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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        2         
f(x) = x *cos(3*x)
f(x)=x2cos⁡(3x)f{\left(x \right)} = x^{2} \cos{\left(3 x \right)} 
f = x^2*cos(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:
x2cos⁡(3x)=0x^{2} \cos{\left(3 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=−5π6x_{2} = - \frac{5 \pi}{6}
x3=−π2x_{3} = - \frac{\pi}{2}
x4=−π6x_{4} = - \frac{\pi}{6}
x5=π6x_{5} = \frac{\pi}{6}
x6=π2x_{6} = \frac{\pi}{2}
x7=5π6x_{7} = \frac{5 \pi}{6}
Solución numérica
x1=−27.7507351067098x_{1} = -27.7507351067098
x2=−47.6474885794452x_{2} = -47.6474885794452
x3=27.7507351067098x_{3} = 27.7507351067098
x4=73.8274273593601x_{4} = 73.8274273593601
x5=−31.9395253114962x_{5} = -31.9395253114962
x6=34.0339204138894x_{6} = 34.0339204138894
x7=−89.5353906273091x_{7} = -89.5353906273091
x8=−41.3643032722656x_{8} = -41.3643032722656
x9=42.4115008234622x_{9} = 42.4115008234622
x10=−5.75958653158129x_{10} = -5.75958653158129
x11=14.1371669411541x_{11} = 14.1371669411541
x12=31.9395253114962x_{12} = 31.9395253114962
x13=−1.5707963267949x_{13} = -1.5707963267949
x14=40.317105721069x_{14} = 40.317105721069
x15=44.5058959258554x_{15} = 44.5058959258554
x16=−67.5442420521806x_{16} = -67.5442420521806
x17=16.2315620435473x_{17} = 16.2315620435473
x18=−100.007366139275x_{18} = -100.007366139275
x19=5.75958653158129x_{19} = 5.75958653158129
x20=89.5353906273091x_{20} = 89.5353906273091
x21=−78.0162175641465x_{21} = -78.0162175641465
x22=−80.1106126665397x_{22} = -80.1106126665397
x23=36.1283155162826x_{23} = 36.1283155162826
x24=−91.6297857297023x_{24} = -91.6297857297023
x25=−60.2138591938044x_{25} = -60.2138591938044
x26=95.8185759344887x_{26} = 95.8185759344887
x27=−3.66519142918809x_{27} = -3.66519142918809
x28=−34.0339204138894x_{28} = -34.0339204138894
x29=71.733032256967x_{29} = 71.733032256967
x30=−69.6386371545737x_{30} = -69.6386371545737
x31=92.6769832808989x_{31} = 92.6769832808989
x32=−97.9129710368819x_{32} = -97.9129710368819
x33=−12.0427718387609x_{33} = -12.0427718387609
x34=60.2138591938044x_{34} = 60.2138591938044
x35=75.9218224617533x_{35} = 75.9218224617533
x36=78.0162175641465x_{36} = 78.0162175641465
x37=−29.845130209103x_{37} = -29.845130209103
x38=0x_{38} = 0
x39=−38.2227106186758x_{39} = -38.2227106186758
x40=−75.9218224617533x_{40} = -75.9218224617533
x41=23.5619449019235x_{41} = 23.5619449019235
x42=20.4203522483337x_{42} = 20.4203522483337
x43=−25.6563400043166x_{43} = -25.6563400043166
x44=3.66519142918809x_{44} = 3.66519142918809
x45=−87.4409955249159x_{45} = -87.4409955249159
x46=53.9306738866248x_{46} = 53.9306738866248
x47=38.2227106186758x_{47} = 38.2227106186758
x48=−32.9867228626928x_{48} = -32.9867228626928
x49=−71.733032256967x_{49} = -71.733032256967
x50=7.85398163397448x_{50} = 7.85398163397448
x51=−17.2787595947439x_{51} = -17.2787595947439
x52=12.0427718387609x_{52} = 12.0427718387609
x53=−63.3554518473942x_{53} = -63.3554518473942
x54=26.7035375555132x_{54} = 26.7035375555132
x55=64.4026493985908x_{55} = 64.4026493985908
x56=−9.94837673636768x_{56} = -9.94837673636768
x57=−14.1371669411541x_{57} = -14.1371669411541
x58=−51.8362787842316x_{58} = -51.8362787842316
x59=−16.2315620435473x_{59} = -16.2315620435473
x60=9.94837673636768x_{60} = 9.94837673636768
x61=58.1194640914112x_{61} = 58.1194640914112
x62=0.523598775598299x_{62} = 0.523598775598299
x63=49.7418836818384x_{63} = 49.7418836818384
x64=18.3259571459405x_{64} = 18.3259571459405
x65=−95.8185759344887x_{65} = -95.8185759344887
x66=−36.1283155162826x_{66} = -36.1283155162826
x67=86.3937979737193x_{67} = 86.3937979737193
x68=62.3082542961976x_{68} = 62.3082542961976
x69=−49.7418836818384x_{69} = -49.7418836818384
x70=97.9129710368819x_{70} = 97.9129710368819
x71=84.2994028713261x_{71} = 84.2994028713261
x72=56.025068989018x_{72} = 56.025068989018
x73=−23.5619449019235x_{73} = -23.5619449019235
x74=66.497044500984x_{74} = 66.497044500984
x75=−82.2050077689329x_{75} = -82.2050077689329
x76=−7.85398163397448x_{76} = -7.85398163397448
x77=67.5442420521806x_{77} = 67.5442420521806
x78=80.1106126665397x_{78} = 80.1106126665397
x79=−93.7241808320955x_{79} = -93.7241808320955
x80=−53.9306738866248x_{80} = -53.9306738866248
x81=29.845130209103x_{81} = 29.845130209103
x82=−73.8274273593601x_{82} = -73.8274273593601
x83=−121.998514714404x_{83} = -121.998514714404
x84=51.8362787842316x_{84} = 51.8362787842316
x85=100.007366139275x_{85} = 100.007366139275
x86=−45.553093477052x_{86} = -45.553093477052
x87=82.2050077689329x_{87} = 82.2050077689329
x88=−2.61799387799149x_{88} = -2.61799387799149
x89=−43.4586983746588x_{89} = -43.4586983746588
x90=22.5147473507269x_{90} = 22.5147473507269
x91=−84.2994028713261x_{91} = -84.2994028713261
x92=−65.4498469497874x_{92} = -65.4498469497874
x93=88.4881930761125x_{93} = 88.4881930761125
x94=−58.1194640914112x_{94} = -58.1194640914112
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*cos(3*x).
02cos⁡(0⋅3)0^{2} \cos{\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
−3x2sin⁡(3x)+2xcos⁡(3x)=0- 3 x^{2} \sin{\left(3 x \right)} + 2 x \cos{\left(3 x \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=65.9768137973912x_{4} = 65.9768137973912
x5=6.31822725550968x_{5} = 6.31822725550968
x6=−35.6109562885689x_{6} = -35.6109562885689
x7=−4.24076625725554x_{7} = -4.24076625725554
x8=61.7882518935787x_{8} = 61.7882518935787
x9=17.8148265565878x_{9} = 17.8148265565878
x10=−72.2597062722654x_{10} = -72.2597062722654
x11=59.6939829539286x_{11} = 59.6939829539286
x12=−87.967120448543x_{12} = -87.967120448543
x13=−81.6841294396421x_{13} = -81.6841294396421
x14=−2.19277791090745x_{14} = -2.19277791090745
x15=−17.8148265565878x_{15} = -17.8148265565878
x16=−51.3170101463671x_{16} = -51.3170101463671
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=52.3641211184374x_{23} = 52.3641211184374
x24=−59.6939829539286x_{24} = -59.6939829539286
x25=54.4583530486096x_{25} = 54.4583530486096
x26=35.6109562885689x_{26} = 35.6109562885689
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=−76.4483279927407x_{36} = -76.4483279927407
x37=−28.2821897477697x_{37} = -28.2821897477697
x38=100.533175319193x_{38} = 100.533175319193
x39=92.1557958386718x_{39} = 92.1557958386718
x40=0x_{40} = 0
x41=−5.27787047164924x_{41} = -5.27787047164924
x42=48.1756998057147x_{42} = 48.1756998057147
x43=−79.5898059196778x_{43} = -79.5898059196778
x44=39.7990900239077x_{44} = 39.7990900239077
x45=8.40396768807019x_{45} = 8.40396768807019
x46=87.967120448543x_{46} = 87.967120448543
x47=−24.0947641678942x_{47} = -24.0947641678942
x48=−54.4583530486096x_{48} = -54.4583530486096
x49=32.4699670568908x_{49} = 32.4699670568908
x50=−46.0815142886463x_{50} = -46.0815142886463
x51=−92.1557958386718x_{51} = -92.1557958386718
x52=−39.7990900239077x_{52} = -39.7990900239077
x53=90.0614568087362x_{53} = 90.0614568087362
x54=63.8825291038655x_{54} = 63.8825291038655
x55=−65.9768137973912x_{55} = -65.9768137973912
x56=−41.8932060932537x_{56} = -41.8932060932537
x57=2.19277791090745x_{57} = 2.19277791090745
x58=19.9079118108102x_{58} = 19.9079118108102
x59=−13.6298592553469x_{59} = -13.6298592553469
x60=26.1884224615332x_{60} = 26.1884224615332
x61=24.0947641678942x_{61} = 24.0947641678942
x62=46.0815142886463x_{62} = 46.0815142886463
x63=−99.4860010336778x_{63} = -99.4860010336778
x64=−94.2501373603978x_{64} = -94.2501373603978
x65=12.5840132115367x_{65} = 12.5840132115367
x66=15.7220892009256x_{66} = 15.7220892009256
x67=56.5525970612491x_{67} = 56.5525970612491
x68=−85.8727869534015x_{68} = -85.8727869534015
x69=98.4388272431212x_{69} = 98.4388272431212
x70=4.24076625725554x_{70} = 4.24076625725554
x71=3.20985344776581x_{71} = 3.20985344776581
x72=41.8932060932537x_{72} = 41.8932060932537
x73=−30.3760435170464x_{73} = -30.3760435170464
x74=−50.2699027802066x_{74} = -50.2699027802066
x75=−83.7784565381466x_{75} = -83.7784565381466
x76=76.4483279927407x_{76} = 76.4483279927407
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=96.3444812114328x_{85} = 96.3444812114328
x86=22.0012459236092x_{86} = 22.0012459236092
x87=−34.5639476991297x_{87} = -34.5639476991297
x88=83.7784565381466x_{88} = 83.7784565381466
x89=43.9873487211332x_{89} = 43.9873487211332
x90=85.8727869534015x_{90} = 85.8727869534015
x91=−33.5169509084747x_{91} = -33.5169509084747
x92=−68.0711052836269x_{92} = -68.0711052836269
x93=74.3540147599617x_{93} = 74.3540147599617
x94=−7.36049192242691x_{94} = -7.36049192242691
Signos de extremos en los puntos:
(-57.5997231867839, -3317.50591129616)

(-11.538411018410194, -132.913261447715)

(94.25013736039777, 8882.86617857006)

(65.97681379739119, -4352.71775364898)

(6.318227255509681, 39.6996119436951)

(-35.61095628856888, 1267.91804395867)

(-4.240766257255545, 17.7659120606034)

(61.788251893578696, -3817.56586924258)

(17.81482655658785, -317.146056148211)

(-72.25970627226545, -5221.2429425173)

(59.69398295392857, -3563.14939946717)

(-87.96712044854304, 7737.9920673583)

(-81.68412943964212, 6672.0747911911)

(-2.1927779109074463, 4.60036024565405)

(-17.81482655658785, -317.146056148211)

(-51.317010146367075, -2633.21333626447)

(-90.0614568087362, 8110.84378942169)

(-77.49548626833665, 6005.32818207724)

(-70.16540295446222, -4922.96156458467)

(37.70500493524829, 1421.44522703497)

(-9.448258956595456, -89.0482014403384)

(78.5426455910908, -6168.72496623232)

(52.364121118437446, 2741.77898529512)

(-59.69398295392857, -3563.14939946717)

(54.45835304860955, 2965.49001951849)

(35.61095628856888, 1267.91804395867)

(-61.788251893578696, -3817.56586924258)

(-43.987348721133195, 1934.66466356844)

(50.2699027802066, 2526.84093261722)

(-19.907911810810152, -396.102917172654)

(-22.001245923609222, -483.832752880189)

(72.25970627226545, -5221.2429425173)

(-37.70500493524829, 1421.44522703497)

(-15.722089200925566, -246.962165843019)

(-26.18842246153318, -685.611356749139)

(-76.44832799274067, -5844.12464333712)

(-28.282189747769714, -799.660127269992)

(100.53317531919265, 10106.6971248661)

(92.1557958386718, 8492.46849315845)

(0, 0)

(-5.27787047164924, -27.6363188099095)

(48.17569980571475, 2320.67586145916)

(-79.58980591967777, 6334.31499580275)

(39.79909002390769, 1583.74539126285)

(8.403967688070194, 70.4054940215946)

(87.96712044854304, 7737.9920673583)

(-24.094764167894162, -580.335565594115)

(-54.45835304860955, 2965.49001951849)

(32.46996705689075, -1054.07660868777)

(-46.08151428864634, 2123.28377178932)

(-92.1557958386718, 8492.46849315845)

(-39.79909002390769, 1583.74539126285)

(90.0614568087362, 8110.84378942169)

(63.88252910386553, -4080.75532063342)

(-65.97681379739119, -4352.71775364898)

(-41.89320609325366, 1754.81853674717)

(2.1927779109074463, 4.60036024565405)

(19.907911810810152, -396.102917172654)

(-13.629859255346949, -185.551239039262)

(26.18842246153318, -685.611356749139)

(24.094764167894162, -580.335565594115)

(46.08151428864634, 2123.28377178932)

(-99.48600103367775, -9897.24218693458)

(-94.25013736039777, 8882.86617857006)

(12.58401321153674, 158.135632959759)

(15.722089200925566, -246.962165843019)

(56.55259706124914, 3197.9740353083)

(-85.87278695340147, 7373.91332696667)

(98.4388272431212, 9689.98049442277)

(4.240766257255545, 17.7659120606034)

(3.20985344776581, -10.0878773357935)

(41.89320609325366, 1754.81853674717)

(-30.376043517046433, -922.481877774411)

(-50.2699027802066, 2526.84093261722)

(-83.7784565381466, 7018.60756824496)

(76.44832799274067, -5844.12464333712)

(-48.17569980571475, 2320.67586145916)

(-63.88252910386553, -4080.75532063342)

(68.07110528362693, -4633.45316829715)

(-55.50547363006835, -3080.63540471643)

(28.282189747769714, -799.660127269992)

(10.493124973438015, 109.884119985328)

(70.16540295446222, -4922.96156458467)

(30.376043517046433, -922.481877774411)

(96.34448121143284, 9282.03684565777)

(22.001245923609222, -483.832752880189)

(-34.56394769912969, -1194.44432031071)

(83.7784565381466, 7018.60756824496)

(43.987348721133195, 1934.66466356844)

(85.87278695340147, 7373.91332696667)

(-33.51695090847467, 1123.16384189537)

(-68.07110528362693, -4633.45316829715)

(74.3540147599617, -5528.29730210002)

(-7.360491922426915, -53.9559771019712)


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=−57.5997231867839x_{1} = -57.5997231867839
x2=−11.5384110184102x_{2} = -11.5384110184102
x3=65.9768137973912x_{3} = 65.9768137973912
x4=61.7882518935787x_{4} = 61.7882518935787
x5=17.8148265565878x_{5} = 17.8148265565878
x6=−72.2597062722654x_{6} = -72.2597062722654
x7=59.6939829539286x_{7} = 59.6939829539286
x8=−17.8148265565878x_{8} = -17.8148265565878
x9=−51.3170101463671x_{9} = -51.3170101463671
x10=−70.1654029544622x_{10} = -70.1654029544622
x11=−9.44825895659546x_{11} = -9.44825895659546
x12=78.5426455910908x_{12} = 78.5426455910908
x13=−59.6939829539286x_{13} = -59.6939829539286
x14=−61.7882518935787x_{14} = -61.7882518935787
x15=−19.9079118108102x_{15} = -19.9079118108102
x16=−22.0012459236092x_{16} = -22.0012459236092
x17=72.2597062722654x_{17} = 72.2597062722654
x18=−15.7220892009256x_{18} = -15.7220892009256
x19=−26.1884224615332x_{19} = -26.1884224615332
x20=−76.4483279927407x_{20} = -76.4483279927407
x21=−28.2821897477697x_{21} = -28.2821897477697
x22=0x_{22} = 0
x23=−5.27787047164924x_{23} = -5.27787047164924
x24=−24.0947641678942x_{24} = -24.0947641678942
x25=32.4699670568908x_{25} = 32.4699670568908
x26=63.8825291038655x_{26} = 63.8825291038655
x27=−65.9768137973912x_{27} = -65.9768137973912
x28=19.9079118108102x_{28} = 19.9079118108102
x29=−13.6298592553469x_{29} = -13.6298592553469
x30=26.1884224615332x_{30} = 26.1884224615332
x31=24.0947641678942x_{31} = 24.0947641678942
x32=−99.4860010336778x_{32} = -99.4860010336778
x33=15.7220892009256x_{33} = 15.7220892009256
x34=3.20985344776581x_{34} = 3.20985344776581
x35=−30.3760435170464x_{35} = -30.3760435170464
x36=76.4483279927407x_{36} = 76.4483279927407
x37=−63.8825291038655x_{37} = -63.8825291038655
x38=68.0711052836269x_{38} = 68.0711052836269
x39=−55.5054736300684x_{39} = -55.5054736300684
x40=28.2821897477697x_{40} = 28.2821897477697
x41=70.1654029544622x_{41} = 70.1654029544622
x42=30.3760435170464x_{42} = 30.3760435170464
x43=22.0012459236092x_{43} = 22.0012459236092
x44=−34.5639476991297x_{44} = -34.5639476991297
x45=−68.0711052836269x_{45} = -68.0711052836269
x46=74.3540147599617x_{46} = 74.3540147599617
x47=−7.36049192242691x_{47} = -7.36049192242691
Puntos máximos de la función:
x47=94.2501373603978x_{47} = 94.2501373603978
x47=6.31822725550968x_{47} = 6.31822725550968
x47=−35.6109562885689x_{47} = -35.6109562885689
x47=−4.24076625725554x_{47} = -4.24076625725554
x47=−87.967120448543x_{47} = -87.967120448543
x47=−81.6841294396421x_{47} = -81.6841294396421
x47=−2.19277791090745x_{47} = -2.19277791090745
x47=−90.0614568087362x_{47} = -90.0614568087362
x47=−77.4954862683366x_{47} = -77.4954862683366
x47=37.7050049352483x_{47} = 37.7050049352483
x47=52.3641211184374x_{47} = 52.3641211184374
x47=54.4583530486096x_{47} = 54.4583530486096
x47=35.6109562885689x_{47} = 35.6109562885689
x47=−43.9873487211332x_{47} = -43.9873487211332
x47=50.2699027802066x_{47} = 50.2699027802066
x47=−37.7050049352483x_{47} = -37.7050049352483
x47=100.533175319193x_{47} = 100.533175319193
x47=92.1557958386718x_{47} = 92.1557958386718
x47=48.1756998057147x_{47} = 48.1756998057147
x47=−79.5898059196778x_{47} = -79.5898059196778
x47=39.7990900239077x_{47} = 39.7990900239077
x47=8.40396768807019x_{47} = 8.40396768807019
x47=87.967120448543x_{47} = 87.967120448543
x47=−54.4583530486096x_{47} = -54.4583530486096
x47=−46.0815142886463x_{47} = -46.0815142886463
x47=−92.1557958386718x_{47} = -92.1557958386718
x47=−39.7990900239077x_{47} = -39.7990900239077
x47=90.0614568087362x_{47} = 90.0614568087362
x47=−41.8932060932537x_{47} = -41.8932060932537
x47=2.19277791090745x_{47} = 2.19277791090745
x47=46.0815142886463x_{47} = 46.0815142886463
x47=−94.2501373603978x_{47} = -94.2501373603978
x47=12.5840132115367x_{47} = 12.5840132115367
x47=56.5525970612491x_{47} = 56.5525970612491
x47=−85.8727869534015x_{47} = -85.8727869534015
x47=98.4388272431212x_{47} = 98.4388272431212
x47=4.24076625725554x_{47} = 4.24076625725554
x47=41.8932060932537x_{47} = 41.8932060932537
x47=−50.2699027802066x_{47} = -50.2699027802066
x47=−83.7784565381466x_{47} = -83.7784565381466
x47=−48.1756998057147x_{47} = -48.1756998057147
x47=10.493124973438x_{47} = 10.493124973438
x47=96.3444812114328x_{47} = 96.3444812114328
x47=83.7784565381466x_{47} = 83.7784565381466
x47=43.9873487211332x_{47} = 43.9873487211332
x47=85.8727869534015x_{47} = 85.8727869534015
x47=−33.5169509084747x_{47} = -33.5169509084747
Decrece en los intervalos
[78.5426455910908,∞)\left[78.5426455910908, \infty\right)
Crece en los intervalos
(−∞,−99.4860010336778]\left(-\infty, -99.4860010336778\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
−9x2cos⁡(3x)−12xsin⁡(3x)+2cos⁡(3x)=0- 9 x^{2} \cos{\left(3 x \right)} - 12 x \sin{\left(3 x \right)} + 2 \cos{\left(3 x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=17.3044117895173x_{1} = 17.3044117895173
x2=−84.3046744803925x_{2} = -84.3046744803925
x3=−16.2588593807667x_{3} = -16.2588593807667
x4=97.9175098295664x_{4} = 97.9175098295664
x5=31.95342940113x_{5} = 31.95342940113
x6=−82.2104136545356x_{6} = -82.2104136545356
x7=47.6568129963247x_{7} = 47.6568129963247
x8=−29.8600083023031x_{8} = -29.8600083023031
x9=3.77988002313343x_{9} = 3.77988002313343
x10=−78.021913622938x_{10} = -78.021913622938
x11=14.1684778276527x_{11} = 14.1684778276527
x12=−49.7508157556872x_{12} = -49.7508157556872
x13=90.5874942021338x_{13} = 90.5874942021338
x14=−8.95060704504339x_{14} = -8.95060704504339
x15=−25.6736416013905x_{15} = -25.6736416013905
x16=82.2104136545356x_{16} = 82.2104136545356
x17=−93.7289224390323x_{17} = -93.7289224390323
x18=−51.8448501894803x_{18} = -51.8448501894803
x19=0.199913807009275x_{19} = 0.199913807009275
x20=56.0329998874158x_{20} = 56.0329998874158
x21=88.493215194763x_{21} = 88.493215194763
x22=−65.4566362695214x_{22} = -65.4566362695214
x23=−5.83494568158038x_{23} = -5.83494568158038
x24=42.4219754185456x_{24} = 42.4219754185456
x25=73.8334465048004x_{25} = 73.8334465048004
x26=−56.0329998874158x_{26} = -56.0329998874158
x27=39.2812198811439x_{27} = 39.2812198811439
x28=78.021913622938x_{28} = 78.021913622938
x29=−63.3624655178403x_{29} = -63.3624655178403
x30=−21.4882164075572x_{30} = -21.4882164075572
x31=36.1406096777871x_{31} = 36.1406096777871
x32=12.0794723044888x_{32} = 12.0794723044888
x33=16.2588593807667x_{33} = 16.2588593807667
x34=51.8448501894803x_{34} = 51.8448501894803
x35=−9.99268998886934x_{35} = -9.99268998886934
x36=−69.6450182357207x_{36} = -69.6450182357207
x37=66.5037269419935x_{37} = 66.5037269419935
x38=−97.9175098295664x_{38} = -97.9175098295664
x39=−27.7667337891131x_{39} = -27.7667337891131
x40=−89.5403540197371x_{40} = -89.5403540197371
x41=−71.7392270890376x_{41} = -71.7392270890376
x42=−91.6346356954313x_{42} = -91.6346356954313
x43=29.8600083023031x_{43} = 29.8600083023031
x44=1.79524321800934x_{44} = 1.79524321800934
x45=−40.3281239196866x_{45} = -40.3281239196866
x46=−58.1271093320215x_{46} = -58.1271093320215
x47=100.011809894359x_{47} = 100.011809894359
x48=60.2212386345931x_{48} = 60.2212386345931
x49=−3.77988002313343x_{49} = -3.77988002313343
x50=7.90984185417305x_{50} = 7.90984185417305
x51=22.5344558708549x_{51} = 22.5344558708549
x52=−2.77132823698164x_{52} = -2.77132823698164
x53=95.8232139183403x_{53} = 95.8232139183403
x54=84.3046744803925x_{54} = 84.3046744803925
x55=−38.234331899335x_{55} = -38.234331899335
x56=−53.938912611874x_{56} = -53.938912611874
x57=−1.79524321800934x_{57} = -1.79524321800934
x58=−12.0794723044888x_{58} = -12.0794723044888
x59=64.4095490707657x_{59} = 64.4095490707657
x60=38.234331899335x_{60} = 38.234331899335
x61=58.1271093320215x_{61} = 58.1271093320215
x62=−43.4689207900395x_{62} = -43.4689207900395
x63=71.7392270890376x_{63} = 71.7392270890376
x64=62.3153857944182x_{64} = 62.3153857944182
x65=27.7667337891131x_{65} = 27.7667337891131
x66=−47.6568129963247x_{66} = -47.6568129963247
x67=93.7289224390323x_{67} = 93.7289224390323
x68=40.3281239196866x_{68} = 40.3281239196866
x69=−67.5508209267318x_{69} = -67.5508209267318
x70=−60.2212386345931x_{70} = -60.2212386345931
x71=−45.5628462738552x_{71} = -45.5628462738552
x72=−14.1684778276527x_{72} = -14.1684778276527
x73=75.9276756093914x_{73} = 75.9276756093914
x74=9.99268998886934x_{74} = 9.99268998886934
x75=−23.5807801067948x_{75} = -23.5807801067948
x76=−75.9276756093914x_{76} = -75.9276756093914
x77=−4.80347720658609x_{77} = -4.80347720658609
x78=−34.0469701077371x_{78} = -34.0469701077371
x79=20.4420746646229x_{79} = 20.4420746646229
x80=−31.95342940113x_{80} = -31.95342940113
x81=53.938912611874x_{81} = 53.938912611874
x82=−100.011809894359x_{82} = -100.011809894359
x83=86.3989418144419x_{83} = 86.3989418144419
x84=49.7508157556872x_{84} = 49.7508157556872
x85=−87.4460777759607x_{85} = -87.4460777759607
x86=5.83494568158038x_{86} = 5.83494568158038
x87=−80.1161598470679x_{87} = -80.1161598470679
x88=−7.90984185417305x_{88} = -7.90984185417305
x89=44.5158780138332x_{89} = 44.5158780138332
x90=80.1161598470679x_{90} = 80.1161598470679
x91=−73.8334465048004x_{91} = -73.8334465048004
x92=−36.1406096777871x_{92} = -36.1406096777871
x93=24.6271783553447x_{93} = 24.6271783553447
x94=34.0469701077371x_{94} = 34.0469701077371
x95=18.3501507736041x_{95} = 18.3501507736041
x96=−95.8232139183403x_{96} = -95.8232139183403

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
[90.5874942021338,∞)\left[90.5874942021338, \infty\right)
Convexa en los intervalos
(−∞,−100.011809894359]\left(-\infty, -100.011809894359\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(x2cos⁡(3x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x^{2} \cos{\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→∞(x2cos⁡(3x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(x^{2} \cos{\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*cos(3*x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(xcos⁡(3x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x \cos{\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→∞(xcos⁡(3x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(x \cos{\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:
x2cos⁡(3x)=x2cos⁡(3x)x^{2} \cos{\left(3 x \right)} = x^{2} \cos{\left(3 x \right)}
- Sí
x2cos⁡(3x)=−x2cos⁡(3x)x^{2} \cos{\left(3 x \right)} = - x^{2} \cos{\left(3 x \right)}
- No
es decir, función
es
par
Gráfico
Gráfico de la función y = x^2*cos(3*x)
    © Sr Examen – Калькулятор