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x*sin(3*x)
Trazado el gráfico de la función/ x*sin(3*x)

Gráfico de la función y = x*sin(3*x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = x*sin(3*x)
f(x)=xsin⁡(3x)f{\left(x \right)} = x \sin{\left(3 x \right)} 
f = x*sin(3*x)
Gráfico de la función
02468-8-6-4-2-1010-2020
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)=0x \sin{\left(3 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=−2π3x_{2} = - \frac{2 \pi}{3}
x3=−π3x_{3} = - \frac{\pi}{3}
x4=π3x_{4} = \frac{\pi}{3}
x5=2π3x_{5} = \frac{2 \pi}{3}
x6=πx_{6} = \pi
Solución numérica
x1=−39.7935069454707x_{1} = -39.7935069454707
x2=−26.1799387799149x_{2} = -26.1799387799149
x3=48.1710873550435x_{3} = 48.1710873550435
x4=32.4631240870945x_{4} = 32.4631240870945
x5=−41.8879020478639x_{5} = -41.8879020478639
x6=90.0589894029074x_{6} = 90.0589894029074
x7=4.18879020478639x_{7} = 4.18879020478639
x8=−53.4070751110265x_{8} = -53.4070751110265
x9=26.1799387799149x_{9} = 26.1799387799149
x10=52.3598775598299x_{10} = 52.3598775598299
x11=−1.0471975511966x_{11} = -1.0471975511966
x12=28.2743338823081x_{12} = 28.2743338823081
x13=19.8967534727354x_{13} = 19.8967534727354
x14=50.2654824574367x_{14} = 50.2654824574367
x15=8.37758040957278x_{15} = 8.37758040957278
x16=−2.0943951023932x_{16} = -2.0943951023932
x17=−86.9173967493176x_{17} = -86.9173967493176
x18=24.0855436775217x_{18} = 24.0855436775217
x19=−11.5191730631626x_{19} = -11.5191730631626
x20=−90.0589894029074x_{20} = -90.0589894029074
x21=−33.5103216382911x_{21} = -33.5103216382911
x22=68.0678408277789x_{22} = 68.0678408277789
x23=65.9734457253857x_{23} = 65.9734457253857
x24=0x_{24} = 0
x25=3.14159265358979x_{25} = 3.14159265358979
x26=−24.0855436775217x_{26} = -24.0855436775217
x27=−30.3687289847013x_{27} = -30.3687289847013
x28=−68.0678408277789x_{28} = -68.0678408277789
x29=78.5398163397448x_{29} = 78.5398163397448
x30=98.4365698124802x_{30} = 98.4365698124802
x31=−70.162235930172x_{31} = -70.162235930172
x32=59.6902604182061x_{32} = 59.6902604182061
x33=17.8023583703422x_{33} = 17.8023583703422
x34=100.530964914873x_{34} = 100.530964914873
x35=−85.870199198121x_{35} = -85.870199198121
x36=−72.2566310325652x_{36} = -72.2566310325652
x37=21.9911485751286x_{37} = 21.9911485751286
x38=−79.5870138909414x_{38} = -79.5870138909414
x39=−37.6991118430775x_{39} = -37.6991118430775
x40=−81.6814089933346x_{40} = -81.6814089933346
x41=−21.9911485751286x_{41} = -21.9911485751286
x42=−46.0766922526503x_{42} = -46.0766922526503
x43=−13.6135681655558x_{43} = -13.6135681655558
x44=−4.18879020478639x_{44} = -4.18879020478639
x45=−87.9645943005142x_{45} = -87.9645943005142
x46=−77.4926187885482x_{46} = -77.4926187885482
x47=41.8879020478639x_{47} = 41.8879020478639
x48=−99.4837673636768x_{48} = -99.4837673636768
x49=1.0471975511966x_{49} = 1.0471975511966
x50=−94.2477796076938x_{50} = -94.2477796076938
x51=15.707963267949x_{51} = 15.707963267949
x52=30.3687289847013x_{52} = 30.3687289847013
x53=−43.9822971502571x_{53} = -43.9822971502571
x54=39.7935069454707x_{54} = 39.7935069454707
x55=−6.28318530717959x_{55} = -6.28318530717959
x56=−8.37758040957278x_{56} = -8.37758040957278
x57=−83.7758040957278x_{57} = -83.7758040957278
x58=−28.2743338823081x_{58} = -28.2743338823081
x59=−95.2949771588904x_{59} = -95.2949771588904
x60=−17.8023583703422x_{60} = -17.8023583703422
x61=94.2477796076938x_{61} = 94.2477796076938
x62=−48.1710873550435x_{62} = -48.1710873550435
x63=46.0766922526503x_{63} = 46.0766922526503
x64=−59.6902604182061x_{64} = -59.6902604182061
x65=87.9645943005142x_{65} = 87.9645943005142
x66=−63.8790506229925x_{66} = -63.8790506229925
x67=−61.7846555205993x_{67} = -61.7846555205993
x68=63.8790506229925x_{68} = 63.8790506229925
x69=−92.1533845053006x_{69} = -92.1533845053006
x70=2.0943951023932x_{70} = 2.0943951023932
x71=74.3510261349584x_{71} = 74.3510261349584
x72=−15.707963267949x_{72} = -15.707963267949
x73=83.7758040957278x_{73} = 83.7758040957278
x74=96.342174710087x_{74} = 96.342174710087
x75=6.28318530717959x_{75} = 6.28318530717959
x76=13.6135681655558x_{76} = 13.6135681655558
x77=−50.2654824574367x_{77} = -50.2654824574367
x78=54.4542726622231x_{78} = 54.4542726622231
x79=−57.5958653158129x_{79} = -57.5958653158129
x80=37.6991118430775x_{80} = 37.6991118430775
x81=−19.8967534727354x_{81} = -19.8967534727354
x82=43.9822971502571x_{82} = 43.9822971502571
x83=56.5486677646163x_{83} = 56.5486677646163
x84=−65.9734457253857x_{84} = -65.9734457253857
x85=58.6430628670095x_{85} = 58.6430628670095
x86=−55.5014702134197x_{86} = -55.5014702134197
x87=76.4454212373516x_{87} = 76.4454212373516
x88=75.398223686155x_{88} = 75.398223686155
x89=92.1533845053006x_{89} = 92.1533845053006
x90=70.162235930172x_{90} = 70.162235930172
x91=−35.6047167406843x_{91} = -35.6047167406843
x92=72.2566310325652x_{92} = 72.2566310325652
x93=10.471975511966x_{93} = 10.471975511966
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 x*sin(3*x).
0sin⁡(0⋅3)0 \sin{\left(0 \cdot 3 \right)}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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)+sin⁡(3x)=03 x \cos{\left(3 x \right)} + \sin{\left(3 x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−14.1450206271366x_{1} = -14.1450206271366
x2=−47.6498203678514x_{2} = -47.6498203678514
x3=−100.008477152089x_{3} = -100.008477152089
x4=23.5666593462033x_{4} = 23.5666593462033
x5=−78.0176417347899x_{5} = -78.0176417347899
x6=93.7253663237826x_{6} = 93.7253663237826
x7=100.008477152089x_{7} = 100.008477152089
x8=95.8197355146347x_{8} = 95.8197355146347
x9=16.2384035725192x_{9} = 16.2384035725192
x10=−87.4422661984441x_{10} = -87.4422661984441
x11=49.7441173016936x_{11} = 49.7441173016936
x12=89.5366315785916x_{12} = 89.5366315785916
x13=−75.9232859178705x_{13} = -75.9232859178705
x14=84.3007208972085x_{14} = 84.3007208972085
x15=−25.6606697768062x_{15} = -25.6606697768062
x16=−12.0519888065122x_{16} = -12.0519888065122
x17=22.5196809462695x_{17} = 22.5196809462695
x18=−3.69517946883234x_{18} = -3.69517946883234
x19=−84.3007208972085x_{19} = -84.3007208972085
x20=−45.5555324591521x_{20} = -45.5555324591521
x21=18.3320175191655x_{21} = 18.3320175191655
x22=66.4987153630436x_{22} = 66.4987153630436
x23=3.69517946883234x_{23} = 3.69517946883234
x24=64.4043745938079x_{24} = 64.4043745938079
x25=82.2063593736386x_{25} = 82.2063593736386
x26=40.3198613996847x_{26} = 40.3198613996847
x27=92.6781821675128x_{27} = 92.6781821675128
x28=−65.4515445438056x_{28} = -65.4515445438056
x29=12.0519888065122x_{29} = 12.0519888065122
x30=−60.2157043931142x_{30} = -60.2157043931142
x31=−53.9327340398572x_{31} = -53.9327340398572
x32=−51.8384221669793x_{32} = -51.8384221669793
x33=−58.1213757786594x_{33} = -58.1213757786594
x34=51.8384221669793x_{34} = 51.8384221669793
x35=60.2157043931142x_{35} = 60.2157043931142
x36=36.1313906251304x_{36} = 36.1313906251304
x37=−29.8488525127497x_{37} = -29.8488525127497
x38=0x_{38} = 0
x39=58.1213757786594x_{39} = 58.1213757786594
x40=29.8488525127497x_{40} = 29.8488525127497
x41=−9.95952883536913x_{41} = -9.95952883536913
x42=73.8289323297373x_{42} = 73.8289323297373
x43=−56.0270521345864x_{43} = -56.0270521345864
x44=−95.8197355146347x_{44} = -95.8197355146347
x45=−91.6309983173967x_{45} = -91.6309983173967
x46=−69.6402326441114x_{46} = -69.6402326441114
x47=54.9798923539233x_{47} = 54.9798923539233
x48=−4.7358122417304x_{48} = -4.7358122417304
x49=−27.7547382346962x_{49} = -27.7547382346962
x50=−38.225617263561x_{50} = -38.225617263561
x51=−89.5366315785916x_{51} = -89.5366315785916
x52=−34.037184713218x_{52} = -34.037184713218
x53=7.8680949243268x_{53} = 7.8680949243268
x54=78.0176417347899x_{54} = 78.0176417347899
x55=0.676252612703478x_{55} = 0.676252612703478
x56=−106.291596785411x_{56} = -106.291596785411
x57=53.9327340398572x_{57} = 53.9327340398572
x58=62.3100374768166x_{58} = 62.3100374768166
x59=34.037184713218x_{59} = 34.037184713218
x60=−67.5458870110976x_{60} = -67.5458870110976
x61=−93.7253663237826x_{61} = -93.7253663237826
x62=−36.1313906251304x_{62} = -36.1313906251304
x63=−23.5666593462033x_{63} = -23.5666593462033
x64=75.9232859178705x_{64} = 75.9232859178705
x65=−97.9141058139495x_{65} = -97.9141058139495
x66=−7.8680949243268x_{66} = -7.8680949243268
x67=20.4257915111899x_{67} = 20.4257915111899
x68=27.7547382346962x_{68} = 27.7547382346962
x69=42.4141204421759x_{69} = 42.4141204421759
x70=88.4894487126566x_{70} = 88.4894487126566
x71=67.5458870110976x_{71} = 67.5458870110976
x72=−5.77879264132779x_{72} = -5.77879264132779
x73=−80.1119996057056x_{73} = -80.1119996057056
x74=38.225617263561x_{74} = 38.225617263561
x75=−71.7345811655909x_{75} = -71.7345811655909
x76=−73.8289323297373x_{76} = -73.8289323297373
x77=−61.2628704049539x_{77} = -61.2628704049539
x78=97.9141058139495x_{78} = 97.9141058139495
x79=86.3950840487432x_{79} = 86.3950840487432
x80=31.9430036030065x_{80} = 31.9430036030065
x81=−16.2384035725192x_{81} = -16.2384035725192
x82=56.0270521345864x_{82} = 56.0270521345864
x83=14.1450206271366x_{83} = 14.1450206271366
x84=9.95952883536913x_{84} = 9.95952883536913
x85=−21.4727239072797x_{85} = -21.4727239072797
x86=71.7345811655909x_{86} = 71.7345811655909
x87=26.7076976049501x_{87} = 26.7076976049501
x88=80.1119996057056x_{88} = 80.1119996057056
x89=−41.3669891991005x_{89} = -41.3669891991005
x90=−49.7441173016936x_{90} = -49.7441173016936
x91=5.77879264132779x_{91} = 5.77879264132779
x92=−31.9430036030065x_{92} = -31.9430036030065
x93=−1.63772681314496x_{93} = -1.63772681314496
x94=−82.2063593736386x_{94} = -82.2063593736386
x95=−43.4612548800528x_{95} = -43.4612548800528
x96=44.5083922872857x_{96} = 44.5083922872857
Signos de extremos en los puntos:
(-14.145020627136628, -14.1410946924197)

(-47.64982036785143, -47.6486544974204)

(-100.00847715208945, -100.007921648254)

(23.566659346203334, 23.564302320531)

(-78.01764173478989, 78.0169296548843)

(93.7253663237826, -93.724773581063)

(100.00847715208945, -100.007921648254)

(95.81973551463474, -95.8191557274852)

(16.238403572519243, -16.234983408456)

(-87.44226619844414, -87.4416308655269)

(49.74411730169364, -49.7430005126604)

(89.53663157859164, -89.5360111065335)

(-75.92328591787046, 75.9225541956887)

(84.3007208972085, 84.3000618885604)

(-25.66066977680624, 25.6585050427546)

(-12.05198880651224, -12.0473817907474)

(22.519680946269467, -22.5172143736575)

(-3.695179468832341, -3.68023600531)

(-84.3007208972085, 84.3000618885604)

(-45.555532459152055, -45.5543129953057)

(18.332017519165465, -18.3289877498992)

(66.49871536304362, -66.4978799407601)

(3.695179468832341, -3.68023600531)

(64.40437459380786, -64.4035120058269)

(82.20635937363859, 82.2056835759154)

(40.319861399684655, 40.318483599613)

(92.6781821675128, 92.6775827274368)

(-65.45154454380558, 65.4506957559693)

(12.05198880651224, -12.0473817907474)

(-60.21570439311422, -60.2147818052389)

(-53.93273403985717, -53.9317039796355)

(-51.83842216697935, -51.8373504940684)

(-58.121375778659406, -58.1204199481347)

(51.83842216697935, -51.8373504940684)

(60.21570439311422, -60.2147818052389)

(36.131390625130436, 36.1298531252298)

(-29.848852512749733, 29.8469914576284)

(0, 0)

(58.121375778659406, -58.1204199481347)

(29.848852512749733, 29.8469914576284)

(-9.959528835369131, -9.9539553863956)

(73.82893232973727, 73.8281798509399)

(-56.027052134586405, -56.0260605764261)

(-95.81973551463474, -95.8191557274852)

(-91.63099831739673, -91.6303920268926)

(-69.64023264411136, 69.6394349069578)

(54.97989235392331, 54.9788819113479)

(-4.735812241730396, 4.72412470459143)

(-27.754738234696216, 27.7527367909844)

(-38.225617263561, 38.2241639871628)

(-89.53663157859164, -89.5360111065335)

(-34.037184713217975, 34.0355526287721)

(7.868094924326803, -7.86104354987779)

(78.01764173478989, 78.0169296548843)

(0.676252612703478, 0.606568580386551)

(-106.29159678541116, -106.291074118075)

(53.93273403985717, -53.9317039796355)

(62.31003747681657, -62.3091458971384)

(34.037184713217975, 34.0355526287721)

(-67.54588701109756, 67.5450645399848)

(-93.7253663237826, -93.724773581063)

(-36.131390625130436, 36.1298531252298)

(-23.566659346203334, 23.564302320531)

(75.92328591787046, 75.9225541956887)

(-97.91410581394948, -97.9135384281556)

(-7.868094924326803, -7.86104354987779)

(20.425791511189885, -20.4230721814922)

(27.754738234696216, 27.7527367909844)

(42.41412044217588, 42.4128106665257)

(88.4894487126566, 88.4888208980964)

(67.54588701109756, 67.5450645399848)

(-5.778792641327787, -5.76920286928617)

(-80.1119996057056, 80.111306141125)

(38.225617263561, 38.2241639871628)

(-71.73458116559091, 71.7338067182464)

(-73.82893232973727, 73.8281798509399)

(-61.262870404953894, 61.2619635861633)

(97.91410581394948, -97.9135384281556)

(86.39508404874319, 86.3944410152197)

(31.943003603006492, 31.9412645361552)

(-16.238403572519243, -16.234983408456)

(56.027052134586405, -56.0260605764261)

(14.145020627136628, -14.1410946924197)

(9.959528835369131, -9.9539553863956)

(-21.47272390727972, 21.4701371131251)

(71.73458116559091, 71.7338067182464)

(26.707697604950084, -26.7056177152197)

(80.1119996057056, 80.111306141125)

(-41.36698919910045, -41.3656462720145)

(-49.74411730169364, -49.7430005126604)

(5.778792641327787, -5.76920286928617)

(-31.943003603006492, 31.9412645361552)

(-1.6377268131449612, -1.60482329657076)

(-82.20635937363859, 82.2056835759154)

(-43.46125488005277, -43.4599766586844)

(44.50839228728569, 44.5071441357397)


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=−14.1450206271366x_{1} = -14.1450206271366
x2=−47.6498203678514x_{2} = -47.6498203678514
x3=−100.008477152089x_{3} = -100.008477152089
x4=93.7253663237826x_{4} = 93.7253663237826
x5=100.008477152089x_{5} = 100.008477152089
x6=95.8197355146347x_{6} = 95.8197355146347
x7=16.2384035725192x_{7} = 16.2384035725192
x8=−87.4422661984441x_{8} = -87.4422661984441
x9=49.7441173016936x_{9} = 49.7441173016936
x10=89.5366315785916x_{10} = 89.5366315785916
x11=−12.0519888065122x_{11} = -12.0519888065122
x12=22.5196809462695x_{12} = 22.5196809462695
x13=−3.69517946883234x_{13} = -3.69517946883234
x14=−45.5555324591521x_{14} = -45.5555324591521
x15=18.3320175191655x_{15} = 18.3320175191655
x16=66.4987153630436x_{16} = 66.4987153630436
x17=3.69517946883234x_{17} = 3.69517946883234
x18=64.4043745938079x_{18} = 64.4043745938079
x19=12.0519888065122x_{19} = 12.0519888065122
x20=−60.2157043931142x_{20} = -60.2157043931142
x21=−53.9327340398572x_{21} = -53.9327340398572
x22=−51.8384221669793x_{22} = -51.8384221669793
x23=−58.1213757786594x_{23} = -58.1213757786594
x24=51.8384221669793x_{24} = 51.8384221669793
x25=60.2157043931142x_{25} = 60.2157043931142
x26=0x_{26} = 0
x27=58.1213757786594x_{27} = 58.1213757786594
x28=−9.95952883536913x_{28} = -9.95952883536913
x29=−56.0270521345864x_{29} = -56.0270521345864
x30=−95.8197355146347x_{30} = -95.8197355146347
x31=−91.6309983173967x_{31} = -91.6309983173967
x32=−89.5366315785916x_{32} = -89.5366315785916
x33=7.8680949243268x_{33} = 7.8680949243268
x34=−106.291596785411x_{34} = -106.291596785411
x35=53.9327340398572x_{35} = 53.9327340398572
x36=62.3100374768166x_{36} = 62.3100374768166
x37=−93.7253663237826x_{37} = -93.7253663237826
x38=−97.9141058139495x_{38} = -97.9141058139495
x39=−7.8680949243268x_{39} = -7.8680949243268
x40=20.4257915111899x_{40} = 20.4257915111899
x41=−5.77879264132779x_{41} = -5.77879264132779
x42=97.9141058139495x_{42} = 97.9141058139495
x43=−16.2384035725192x_{43} = -16.2384035725192
x44=56.0270521345864x_{44} = 56.0270521345864
x45=14.1450206271366x_{45} = 14.1450206271366
x46=9.95952883536913x_{46} = 9.95952883536913
x47=26.7076976049501x_{47} = 26.7076976049501
x48=−41.3669891991005x_{48} = -41.3669891991005
x49=−49.7441173016936x_{49} = -49.7441173016936
x50=5.77879264132779x_{50} = 5.77879264132779
x51=−1.63772681314496x_{51} = -1.63772681314496
x52=−43.4612548800528x_{52} = -43.4612548800528
Puntos máximos de la función:
x52=23.5666593462033x_{52} = 23.5666593462033
x52=−78.0176417347899x_{52} = -78.0176417347899
x52=−75.9232859178705x_{52} = -75.9232859178705
x52=84.3007208972085x_{52} = 84.3007208972085
x52=−25.6606697768062x_{52} = -25.6606697768062
x52=−84.3007208972085x_{52} = -84.3007208972085
x52=82.2063593736386x_{52} = 82.2063593736386
x52=40.3198613996847x_{52} = 40.3198613996847
x52=92.6781821675128x_{52} = 92.6781821675128
x52=−65.4515445438056x_{52} = -65.4515445438056
x52=36.1313906251304x_{52} = 36.1313906251304
x52=−29.8488525127497x_{52} = -29.8488525127497
x52=29.8488525127497x_{52} = 29.8488525127497
x52=73.8289323297373x_{52} = 73.8289323297373
x52=−69.6402326441114x_{52} = -69.6402326441114
x52=54.9798923539233x_{52} = 54.9798923539233
x52=−4.7358122417304x_{52} = -4.7358122417304
x52=−27.7547382346962x_{52} = -27.7547382346962
x52=−38.225617263561x_{52} = -38.225617263561
x52=−34.037184713218x_{52} = -34.037184713218
x52=78.0176417347899x_{52} = 78.0176417347899
x52=0.676252612703478x_{52} = 0.676252612703478
x52=34.037184713218x_{52} = 34.037184713218
x52=−67.5458870110976x_{52} = -67.5458870110976
x52=−36.1313906251304x_{52} = -36.1313906251304
x52=−23.5666593462033x_{52} = -23.5666593462033
x52=75.9232859178705x_{52} = 75.9232859178705
x52=27.7547382346962x_{52} = 27.7547382346962
x52=42.4141204421759x_{52} = 42.4141204421759
x52=88.4894487126566x_{52} = 88.4894487126566
x52=67.5458870110976x_{52} = 67.5458870110976
x52=−80.1119996057056x_{52} = -80.1119996057056
x52=38.225617263561x_{52} = 38.225617263561
x52=−71.7345811655909x_{52} = -71.7345811655909
x52=−73.8289323297373x_{52} = -73.8289323297373
x52=−61.2628704049539x_{52} = -61.2628704049539
x52=86.3950840487432x_{52} = 86.3950840487432
x52=31.9430036030065x_{52} = 31.9430036030065
x52=−21.4727239072797x_{52} = -21.4727239072797
x52=71.7345811655909x_{52} = 71.7345811655909
x52=80.1119996057056x_{52} = 80.1119996057056
x52=−31.9430036030065x_{52} = -31.9430036030065
x52=−82.2063593736386x_{52} = -82.2063593736386
x52=44.5083922872857x_{52} = 44.5083922872857
Decrece en los intervalos
[100.008477152089,∞)\left[100.008477152089, \infty\right)
Crece en los intervalos
(−∞,−106.291596785411]\left(-\infty, -106.291596785411\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)+2cos⁡(3x))=03 \left(- 3 x \sin{\left(3 x \right)} + 2 \cos{\left(3 x \right)}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−57.5997231867839x_{1} = -57.5997231867839
x2=−11.5384110184102x_{2} = -11.5384110184102
x3=94.2501373603978x_{3} = 94.2501373603978
x4=105.769053656653x_{4} = 105.769053656653
x5=65.9768137973912x_{5} = 65.9768137973912
x6=6.31822725550968x_{6} = 6.31822725550968
x7=−35.6109562885689x_{7} = -35.6109562885689
x8=−4.24076625725554x_{8} = -4.24076625725554
x9=61.7882518935787x_{9} = 61.7882518935787
x10=17.8148265565878x_{10} = 17.8148265565878
x11=−72.2597062722654x_{11} = -72.2597062722654
x12=59.6939829539286x_{12} = 59.6939829539286
x13=−87.967120448543x_{13} = -87.967120448543
x14=−81.6841294396421x_{14} = -81.6841294396421
x15=−2.19277791090745x_{15} = -2.19277791090745
x16=−17.8148265565878x_{16} = -17.8148265565878
x17=−90.0614568087362x_{17} = -90.0614568087362
x18=−77.4954862683366x_{18} = -77.4954862683366
x19=−70.1654029544622x_{19} = -70.1654029544622
x20=37.7050049352483x_{20} = 37.7050049352483
x21=−9.44825895659546x_{21} = -9.44825895659546
x22=78.5426455910908x_{22} = 78.5426455910908
x23=−0.358957995437268x_{23} = -0.358957995437268
x24=52.3641211184374x_{24} = 52.3641211184374
x25=−59.6939829539286x_{25} = -59.6939829539286
x26=54.4583530486096x_{26} = 54.4583530486096
x27=−61.7882518935787x_{27} = -61.7882518935787
x28=−43.9873487211332x_{28} = -43.9873487211332
x29=50.2699027802066x_{29} = 50.2699027802066
x30=−19.9079118108102x_{30} = -19.9079118108102
x31=−22.0012459236092x_{31} = -22.0012459236092
x32=72.2597062722654x_{32} = 72.2597062722654
x33=−37.7050049352483x_{33} = -37.7050049352483
x34=−15.7220892009256x_{34} = -15.7220892009256
x35=−26.1884224615332x_{35} = -26.1884224615332
x36=0.358957995437268x_{36} = 0.358957995437268
x37=−28.2821897477697x_{37} = -28.2821897477697
x38=100.533175319193x_{38} = 100.533175319193
x39=92.1557958386718x_{39} = 92.1557958386718
x40=48.1756998057147x_{40} = 48.1756998057147
x41=−79.5898059196778x_{41} = -79.5898059196778
x42=39.7990900239077x_{42} = 39.7990900239077
x43=8.40396768807019x_{43} = 8.40396768807019
x44=87.967120448543x_{44} = 87.967120448543
x45=−24.0947641678942x_{45} = -24.0947641678942
x46=−3.20985344776581x_{46} = -3.20985344776581
x47=32.4699670568908x_{47} = 32.4699670568908
x48=−46.0815142886463x_{48} = -46.0815142886463
x49=−92.1557958386718x_{49} = -92.1557958386718
x50=−53.4112354844189x_{50} = -53.4112354844189
x51=−39.7990900239077x_{51} = -39.7990900239077
x52=90.0614568087362x_{52} = 90.0614568087362
x53=63.8825291038655x_{53} = 63.8825291038655
x54=−65.9768137973912x_{54} = -65.9768137973912
x55=−41.8932060932537x_{55} = -41.8932060932537
x56=2.19277791090745x_{56} = 2.19277791090745
x57=19.9079118108102x_{57} = 19.9079118108102
x58=−13.6298592553469x_{58} = -13.6298592553469
x59=26.1884224615332x_{59} = 26.1884224615332
x60=24.0947641678942x_{60} = 24.0947641678942
x61=46.0815142886463x_{61} = 46.0815142886463
x62=−99.4860010336778x_{62} = -99.4860010336778
x63=−94.2501373603978x_{63} = -94.2501373603978
x64=12.5840132115367x_{64} = 12.5840132115367
x65=15.7220892009256x_{65} = 15.7220892009256
x66=56.5525970612491x_{66} = 56.5525970612491
x67=−85.8727869534015x_{67} = -85.8727869534015
x68=98.4388272431212x_{68} = 98.4388272431212
x69=4.24076625725554x_{69} = 4.24076625725554
x70=34.5639476991297x_{70} = 34.5639476991297
x71=41.8932060932537x_{71} = 41.8932060932537
x72=−30.3760435170464x_{72} = -30.3760435170464
x73=−50.2699027802066x_{73} = -50.2699027802066
x74=−83.7784565381466x_{74} = -83.7784565381466
x75=76.4483279927407x_{75} = 76.4483279927407
x76=−95.2973090043489x_{76} = -95.2973090043489
x77=−48.1756998057147x_{77} = -48.1756998057147
x78=−63.8825291038655x_{78} = -63.8825291038655
x79=68.0711052836269x_{79} = 68.0711052836269
x80=−55.5054736300684x_{80} = -55.5054736300684
x81=28.2821897477697x_{81} = 28.2821897477697
x82=10.493124973438x_{82} = 10.493124973438
x83=70.1654029544622x_{83} = 70.1654029544622
x84=30.3760435170464x_{84} = 30.3760435170464
x85=−1.2145323891418x_{85} = -1.2145323891418
x86=96.3444812114328x_{86} = 96.3444812114328
x87=22.0012459236092x_{87} = 22.0012459236092
x88=83.7784565381466x_{88} = 83.7784565381466
x89=43.9873487211332x_{89} = 43.9873487211332
x90=85.8727869534015x_{90} = 85.8727869534015
x91=1.2145323891418x_{91} = 1.2145323891418
x92=−33.5169509084747x_{92} = -33.5169509084747
x93=−68.0711052836269x_{93} = -68.0711052836269
x94=74.3540147599617x_{94} = 74.3540147599617

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
[105.769053656653,∞)\left[105.769053656653, \infty\right)
Convexa en los intervalos
(−∞,−94.2501373603978]\left(-\infty, -94.2501373603978\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(xsin⁡(3x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x \sin{\left(3 x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(xsin⁡(3x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(x \sin{\left(3 x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función x*sin(3*x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞sin⁡(3x)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(3 x \right)} = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=⟨−1,1⟩xy = \left\langle -1, 1\right\rangle x
lim⁡x→∞sin⁡(3x)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(3 x \right)} = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=⟨−1,1⟩xy = \left\langle -1, 1\right\rangle 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)=xsin⁡(3x)x \sin{\left(3 x \right)} = x \sin{\left(3 x \right)}
- Sí
xsin⁡(3x)=−xsin⁡(3x)x \sin{\left(3 x \right)} = - x \sin{\left(3 x \right)}
- No
es decir, función
es
par
Gráfico
Gráfico de la función y = x*sin(3*x)
    © Sr Examen – Калькулятор