723300965849x_{37} = -17.2723300965849
x38=−49.7485839674963x_{38} = -49.7485839674963
x39=−58.1251987998863x_{39} = -58.1251987998863
x40=−47.6544832880973x_{40} = -47.6544832880973
x41=80.1147733747798x_{41} = 80.1147733747798
x42=75.9262127024484x_{42} = 75.9262127024484
x43=−27.7467309956061x_{43} = -27.7467309956061
x44=93.7229952896613x_{44} = 93.7229952896613
x45=27.7627423539192x_{45} = 27.7627423539192
x46=36.137539795388x_{46} = 36.137539795388
x47=42.4193590081437x_{47} = 42.4193590081437
x48=16.2247054307724x_{48} = 16.2247054307724
x49=92.6805798692254x_{49} = 92.6805798692254
x50=−71.7314832647885x_{50} = -71.7314832647885
x51=60.212013791028x_{51} = 60.212013791028
x52=29.856295359853x_{52} = 29.856295359853
x53=−5.81650479334164x_{53} = -5.81650479334164
x54=−45.5604096674964x_{54} = -45.5604096674964
x55=−25.6520090653707x_{55} = -25.6520090653707
x56=−1.71221607813122x_{56} = -1.71221607813122
x57=20.414906074639x_{57} = 20.414906074639
x58=5.7397730916088x_{58} = 5.7397730916088
x59=−14.160695216171x_{59} = -14.160695216171
x60=82.2090624816772x_{60} = 82.2090624816772
x61=−36.1252398830002x_{61} = -36.1252398830002
x62=88.4919599039137x_{62} = 88.4919599039137
x63=58.1175521765418x_{63} = 58.1175521765418
x64=71.7376788324169x_{64} = 71.7376788324169
x65=14.1292889745525x_{65} = 14.1292889745525
x66=−43.4663670146734x_{66} = -43.4663670146734
x67=−60.2193944752711x_{67} = -60.2193944752711
x68=84.3033568547452x_{68} = 84.3033568547452
x69=−7.89608208530915x_{69} = -7.89608208530915
x70=−82.2036561072884x_{70} = -82.2036561072884
x71=−93.7277372264291x_{71} = -93.7277372264291
x72=66.4953734902289x_{72} = 66.4953734902289
x73=31.9499587197976x_{73} = 31.9499587197976
x74=−0.95989719783345x_{74} = -0.95989719783345
x75=−21.4623740434466x_{75} = -21.4623740434466
x76=100.006255084988x_{76} = 100.006255084988
x77=−75.9203589343812x_{77} = -75.9203589343812
x78=−97.9163752974962x_{78} = -97.9163752974962
x79=64.4009240388803x_{79} = 64.4009240388803
x80=−56.0310180297196x_{80} = -56.0310180297196
x81=48.7015306215084x_{81} = 48.7015306215084
x82=−95.8220545994953x_{82} = -95.8220545994953
x83=18.3198868045554x_{83} = 18.3198868045554
x84=−31.9360463105638x_{84} = -31.9360463105638
x85=−100.010699111567x_{85} = -100.010699111567
x86=34.0437120782042x_{86} = 34.0437120782042
x87=56.0230855877076x_{87} = 56.0230855877076
x88=−9.98172626669137x_{88} = -9.98172626669137
x89=−65.448149247889x_{89} = -65.448149247889
x90=24.60462482455x_{90} = 24.60462482455
x91=95.8174163069689x_{91} = 95.8174163069689
x92=62.3064709329742x_{92} = 62.3064709329742
x93=9.93714128227479x_{93} = 9.93714128227479
x94=49.7396496884211x_{94} = 49.7396496884211
Signos 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=89.5341496175094x_{1} = 89.5341496175094
x2=7.8396736012599x_{2} = 7.8396736012599
x3=−3.75177195803661x_{3} = -3.75177195803661
x4=−16.2520658835412x_{4} = -16.2520658835412
x5=35.0779499321882x_{5} = 35.0779499321882
x6=3.63183452315228x_{6} = 3.63183452315228
x7=−51.842708427932x_{7} = -51.842708427932
x8=12.0335124171734x_{8} = 12.0335124171734
x9=−87.4448074455823x_{9} = -87.4448074455823
x10=97.9118362155217x_{10} = 97.9118362155217
x11=22.5098087774546x_{11} = 22.5098087774546
x12=−91.6334234062544x_{12} = -91.6334234062544
x13=−53.9368539004158x_{13} = -53.9368539004158
x14=−12.0703667254341x_{14} = -12.0703667254341
x15=53.9286134447172x_{15} = 53.9286134447172
x16=−41.3723600294113x_{16} = -41.3723600294113
x17=51.8341350739919x_{17} = 51.8341350739919
x18=−89.5391133882524x_{18} = -89.5391133882524
x19=−49.7485839674963x_{19} = -49.7485839674963
x20=−58.1251987998863x_{20} = -58.1251987998863
x21=−47.6544832880973x_{21} = -47.6544832880973
x22=93.7229952896613x_{22} = 93.7229952896613
x23=16.2247054307724x_{23} = 16.2247054307724
x24=60.212013791028x_{24} = 60.212013791028
x25=−5.81650479334164x_{25} = -5.81650479334164
x26=−45.5604096674964x_{26} = -45.5604096674964
x27=−1.71221607813122x_{27} = -1.71221607813122
x28=20.414906074639x_{28} = 20.414906074639
x29=5.7397730916088x_{29} = 5.7397730916088
x30=−14.160695216171x_{30} = -14.160695216171
x31=58.1175521765418x_{31} = 58.1175521765418
x32=14.1292889745525x_{32} = 14.1292889745525
x33=−43.4663670146734x_{33} = -43.4663670146734
x34=−60.2193944752711x_{34} = -60.2193944752711
x35=−7.89608208530915x_{35} = -7.89608208530915
x36=−93.7277372264291x_{36} = -93.7277372264291
x37=66.4953734902289x_{37} = 66.4953734902289
x38=100.006255084988x_{38} = 100.006255084988
x39=−97.9163752974962x_{39} = -97.9163752974962
x40=64.4009240388803x_{40} = 64.4009240388803
x41=−56.0310180297196x_{41} = -56.0310180297196
x42=−95.8220545994953x_{42} = -95.8220545994953
x43=18.3198868045554x_{43} = 18.3198868045554
x44=−100.010699111567x_{44} = -100.010699111567
x45=56.0230855877076x_{45} = 56.0230855877076
x46=−9.98172626669137x_{46} = -9.98172626669137
x47=24.60462482455x_{47} = 24.60462482455
x48=95.8174163069689x_{48} = 95.8174163069689
x49=62.3064709329742x_{49} = 62.3064709329742
x50=9.93714128227479x_{50} = 9.93714128227479
x51=49.7396496884211x_{51} = 49.7396496884211
Puntos máximos de la función:
x51=73.8319421320736x_{51} = 73.8319421320736
x51=−29.8414070727572x_{51} = -29.8414070727572
x51=67.5491767500035x_{51} = 67.5491767500035
x51=−78.0147933275031x_{51} = -78.0147933275031
x51=−84.2980847924199x_{51} = -84.2980847924199
x51=86.3976561110523x_{51} = 86.3976561110523
x51=57.0781065328832x_{51} = 57.0781065328832
x51=−67.5425969944273x_{51} = -67.5425969944273
x51=−38.2198035188413x_{51} = -38.2198035188413
x51=−73.8259223119028x_{51} = -73.8259223119028
x51=−23.5572290697416x_{51} = -23.5572290697416
x51=−69.6370415742658x_{51} = -69.6370415742658
x51=−34.0306555064577x_{51} = -34.0306555064577
x51=44.5133844234738x_{51} = 44.5133844234738
x51=−80.1092256661263x_{51} = -80.1092256661263
x51=38.2314296566913x_{51} = 38.2314296566913
x51=40.3253719836236x_{51} = 40.3253719836236
x51=78.020489958096x_{51} = 78.020489958096
x51=−17.2723300965849x_{51} = -17.2723300965849
x51=80.1147733747798x_{51} = 80.1147733747798
x51=75.9262127024484x_{51} = 75.9262127024484
x51=−27.7467309956061x_{51} = -27.7467309956061
x51=27.7627423539192x_{51} = 27.7627423539192
x51=36.137539795388x_{51} = 36.137539795388
x51=42.4193590081437x_{51} = 42.4193590081437
x51=92.6805798692254x_{51} = 92.6805798692254
x51=−71.7314832647885x_{51} = -71.7314832647885
x51=29.856295359853x_{51} = 29.856295359853
x51=−25.6520090653707x_{51} = -25.6520090653707
x51=82.2090624816772x_{51} = 82.2090624816772
x51=−36.1252398830002x_{51} = -36.1252398830002
x51=88.4919599039137x_{51} = 88.4919599039137
x51=71.7376788324169x_{51} = 71.7376788324169
x51=84.3033568547452x_{51} = 84.3033568547452
x51=−82.2036561072884x_{51} = -82.2036561072884
x51=31.9499587197976x_{51} = 31.9499587197976
x51=−0.95989719783345x_{51} = -0.95989719783345
x51=−21.4623740434466x_{51} = -21.4623740434466
x51=−75.9203589343812x_{51} = -75.9203589343812
x51=48.7015306215084x_{51} = 48.7015306215084
x51=−31.9360463105638x_{51} = -31.9360463105638
x51=34.0437120782042x_{51} = 34.0437120782042
x51=−65.448149247889x_{51} = -65.448149247889
Decrece en los intervalos
[100.006255084988,∞)\left[100.006255084988, \infty\right)
Crece en los intervalos
(−∞,−100.010699111567]\left(-\infty, -100.010699111567\right]
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
3(−3xsin⁡(3x)2+cos⁡(3x)−2x4)=03 \left(- \frac{3 x \sin{\left(3 x \right)}}{2} + \cos{\left(3 x \right)} - \frac{2}{x^{4}}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−19.907911952782x_{1} = -19.907911952782
x2=−39.799090019458x_{2} = -39.799090019458
x3=24.0947642225796x_{3} = 24.0947642225796
x4=−37.705004929418x_{4} = -37.705004929418
x5=90.0614568086612x_{5} = 90.0614568086612
x6=17.8148268039322x_{6} = 17.8148268039322
x7=48.1756998040024x_{7} = 48.1756998040024
x8=−50.2699027788224x_{8} = -50.2699027788224
x9=−2.18458133232718x_{9} = -2.18458133232718
x10=83.7784565380389x_{10} = 83.7784565380389
x11=−43.987348718435x_{11} = -43.987348718435
x12=−53.4112354854412x_{12} = -53.4112354854412
x13=70.1654029547235x_{13} = 70.1654029547235
x14=−61.7882518940721x_{14} = -61.7882518940721
x15=4.24044984746248x_{15} = 4.24044984746248
x16=22.0012460097448x_{16} = 22.0012460097448
x17=26.1884224975902x_{17} = 26.1884224975902
x18=−70.1654029547235x_{18} = -70.1654029547235
x19=63.8825291042832x_{19} = 63.8825291042832
x20=−95.2973090044054x_{20} = -95.2973090044054
x21=−35.6109562808109x_{21} = -35.6109562808109
x22=2.18458133232718x_{22} = 2.18458133232718
x23=37.705004929418x_{23} = 37.705004929418
x24=−68.071105283931x_{24} = -68.071105283931
x25=−15.7220896627602x_{25} = -15.7220896627602
x26=76.4483279929109x_{26} = 76.4483279929109
x27=−33.5169508979714x_{27} = -33.5169508979714
x28=−1.30898709351237x_{28} = -1.30898709351237
x29=−77.4954862681777x_{29} = -77.4954862681777
x30=−4.24044984746248x_{30} = -4.24044984746248
x31=−24.0947642225796x_{31} = -24.0947642225796
x32=46.0815142865079x_{32} = 46.0815142865079
x33=−59.6939829545149x_{33} = -59.6939829545149
x34=68.071105283931x_{34} = 68.071105283931
x35=98.4388272430731x_{35} = 98.4388272430731
x36=100.533175319149x_{36} = 100.533175319149
x37=−26.1884224975902x_{37} = -26.1884224975902
x38=−57.5997231874848x_{38} = -57.5997231874848
x39=−90.0614568086612x_{39} = -90.0614568086612
x40=−28.2821897723174x_{40} = -28.2821897723174
x41=85.8727869533063x_{41} = 85.8727869533063
x42=−94.250137360338x_{42} = -94.250137360338
x43=65.9768137977467x_{43} = 65.9768137977467
x44=87.9671204484587x_{44} = 87.9671204484587
x45=−6.31818359713174x_{45} = -6.31818359713174
x46=−55.5054736309118x_{46} = -55.5054736309118
x47=−81.6841294395199x_{47} = -81.6841294395199
x48=−72.259706272491x_{48} = -72.259706272491
x49=−22.0012460097448x_{49} = -22.0012460097448
x50=−85.8727869533063x_{50} = -85.8727869533063
x51=−41.8932060898102x_{51} = -41.8932060898102
x52=9.44826483019524x_{52} = 9.44826483019524
x53=8.4039571519597x_{53} = 8.4039571519597
x54=−99.4860010337234x_{54} = -99.4860010337234
x55=−87.9671204484587x_{55} = -87.9671204484587
x56=−46.0815142865079x_{56} = -46.0815142865079
x57=−30.3760435342236x_{57} = -30.3760435342236
x58=−83.7784565380389x_{58} = -83.7784565380389
x59=−11.5384131843255x_{59} = -11.5384131843255
x60=15.7220896627602x_{60} = 15.7220896627602
x61=28.2821897723174x_{61} = 28.2821897723174
x62=74.3540147601572x_{62} = 74.3540147601572
x63=6.31818359713174x_{63} = 6.31818359713174
x64=50.2699027788224x_{64} = 50.2699027788224
x65=−3.21110226651524x_{65} = -3.21110226651524
x66=92.1557958386049x_{66} = 92.1557958386049
x67=59.6939829545149x_{67} = 59.6939829545149
x68=−13.6298601979387x_{68} = -13.6298601979387
x69=−65.9768137977467x_{69} = -65.9768137977467
x70=−63.8825291042832x_{70} = -63.8825291042832
x71=94.250137360338x_{71} = 94.250137360338
x72=−17.8148268039322x_{72} = -17.8148268039322
x73=30.3760435342236x_{73} = 30.3760435342236
x74=19.907911952782x_{74} = 19.907911952782
x75=78.5426455912395x_{75} = 78.5426455912395
x76=54.4583530476818x_{76} = 54.4583530476818
x77=52.3641211173087x_{77} = 52.3641211173087
x78=12.5840118070877x_{78} = 12.5840118070877
x79=−9.44826483019524x_{79} = -9.44826483019524
x80=39.799090019458x_{80} = 39.799090019458
x81=41.8932060898102x_{81} = 41.8932060898102
x82=−79.5898059195386x_{82} = -79.5898059195386
x83=−92.1557958386049x_{83} = -92.1557958386049
x84=32.4699670691998x_{84} = 32.4699670691998
x85=56.5525970604809x_{85} = 56.5525970604809
x86=43.987348718435x_{86} = 43.987348718435
x87=10.4931214936956x_{87} = 10.4931214936956
x88=61.7882518940721x_{88} = 61.7882518940721
x89=−48.1756998040024x_{89} = -48.1756998040024
x90=72.259706272491x_{90} = 72.259706272491
x91=96.3444812113793x_{91} = 96.3444812113793
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0

lim⁡x→0−(3(−3xsin⁡(3x)2+cos⁡(3x)−2x4))=−∞\lim_{x \to 0^-}\left(3 \left(- \frac{3 x \sin{\left(3 x \right)}}{2} + \cos{\left(3 x \right)} - \frac{2}{x^{4}}\right)\right) = -\infty
lim⁡x→0+(3(−3xsin⁡(3x)2+cos⁡(3x)−2x4))=−∞\lim_{x \to 0^+}\left(3 \left(- \frac{3 x \sin{\left(3 x \right)}}{2} + \cos{\left(3 x \right)} - \frac{2}{x^{4}}\right)\right) = -\infty
- los límites son iguales, es decir omitimos el punto correspondiente

Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
[78.5426455912395,∞)\left[78.5426455912395, \infty\right)
Convexa en los intervalos
(−∞,−94.250137360338]\left(-\infty, -94.250137360338\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(((xsin⁡(3x)2+x)−1x2)−4)=−∞\lim_{x \to -\infty}\left(\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞(((xsin⁡(3x)2+x)−1x2)−4)=∞\lim_{x \to \infty}\left(\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (sin(3*x)/2)*x + x - 1/x^2 - 4, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(((xsin⁡(3x)2+x)−1x2)−4x)=1\lim_{x \to -\infty}\left(\frac{\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4}{x}\right) = 1
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xy = x
lim⁡x→∞(((xsin⁡(3x)2+x)−1x2)−4x)=1\lim_{x \to \infty}\left(\frac{\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4}{x}\right) = 1
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xy = x
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
((xsin⁡(3x)2+x)−1x2)−4=xsin⁡(3x)2−x−4−1x2\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4 = \frac{x \sin{\left(3 x \right)}}{2} - x - 4 - \frac{1}{x^{2}}
- No
((xsin⁡(3x)2+x)−1x2)−4=−xsin⁡(3x)2+x+4+1x2\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4 = - \frac{x \sin{\left(3 x \right)}}{2} + x + 4 + \frac{1}{x^{2}}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор