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x^2*sin(3*x)
Trazado el gráfico de la función/ x^2*sin(3*x)

Gráfico de la función y = x^2*sin(3*x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        2         
f(x) = x *sin(3*x)
f(x)=x2sin⁡(3x)f{\left(x \right)} = x^{2} \sin{\left(3 x \right)} 
f = x^2*sin(3*x)
Gráfico de la función
02468-8-6-4-2-1010-200200
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:
x2sin⁡(3x)=0x^{2} \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=14.6607657167524x_{8} = 14.6607657167524
x9=26.1799387799149x_{9} = 26.1799387799149
x10=52.3598775598299x_{10} = 52.3598775598299
x11=−9.42477796076938x_{11} = -9.42477796076938
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=−84.8230016469244x_{16} = -84.8230016469244
x17=−2.0943951023932x_{17} = -2.0943951023932
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=61.7846555205993x_{34} = 61.7846555205993
x35=100.530964914873x_{35} = 100.530964914873
x36=−85.870199198121x_{36} = -85.870199198121
x37=−72.2566310325652x_{37} = -72.2566310325652
x38=21.9911485751286x_{38} = 21.9911485751286
x39=−79.5870138909414x_{39} = -79.5870138909414
x40=−37.6991118430775x_{40} = -37.6991118430775
x41=−81.6814089933346x_{41} = -81.6814089933346
x42=−21.9911485751286x_{42} = -21.9911485751286
x43=−46.0766922526503x_{43} = -46.0766922526503
x44=−13.6135681655558x_{44} = -13.6135681655558
x45=−4.18879020478639x_{45} = -4.18879020478639
x46=−87.9645943005142x_{46} = -87.9645943005142
x47=−3.14159265358979x_{47} = -3.14159265358979
x48=−77.4926187885482x_{48} = -77.4926187885482
x49=41.8879020478639x_{49} = 41.8879020478639
x50=−99.4837673636768x_{50} = -99.4837673636768
x51=1.0471975511966x_{51} = 1.0471975511966
x52=−94.2477796076938x_{52} = -94.2477796076938
x53=−80.634211442138x_{53} = -80.634211442138
x54=15.707963267949x_{54} = 15.707963267949
x55=30.3687289847013x_{55} = 30.3687289847013
x56=−43.9822971502571x_{56} = -43.9822971502571
x57=39.7935069454707x_{57} = 39.7935069454707
x58=−83.7758040957278x_{58} = -83.7758040957278
x59=−28.2743338823081x_{59} = -28.2743338823081
x60=−95.2949771588904x_{60} = -95.2949771588904
x61=−17.8023583703422x_{61} = -17.8023583703422
x62=94.2477796076938x_{62} = 94.2477796076938
x63=−48.1710873550435x_{63} = -48.1710873550435
x64=46.0766922526503x_{64} = 46.0766922526503
x65=−59.6902604182061x_{65} = -59.6902604182061
x66=87.9645943005142x_{66} = 87.9645943005142
x67=−63.8790506229925x_{67} = -63.8790506229925
x68=−61.7846555205993x_{68} = -61.7846555205993
x69=63.8790506229925x_{69} = 63.8790506229925
x70=−92.1533845053006x_{70} = -92.1533845053006
x71=2.0943951023932x_{71} = 2.0943951023932
x72=74.3510261349584x_{72} = 74.3510261349584
x73=−15.707963267949x_{73} = -15.707963267949
x74=83.7758040957278x_{74} = 83.7758040957278
x75=−52.3598775598299x_{75} = -52.3598775598299
x76=96.342174710087x_{76} = 96.342174710087
x77=6.28318530717959x_{77} = 6.28318530717959
x78=97.3893722612836x_{78} = 97.3893722612836
x79=−50.2654824574367x_{79} = -50.2654824574367
x80=54.4542726622231x_{80} = 54.4542726622231
x81=−57.5958653158129x_{81} = -57.5958653158129
x82=37.6991118430775x_{82} = 37.6991118430775
x83=−19.8967534727354x_{83} = -19.8967534727354
x84=85.870199198121x_{84} = 85.870199198121
x85=43.9822971502571x_{85} = 43.9822971502571
x86=56.5486677646163x_{86} = 56.5486677646163
x87=−65.9734457253857x_{87} = -65.9734457253857
x88=−55.5014702134197x_{88} = -55.5014702134197
x89=76.4454212373516x_{89} = 76.4454212373516
x90=92.1533845053006x_{90} = 92.1533845053006
x91=70.162235930172x_{91} = 70.162235930172
x92=−35.6047167406843x_{92} = -35.6047167406843
x93=72.2566310325652x_{93} = 72.2566310325652
x94=10.471975511966x_{94} = 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^2*sin(3*x).
02sin⁡(0⋅3)0^{2} \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
3x2cos⁡(3x)+2xsin⁡(3x)=03 x^{2} \cos{\left(3 x \right)} + 2 x \sin{\left(3 x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−67.5475318097959x_{1} = -67.5475318097959
x2=−95.820895038653x_{2} = -95.820895038653
x3=−25.6649966297225x_{3} = -25.6649966297225
x4=−70.6889782741115x_{4} = -70.6889782741115
x5=12.0611776969175x_{5} = 12.0611776969175
x6=−93.7265517554948x_{6} = -93.7265517554948
x7=48.6992490006399x_{7} = 48.6992490006399
x8=88.4907042779379x_{8} = 88.4907042779379
x9=−69.6418279874538x_{9} = -69.6418279874538
x10=−23.5713700221828x_{10} = -23.5713700221828
x11=95.820895038653x_{11} = 95.820895038653
x12=−43.4638107841198x_{12} = -43.4638107841198
x13=73.8304371774155x_{13} = 73.8304371774155
x14=24.6181670049683x_{14} = 24.6181670049683
x15=58.1232872144417x_{15} = 58.1232872144417
x16=9.97063128985059x_{16} = 9.97063128985059
x17=−7.88210793986787x_{17} = -7.88210793986787
x18=51.8405651953147x_{18} = 51.8405651953147
x19=78.0190658014538x_{19} = 78.0190658014538
x20=−91.6322108409093x_{20} = -91.6322108409093
x21=86.3963700471954x_{21} = 86.3963700471954
x22=16.2452335983018x_{22} = 16.2452335983018
x23=−3.72423528944333x_{23} = -3.72423528944333
x24=31.946480380411x_{24} = 31.946480380411
x25=−65.4532419617293x_{25} = -65.4532419617293
x26=3.72423528944333x_{26} = 3.72423528944333
x27=−78.0190658014538x_{27} = -78.0190658014538
x28=−14.1528569238997x_{28} = -14.1528569238997
x29=−46.6050589026417x_{29} = -46.6050589026417
x30=−53.9347938783761x_{30} = -53.9347938783761
x31=5.79774798819825x_{31} = 5.79774798819825
x32=29.8525729609081x_{32} = 29.8525729609081
x33=−29.8525729609081x_{33} = -29.8525729609081
x34=71.7361299404533x_{34} = 71.7361299404533
x35=−36.1344646875895x_{35} = -36.1344646875895
x36=56.0290349994261x_{36} = 56.0290349994261
x37=−45.5579709190865x_{37} = -45.5579709190865
x38=−100.009588115537x_{38} = -100.009588115537
x39=23.5713700221828x_{39} = 23.5713700221828
x40=−12.0611776969175x_{40} = -12.0611776969175
x41=−21.4778930345772x_{41} = -21.4778930345772
x42=−89.5378724610825x_{42} = -89.5378724610825
x43=−47.6521516999475x_{43} = -47.6521516999475
x44=84.3020388406696x_{44} = 84.3020388406696
x45=−1.69566169803409x_{45} = -1.69566169803409
x46=68.5946791436526x_{46} = 68.5946791436526
x47=−16.2452335983018x_{47} = -16.2452335983018
x48=−41.3696744286036x_{48} = -41.3696744286036
x49=−87.4435367981187x_{49} = -87.4435367981187
x50=36.1344646875895x_{50} = 36.1344646875895
x51=0x_{51} = 0
x52=42.4167394139284x_{52} = 42.4167394139284
x53=−34.0404477609703x_{53} = -34.0404477609703
x54=62.3118204533444x_{54} = 62.3118204533444
x55=−5.79774798819825x_{55} = -5.79774798819825
x56=−71.7361299404533x_{56} = -71.7361299404533
x57=90.5850413231642x_{57} = 90.5850413231642
x58=7.88210793986787x_{58} = 7.88210793986787
x59=66.5003860571966x_{59} = 66.5003860571966
x60=38.2285230247401x_{60} = 38.2285230247401
x61=75.9247492611644x_{61} = 75.9247492611644
x62=−56.0290349994261x_{62} = -56.0290349994261
x63=80.1133864488351x_{63} = 80.1133864488351
x64=49.7463505204702x_{64} = 49.7463505204702
x65=−73.8304371774155x_{65} = -73.8304371774155
x66=14.1528569238997x_{66} = 14.1528569238997
x67=1.69566169803409x_{67} = 1.69566169803409
x68=−80.1133864488351x_{68} = -80.1133864488351
x69=−38.2285230247401x_{69} = -38.2285230247401
x70=53.9347938783761x_{70} = 53.9347938783761
x71=44.5108880888307x_{71} = 44.5108880888307
x72=−27.7587390549925x_{72} = -27.7587390549925
x73=64.4060996042015x_{73} = 64.4060996042015
x74=27.7587390549925x_{74} = 27.7587390549925
x75=18.338069892946x_{75} = 18.338069892946
x76=93.7265517554948x_{76} = 93.7265517554948
x77=−97.9152405384144x_{77} = -97.9152405384144
x78=20.4312249887476x_{78} = 20.4312249887476
x79=82.2077108894621x_{79} = 82.2077108894621
x80=26.7118550646915x_{80} = 26.7118550646915
x81=−75.9247492611644x_{81} = -75.9247492611644
x82=−51.8405651953147x_{82} = -51.8405651953147
x83=−9.97063128985059x_{83} = -9.97063128985059
x84=−82.2077108894621x_{84} = -82.2077108894621
x85=22.5246102236197x_{85} = 22.5246102236197
x86=−60.2175493662913x_{86} = -60.2175493662913
x87=−49.7463505204702x_{87} = -49.7463505204702
x88=−6.8391743033139x_{88} = -6.8391743033139
x89=−84.3020388406696x_{89} = -84.3020388406696
x90=−31.946480380411x_{90} = -31.946480380411
x91=40.3226163252311x_{91} = 40.3226163252311
x92=100.009588115537x_{92} = 100.009588115537
x93=−58.1232872144417x_{93} = -58.1232872144417
x94=97.9152405384144x_{94} = 97.9152405384144
x95=25.6649966297225x_{95} = 25.6649966297225
x96=60.2175493662913x_{96} = 60.2175493662913
x97=34.0404477609703x_{97} = 34.0404477609703
Signos de extremos en los puntos:
(-67.54753180979594, -4562.44684760667)

(-95.82089503865303, 9181.42171185364)

(-25.664996629722534, -658.469942174557)

(-70.68897827411146, 4996.70944203839)

(12.061177696917529, -145.250293119571)

(-93.72655175549478, 8784.44429018506)

(48.69924900063987, 2371.3946622328)

(88.49070427793794, 7830.38253084235)

(-69.64182798745382, -4849.76199848377)

(-23.571370022182833, -555.38739573202)

(95.82089503865303, -9181.42171185364)

(-43.463810784119765, 1888.8806648591)

(73.83043717741552, 5450.7112451744)

(24.61816700496828, -605.832046611167)

(58.12328721444166, -3378.09431631418)

(9.97063128985059, -99.1920084416929)

(-7.882107939867874, 61.9065885787803)

(51.84056519531469, -2687.22200510667)

(78.01906580145378, 6086.75241847789)

(-91.6322108409093, 8396.2398501923)

(86.39637004719542, 7464.11054503295)

(16.245233598301798, -263.685672729676)

(-3.724235289443328, 13.6529081682129)

(31.946480380411014, 1020.35545902797)

(-65.45324196172929, -4283.90467836732)

(3.724235289443328, -13.6529081682129)

(-78.01906580145378, -6086.75241847789)

(-14.152856923899682, 200.081506013128)

(-46.60505890264171, -2171.80932719426)

(-53.934793878376055, 2908.73979394154)

(5.7977479881982505, -33.3938391842218)

(29.85257296090814, 890.953973249049)

(-29.85257296090814, -890.953973249049)

(71.73612994045328, 5145.85013100463)

(-36.134464687589464, -1305.47737275167)

(56.029034999426095, -3139.03056433799)

(-45.55797091908653, 2075.306527725)

(-100.00958811553703, 10001.6955002229)

(23.571370022182833, 555.38739573202)

(-12.061177696917529, 145.250293119571)

(-21.47789303457719, -461.077827430531)

(-89.53787246108253, 8016.80839187402)

(-47.65215169994751, 2270.50537202856)

(84.30203884066964, 7106.61154089394)

(-1.6956616980340902, 2.67588446922981)

(68.59467914365256, -4705.0078003401)

(-16.245233598301798, 263.685672729676)

(-41.36967442860363, 1711.22778337855)

(-87.44353679811874, 7646.14991522874)

(36.134464687589464, 1305.47737275167)

(0, 0)

(42.416739413928404, 1798.95760144944)

(-34.04044776097026, -1158.52992545043)

(62.31182045334436, -3882.54076506345)

(-5.7977479881982505, 33.3938391842218)

(-71.73612994045328, -5145.85013100463)

(90.5850413231642, 8205.42749832394)

(7.882107939867874, -61.9065885787803)

(66.50038605719662, -4422.07914028269)

(38.22852302474006, 1461.19780110439)

(75.92474926116441, 5764.34534099753)

(-56.029034999426095, 3139.03056433799)

(80.11338644883509, 6417.93247761884)

(49.74635052047024, -2474.47719781134)

(-73.83043717741552, -5450.7112451744)

(14.152856923899682, -200.081506013128)

(1.6956616980340902, -2.67588446922981)

(-80.11338644883509, -6417.93247761884)

(-38.22852302474006, -1461.19780110439)

(53.934793878376055, -2908.73979394154)

(44.51088808883066, 1980.99697361532)

(-27.758739054992514, -770.325467786492)

(64.40609960420149, -4147.92346185966)

(27.758739054992514, 770.325467786492)

(18.33806989294604, -336.062805205876)

(93.72655175549478, -8784.44429018506)

(-97.91524053841435, 9587.17211519922)

(20.431224988747633, -417.212909611885)

(82.20771088946208, 6757.88551842333)

(26.711855064691484, -713.301082535532)

(-75.92474926116441, -5764.34534099753)

(-51.84056519531469, 2687.22200510667)

(-9.97063128985059, 99.1920084416929)

(-82.20771088946208, -6757.88551842333)

(22.524610223619664, -507.13598939687)

(-60.21754936629126, 3625.93104988514)

(-49.74635052047024, 2474.47719781134)

(-6.839174303313896, -46.5536541413815)

(-84.30203884066964, -7106.61154089394)

(-31.946480380411014, -1020.35545902797)

(40.322616325231095, 1625.69121063763)

(100.00958811553703, -10001.6955002229)

(-58.12328721444166, 3378.09431631418)

(97.91524053841435, -9587.17211519922)

(25.664996629722534, 658.469942174557)

(60.21754936629126, -3625.93104988514)

(34.04044776097026, 1158.52992545043)


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=−67.5475318097959x_{1} = -67.5475318097959
x2=−25.6649966297225x_{2} = -25.6649966297225
x3=12.0611776969175x_{3} = 12.0611776969175
x4=−69.6418279874538x_{4} = -69.6418279874538
x5=−23.5713700221828x_{5} = -23.5713700221828
x6=95.820895038653x_{6} = 95.820895038653
x7=24.6181670049683x_{7} = 24.6181670049683
x8=58.1232872144417x_{8} = 58.1232872144417
x9=9.97063128985059x_{9} = 9.97063128985059
x10=51.8405651953147x_{10} = 51.8405651953147
x11=16.2452335983018x_{11} = 16.2452335983018
x12=−65.4532419617293x_{12} = -65.4532419617293
x13=3.72423528944333x_{13} = 3.72423528944333
x14=−78.0190658014538x_{14} = -78.0190658014538
x15=−46.6050589026417x_{15} = -46.6050589026417
x16=5.79774798819825x_{16} = 5.79774798819825
x17=−29.8525729609081x_{17} = -29.8525729609081
x18=−36.1344646875895x_{18} = -36.1344646875895
x19=56.0290349994261x_{19} = 56.0290349994261
x20=−21.4778930345772x_{20} = -21.4778930345772
x21=68.5946791436526x_{21} = 68.5946791436526
x22=−34.0404477609703x_{22} = -34.0404477609703
x23=62.3118204533444x_{23} = 62.3118204533444
x24=−71.7361299404533x_{24} = -71.7361299404533
x25=7.88210793986787x_{25} = 7.88210793986787
x26=66.5003860571966x_{26} = 66.5003860571966
x27=49.7463505204702x_{27} = 49.7463505204702
x28=−73.8304371774155x_{28} = -73.8304371774155
x29=14.1528569238997x_{29} = 14.1528569238997
x30=1.69566169803409x_{30} = 1.69566169803409
x31=−80.1133864488351x_{31} = -80.1133864488351
x32=−38.2285230247401x_{32} = -38.2285230247401
x33=53.9347938783761x_{33} = 53.9347938783761
x34=−27.7587390549925x_{34} = -27.7587390549925
x35=64.4060996042015x_{35} = 64.4060996042015
x36=18.338069892946x_{36} = 18.338069892946
x37=93.7265517554948x_{37} = 93.7265517554948
x38=20.4312249887476x_{38} = 20.4312249887476
x39=26.7118550646915x_{39} = 26.7118550646915
x40=−75.9247492611644x_{40} = -75.9247492611644
x41=−82.2077108894621x_{41} = -82.2077108894621
x42=22.5246102236197x_{42} = 22.5246102236197
x43=−6.8391743033139x_{43} = -6.8391743033139
x44=−84.3020388406696x_{44} = -84.3020388406696
x45=−31.946480380411x_{45} = -31.946480380411
x46=100.009588115537x_{46} = 100.009588115537
x47=97.9152405384144x_{47} = 97.9152405384144
x48=60.2175493662913x_{48} = 60.2175493662913
Puntos máximos de la función:
x48=−95.820895038653x_{48} = -95.820895038653
x48=−70.6889782741115x_{48} = -70.6889782741115
x48=−93.7265517554948x_{48} = -93.7265517554948
x48=48.6992490006399x_{48} = 48.6992490006399
x48=88.4907042779379x_{48} = 88.4907042779379
x48=−43.4638107841198x_{48} = -43.4638107841198
x48=73.8304371774155x_{48} = 73.8304371774155
x48=−7.88210793986787x_{48} = -7.88210793986787
x48=78.0190658014538x_{48} = 78.0190658014538
x48=−91.6322108409093x_{48} = -91.6322108409093
x48=86.3963700471954x_{48} = 86.3963700471954
x48=−3.72423528944333x_{48} = -3.72423528944333
x48=31.946480380411x_{48} = 31.946480380411
x48=−14.1528569238997x_{48} = -14.1528569238997
x48=−53.9347938783761x_{48} = -53.9347938783761
x48=29.8525729609081x_{48} = 29.8525729609081
x48=71.7361299404533x_{48} = 71.7361299404533
x48=−45.5579709190865x_{48} = -45.5579709190865
x48=−100.009588115537x_{48} = -100.009588115537
x48=23.5713700221828x_{48} = 23.5713700221828
x48=−12.0611776969175x_{48} = -12.0611776969175
x48=−89.5378724610825x_{48} = -89.5378724610825
x48=−47.6521516999475x_{48} = -47.6521516999475
x48=84.3020388406696x_{48} = 84.3020388406696
x48=−1.69566169803409x_{48} = -1.69566169803409
x48=−16.2452335983018x_{48} = -16.2452335983018
x48=−41.3696744286036x_{48} = -41.3696744286036
x48=−87.4435367981187x_{48} = -87.4435367981187
x48=36.1344646875895x_{48} = 36.1344646875895
x48=42.4167394139284x_{48} = 42.4167394139284
x48=−5.79774798819825x_{48} = -5.79774798819825
x48=90.5850413231642x_{48} = 90.5850413231642
x48=38.2285230247401x_{48} = 38.2285230247401
x48=75.9247492611644x_{48} = 75.9247492611644
x48=−56.0290349994261x_{48} = -56.0290349994261
x48=80.1133864488351x_{48} = 80.1133864488351
x48=44.5108880888307x_{48} = 44.5108880888307
x48=27.7587390549925x_{48} = 27.7587390549925
x48=−97.9152405384144x_{48} = -97.9152405384144
x48=82.2077108894621x_{48} = 82.2077108894621
x48=−51.8405651953147x_{48} = -51.8405651953147
x48=−9.97063128985059x_{48} = -9.97063128985059
x48=−60.2175493662913x_{48} = -60.2175493662913
x48=−49.7463505204702x_{48} = -49.7463505204702
x48=40.3226163252311x_{48} = 40.3226163252311
x48=−58.1232872144417x_{48} = -58.1232872144417
x48=25.6649966297225x_{48} = 25.6649966297225
x48=34.0404477609703x_{48} = 34.0404477609703
Decrece en los intervalos
[100.009588115537,∞)\left[100.009588115537, \infty\right)
Crece en los intervalos
(−∞,−84.3020388406696]\left(-\infty, -84.3020388406696\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
−9x2sin⁡(3x)+12xcos⁡(3x)+2sin⁡(3x)=0- 9 x^{2} \sin{\left(3 x \right)} + 12 x \cos{\left(3 x \right)} + 2 \sin{\left(3 x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−91.1110647877441x_{1} = -91.1110647877441
x2=80.6397226110677x_{2} = 80.6397226110677
x3=81.686849523616x_{3} = 81.686849523616
x4=−37.7108943469214x_{4} = -37.7108943469214
x5=−15.7361646611065x_{5} = -15.7361646611065
x6=5.3184884765837x_{6} = 5.3184884765837
x7=37.7108943469214x_{7} = 37.7108943469214
x8=−70.1685694071259x_{8} = -70.1685694071259
x9=−48.1803104910586x_{9} = -48.1803104910586
x10=−61.7918474295701x_{10} = -61.7918474295701
x11=−39.8046699723864x_{11} = -39.8046699723864
x12=83.7811086447271x_{12} = 83.7811086447271
x13=59.6977045614732x_{13} = 59.6977045614732
x14=22.0113247907108x_{14} = 22.0113247907108
x15=−94.2524948772148x_{15} = -94.2524948772148
x16=50.2743215491169x_{16} = 50.2743215491169
x17=28.2900369005496x_{17} = 28.2900369005496
x18=−41.8985074546982x_{18} = -41.8985074546982
x19=−59.6977045614732x_{19} = -59.6977045614732
x20=−17.8272599960079x_{20} = -17.8272599960079
x21=19.9190452227554x_{21} = 19.9190452227554
x22=46.086334307675x_{22} = 46.086334307675
x23=−52.3683633021869x_{23} = -52.3683633021869
x24=−81.686849523616x_{24} = -81.686849523616
x25=−87.9696463064529x_{25} = -87.9696463064529
x26=17.8272599960079x_{26} = 17.8272599960079
x27=65.9801811818736x_{27} = 65.9801811818736
x28=94.2524948772148x_{28} = 94.2524948772148
x29=−83.7811086447271x_{29} = -83.7811086447271
x30=76.4512343061507x_{30} = 76.4512343061507
x31=−26.1968951731632x_{31} = -26.1968951731632
x32=−99.4882345031056x_{32} = -99.4882345031056
x33=92.1582069197077x_{33} = 92.1582069197077
x34=15.7361646611065x_{34} = 15.7361646611065
x35=−57.6035800246463x_{35} = -57.6035800246463
x36=−68.0743691134619x_{36} = -68.0743691134619
x37=−1.33148157191381x_{37} = -1.33148157191381
x38=48.1803104910586x_{38} = 48.1803104910586
x39=63.8860068273767x_{39} = 63.8860068273767
x40=40.8515815750295x_{40} = 40.8515815750295
x41=−92.1582069197077x_{41} = -92.1582069197077
x42=0x_{42} = 0
x43=39.8046699723864x_{43} = 39.8046699723864
x44=10.5141061239753x_{44} = 10.5141061239753
x45=−79.5925975567234x_{45} = -79.5925975567234
x46=61.7918474295701x_{46} = 61.7918474295701
x47=1.33148157191381x_{47} = 1.33148157191381
x48=43.9923979732211x_{48} = 43.9923979732211
x49=−65.9801811818736x_{49} = -65.9801811818736
x50=−78.5454744348727x_{50} = -78.5454744348727
x51=78.5454744348727x_{51} = 78.5454744348727
x52=70.1685694071259x_{52} = 70.1685694071259
x53=6.35251871791284x_{53} = 6.35251871791284
x54=4.29038019281278x_{54} = 4.29038019281278
x55=24.1039705794275x_{55} = 24.1039705794275
x56=32.4768042660169x_{56} = 32.4768042660169
x57=−33.5235749406152x_{57} = -33.5235749406152
x58=−22.0113247907108x_{58} = -22.0113247907108
x59=−63.8860068273767x_{59} = -63.8860068273767
x60=−24.1039705794275x_{60} = -24.1039705794275
x61=−11.5575220454079x_{61} = -11.5575220454079
x62=41.8985074546982x_{62} = 41.8985074546982
x63=100.535385529141x_{63} = 100.535385529141
x64=−90.0639239442159x_{64} = -90.0639239442159
x65=68.0743691134619x_{65} = 68.0743691134619
x66=26.1968951731632x_{66} = 26.1968951731632
x67=−46.086334307675x_{67} = -46.086334307675
x68=72.2627809886043x_{68} = 72.2627809886043
x69=−85.8753743968175x_{69} = -85.8753743968175
x70=8.43003016536006x_{70} = 8.43003016536006
x71=−13.6460730571718x_{71} = -13.6460730571718
x72=−77.4983533238184x_{72} = -77.4983533238184
x73=−9.47151023238751x_{73} = -9.47151023238751
x74=55.5094758922482x_{74} = 55.5094758922482
x75=54.4624322126712x_{75} = 54.4624322126712
x76=90.0639239442159x_{76} = 90.0639239442159
x77=−72.2627809886043x_{77} = -72.2627809886043
x78=85.8753743968175x_{78} = 85.8753743968175
x79=74.3570029045849x_{79} = 74.3570029045849
x80=−20.965132411557x_{80} = -20.965132411557
x81=−5.3184884765837x_{81} = -5.3184884765837
x82=30.3833510150596x_{82} = 30.3833510150596
x83=−35.6171914683959x_{83} = -35.6171914683959
x84=−55.5094758922482x_{84} = -55.5094758922482
x85=96.3467874919411x_{85} = 96.3467874919411
x86=−28.2900369005496x_{86} = -28.2900369005496
x87=35.6171914683959x_{87} = 35.6171914683959
x88=87.9696463064529x_{88} = 87.9696463064529
x89=98.4410844667206x_{89} = 98.4410844667206
x90=52.3683633021869x_{90} = 52.3683633021869
x91=−74.3570029045849x_{91} = -74.3570029045849
x92=−43.9923979732211x_{92} = -43.9923979732211
x93=−50.2743215491169x_{93} = -50.2743215491169
x94=−19.9190452227554x_{94} = -19.9190452227554
x95=2.27738191564373x_{95} = 2.27738191564373
x96=−2.27738191564373x_{96} = -2.27738191564373

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
[80.6397226110677,∞)\left[80.6397226110677, \infty\right)
Convexa en los intervalos
(−∞,−99.4882345031056]\left(-\infty, -99.4882345031056\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(x2sin⁡(3x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x^{2} \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→∞(x2sin⁡(3x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(x^{2} \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^2*sin(3*x), dividida por x 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 inclinada a la izquierda:
y=⟨−∞,∞⟩xy = \left\langle -\infty, \infty\right\rangle x
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 inclinada a la derecha:
y=⟨−∞,∞⟩xy = \left\langle -\infty, \infty\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:
x2sin⁡(3x)=−x2sin⁡(3x)x^{2} \sin{\left(3 x \right)} = - x^{2} \sin{\left(3 x \right)}
- No
x2sin⁡(3x)=x2sin⁡(3x)x^{2} \sin{\left(3 x \right)} = x^{2} \sin{\left(3 x \right)}
- Sí
es decir, función
es
impar
Gráfico
Gráfico de la función y = x^2*sin(3*x)
    © Sr Examen – Калькулятор