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Plotting a function graph/ x^3sinx

Graphing y = x^3sinx

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
        3       
f(x) = x *sin(x)
f(x)=x3sin⁡(x)f{\left(x \right)} = x^{3} \sin{\left(x \right)} 
f = x^3*sin(x)
The graph of the function
02468-8-6-4-2-1010-10001000
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
x3sin⁡(x)=0x^{3} \sin{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=πx_{2} = \pi
Numerical solution
x1=62.8318530717959x_{1} = 62.8318530717959
x2=−50.2654824574367x_{2} = -50.2654824574367
x3=47.1238898038469x_{3} = 47.1238898038469
x4=84.8230016469244x_{4} = 84.8230016469244
x5=−53.4070751110265x_{5} = -53.4070751110265
x6=91.106186954104x_{6} = 91.106186954104
x7=−84.8230016469244x_{7} = -84.8230016469244
x8=−135.088484104361x_{8} = -135.088484104361
x9=25.1327412287183x_{9} = 25.1327412287183
x10=−3.14159265358979x_{10} = -3.14159265358979
x11=−6.28318530717959x_{11} = -6.28318530717959
x12=−40.8407044966673x_{12} = -40.8407044966673
x13=−18.8495559215388x_{13} = -18.8495559215388
x14=78.5398163397448x_{14} = 78.5398163397448
x15=−75.398223686155x_{15} = -75.398223686155
x16=−9.42477796076938x_{16} = -9.42477796076938
x17=72.2566310325652x_{17} = 72.2566310325652
x18=−43.9822971502571x_{18} = -43.9822971502571
x19=31.4159265358979x_{19} = 31.4159265358979
x20=9.42477796076938x_{20} = 9.42477796076938
x21=40.8407044966673x_{21} = 40.8407044966673
x22=−69.1150383789755x_{22} = -69.1150383789755
x23=12.5663706143592x_{23} = 12.5663706143592
x24=87.9645943005142x_{24} = 87.9645943005142
x25=59.6902604182061x_{25} = 59.6902604182061
x26=−37.6991118430775x_{26} = -37.6991118430775
x27=−100.530964914873x_{27} = -100.530964914873
x28=−91.106186954104x_{28} = -91.106186954104
x29=97.3893722612836x_{29} = 97.3893722612836
x30=0x_{30} = 0
x31=−12.5663706143592x_{31} = -12.5663706143592
x32=−78.5398163397448x_{32} = -78.5398163397448
x33=18.8495559215388x_{33} = 18.8495559215388
x34=34.5575191894877x_{34} = 34.5575191894877
x35=−94.2477796076938x_{35} = -94.2477796076938
x36=43.9822971502571x_{36} = 43.9822971502571
x37=−31.4159265358979x_{37} = -31.4159265358979
x38=−81.6814089933346x_{38} = -81.6814089933346
x39=−65.9734457253857x_{39} = -65.9734457253857
x40=75.398223686155x_{40} = 75.398223686155
x41=56.5486677646163x_{41} = 56.5486677646163
x42=3.14159265358979x_{42} = 3.14159265358979
x43=15.707963267949x_{43} = 15.707963267949
x44=−56.5486677646163x_{44} = -56.5486677646163
x45=−21.9911485751286x_{45} = -21.9911485751286
x46=50.2654824574367x_{46} = 50.2654824574367
x47=−15.707963267949x_{47} = -15.707963267949
x48=28.2743338823081x_{48} = 28.2743338823081
x49=94.2477796076938x_{49} = 94.2477796076938
x50=−59.6902604182061x_{50} = -59.6902604182061
x51=−62.8318530717959x_{51} = -62.8318530717959
x52=69.1150383789755x_{52} = 69.1150383789755
x53=−34.5575191894877x_{53} = -34.5575191894877
x54=−97.3893722612836x_{54} = -97.3893722612836
x55=21.9911485751286x_{55} = 21.9911485751286
x56=65.9734457253857x_{56} = 65.9734457253857
x57=37.6991118430775x_{57} = 37.6991118430775
x58=−87.9645943005142x_{58} = -87.9645943005142
x59=−72.2566310325652x_{59} = -72.2566310325652
x60=−125.663706143592x_{60} = -125.663706143592
x61=−25.1327412287183x_{61} = -25.1327412287183
x62=−28.2743338823081x_{62} = -28.2743338823081
x63=81.6814089933346x_{63} = 81.6814089933346
x64=6.28318530717959x_{64} = 6.28318530717959
x65=100.530964914873x_{65} = 100.530964914873
x66=53.4070751110265x_{66} = 53.4070751110265
x67=−47.1238898038469x_{67} = -47.1238898038469
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x^3*sin(x).
03sin⁡(0)0^{3} \sin{\left(0 \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
x3cos⁡(x)+3x2sin⁡(x)=0x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=92.7093311956205x_{1} = 92.7093311956205
x2=0x_{2} = 0
x3=67.5885991217338x_{3} = 67.5885991217338
x4=−61.3099494475655x_{4} = -61.3099494475655
x5=−86.4284948180722x_{5} = -86.4284948180722
x6=77.0079573362515x_{6} = 77.0079573362515
x7=−17.4490243427188x_{7} = -17.4490243427188
x8=14.3433507883915x_{8} = 14.3433507883915
x9=−11.2560430143535x_{9} = -11.2560430143535
x10=55.0323309441547x_{10} = 55.0323309441547
x11=−14.3433507883915x_{11} = -14.3433507883915
x12=36.2109745555852x_{12} = 36.2109745555852
x13=95.8498646688189x_{13} = 95.8498646688189
x14=11.2560430143535x_{14} = 11.2560430143535
x15=8.20453136258127x_{15} = 8.20453136258127
x16=−2.45564386287944x_{16} = -2.45564386287944
x17=20.5652079398333x_{17} = 20.5652079398333
x18=−23.6879210560017x_{18} = -23.6879210560017
x19=89.5688718899173x_{19} = 89.5688718899173
x20=48.75613936684x_{20} = 48.75613936684
x21=−64.4491641378738x_{21} = -64.4491641378738
x22=2.45564386287944x_{22} = 2.45564386287944
x23=−83.2882092591146x_{23} = -83.2882092591146
x24=45.6187613383417x_{24} = 45.6187613383417
x25=33.0771723843072x_{25} = 33.0771723843072
x26=−36.2109745555852x_{26} = -36.2109745555852
x27=−80.1480259413025x_{27} = -80.1480259413025
x28=64.4491641378738x_{28} = 64.4491641378738
x29=58.170990540028x_{29} = 58.170990540028
x30=5.23293845351241x_{30} = 5.23293845351241
x31=86.4284948180722x_{31} = 86.4284948180722
x32=−20.5652079398333x_{32} = -20.5652079398333
x33=−26.814952130975x_{33} = -26.814952130975
x34=−8.20453136258127x_{34} = -8.20453136258127
x35=61.3099494475655x_{35} = 61.3099494475655
x36=83.2882092591146x_{36} = 83.2882092591146
x37=−89.5688718899173x_{37} = -89.5688718899173
x38=−95.8498646688189x_{38} = -95.8498646688189
x39=39.3460075465194x_{39} = 39.3460075465194
x40=70.7282251775385x_{40} = 70.7282251775385
x41=−58.170990540028x_{41} = -58.170990540028
x42=−48.75613936684x_{42} = -48.75613936684
x43=−45.6187613383417x_{43} = -45.6187613383417
x44=−77.0079573362515x_{44} = -77.0079573362515
x45=80.1480259413025x_{45} = 80.1480259413025
x46=−67.5885991217338x_{46} = -67.5885991217338
x47=51.894024636399x_{47} = 51.894024636399
x48=−33.0771723843072x_{48} = -33.0771723843072
x49=−39.3460075465194x_{49} = -39.3460075465194
x50=29.9449807735163x_{50} = 29.9449807735163
x51=−42.4820019253669x_{51} = -42.4820019253669
x52=−5.23293845351241x_{52} = -5.23293845351241
x53=17.4490243427188x_{53} = 17.4490243427188
x54=−51.894024636399x_{54} = -51.894024636399
x55=−73.8680180276454x_{55} = -73.8680180276454
x56=23.6879210560017x_{56} = 23.6879210560017
x57=26.814952130975x_{57} = 26.814952130975
x58=−98.9904652640992x_{58} = -98.9904652640992
x59=98.9904652640992x_{59} = 98.9904652640992
x60=−92.7093311956205x_{60} = -92.7093311956205
x61=−29.9449807735163x_{61} = -29.9449807735163
x62=−55.0323309441547x_{62} = -55.0323309441547
x63=−70.7282251775385x_{63} = -70.7282251775385
x64=73.8680180276454x_{64} = 73.8680180276454
x65=42.4820019253669x_{65} = 42.4820019253669
The values of the extrema at the points:
(92.70933119562048, -796421.699586266)

(0, 0)

(67.5885991217338, -308455.804503574)

(-61.309949447565465, -230183.175698878)

(-86.42849481807224, -645222.315380553)

(77.00795733625147, 456328.409900699)

(-17.449024342718843, -5235.85577950966)

(14.34335078839151, 2888.3803804149)

(-11.256043014353493, -1378.01976203725)

(55.032330944154715, -166421.48092055)

(-14.34335078839151, 2888.3803804149)

(36.21097455558523, -47318.9702503321)

(95.84986466881885, 880160.538929613)

(11.256043014353493, -1378.01976203725)

(8.204531362581267, 518.694993552911)

(-2.45564386287944, 9.37949248744233)

(20.56520793983334, 8606.50554943)

(-23.687921056001688, -13186.37925766)

(89.56887188991735, 718170.970965642)

(48.756139366839975, -115682.417566907)

(-64.44916413787378, 267412.604455205)

(2.45564386287944, 9.37949248744233)

(-83.28820925911458, 577389.695139745)

(45.6187613383417, 94731.2779158677)

(33.07717238430719, 36041.7770225777)

(-36.21097455558523, -47318.9702503321)

(-80.14802594130248, -514487.072547109)

(64.44916413787378, 267412.604455205)

(58.17099054002796, 196581.480455827)

(5.232938453512406, -124.316680634702)

(86.42849481807224, -645222.315380553)

(-20.56520793983334, 8606.50554943)

(-26.81495213097502, 19161.5214252829)

(-8.204531362581267, 518.694993552911)

(61.309949447565465, -230183.175698878)

(83.28820925911458, 577389.695139745)

(-89.56887188991735, 718170.970965642)

(-95.84986466881885, 880160.538929613)

(39.34600754651944, 60735.5924841558)

(70.72822517753846, 353498.813601871)

(-58.17099054002796, 196581.480455827)

(-48.756139366839975, -115682.417566907)

(-45.6187613383417, 94731.2779158677)

(-77.00795733625147, 456328.409900699)

(80.14802594130248, -514487.072547109)

(-67.5885991217338, -308455.804503574)

(51.894024636399, 139517.139252855)

(-33.07717238430719, 36041.7770225777)

(-39.34600754651944, 60735.5924841558)

(29.944980773516342, -26717.9738988985)

(-42.48200192536688, -76477.6822699254)

(-5.232938453512406, -124.316680634702)

(17.449024342718843, -5235.85577950966)

(-51.894024636399, 139517.139252855)

(-73.86801802764536, -402727.669491498)

(23.687921056001688, -13186.37925766)

(26.81495213097502, 19161.5214252829)

(-98.99046526409923, -969573.526679447)

(98.99046526409923, -969573.526679447)

(-92.70933119562048, -796421.699586266)

(-29.944980773516342, -26717.9738988985)

(-55.032330944154715, -166421.48092055)

(-70.72822517753846, 353498.813601871)

(73.86801802764536, -402727.669491498)

(42.48200192536688, -76477.6822699254)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
x1=92.7093311956205x_{1} = 92.7093311956205
x2=0x_{2} = 0
x3=67.5885991217338x_{3} = 67.5885991217338
x4=−61.3099494475655x_{4} = -61.3099494475655
x5=−86.4284948180722x_{5} = -86.4284948180722
x6=−17.4490243427188x_{6} = -17.4490243427188
x7=−11.2560430143535x_{7} = -11.2560430143535
x8=55.0323309441547x_{8} = 55.0323309441547
x9=36.2109745555852x_{9} = 36.2109745555852
x10=11.2560430143535x_{10} = 11.2560430143535
x11=−23.6879210560017x_{11} = -23.6879210560017
x12=48.75613936684x_{12} = 48.75613936684
x13=−36.2109745555852x_{13} = -36.2109745555852
x14=−80.1480259413025x_{14} = -80.1480259413025
x15=5.23293845351241x_{15} = 5.23293845351241
x16=86.4284948180722x_{16} = 86.4284948180722
x17=61.3099494475655x_{17} = 61.3099494475655
x18=−48.75613936684x_{18} = -48.75613936684
x19=80.1480259413025x_{19} = 80.1480259413025
x20=−67.5885991217338x_{20} = -67.5885991217338
x21=29.9449807735163x_{21} = 29.9449807735163
x22=−42.4820019253669x_{22} = -42.4820019253669
x23=−5.23293845351241x_{23} = -5.23293845351241
x24=17.4490243427188x_{24} = 17.4490243427188
x25=−73.8680180276454x_{25} = -73.8680180276454
x26=23.6879210560017x_{26} = 23.6879210560017
x27=−98.9904652640992x_{27} = -98.9904652640992
x28=98.9904652640992x_{28} = 98.9904652640992
x29=−92.7093311956205x_{29} = -92.7093311956205
x30=−29.9449807735163x_{30} = -29.9449807735163
x31=−55.0323309441547x_{31} = -55.0323309441547
x32=73.8680180276454x_{32} = 73.8680180276454
x33=42.4820019253669x_{33} = 42.4820019253669
Maxima of the function at points:
x33=77.0079573362515x_{33} = 77.0079573362515
x33=14.3433507883915x_{33} = 14.3433507883915
x33=−14.3433507883915x_{33} = -14.3433507883915
x33=95.8498646688189x_{33} = 95.8498646688189
x33=8.20453136258127x_{33} = 8.20453136258127
x33=−2.45564386287944x_{33} = -2.45564386287944
x33=20.5652079398333x_{33} = 20.5652079398333
x33=89.5688718899173x_{33} = 89.5688718899173
x33=−64.4491641378738x_{33} = -64.4491641378738
x33=2.45564386287944x_{33} = 2.45564386287944
x33=−83.2882092591146x_{33} = -83.2882092591146
x33=45.6187613383417x_{33} = 45.6187613383417
x33=33.0771723843072x_{33} = 33.0771723843072
x33=64.4491641378738x_{33} = 64.4491641378738
x33=58.170990540028x_{33} = 58.170990540028
x33=−20.5652079398333x_{33} = -20.5652079398333
x33=−26.814952130975x_{33} = -26.814952130975
x33=−8.20453136258127x_{33} = -8.20453136258127
x33=83.2882092591146x_{33} = 83.2882092591146
x33=−89.5688718899173x_{33} = -89.5688718899173
x33=−95.8498646688189x_{33} = -95.8498646688189
x33=39.3460075465194x_{33} = 39.3460075465194
x33=70.7282251775385x_{33} = 70.7282251775385
x33=−58.170990540028x_{33} = -58.170990540028
x33=−45.6187613383417x_{33} = -45.6187613383417
x33=−77.0079573362515x_{33} = -77.0079573362515
x33=51.894024636399x_{33} = 51.894024636399
x33=−33.0771723843072x_{33} = -33.0771723843072
x33=−39.3460075465194x_{33} = -39.3460075465194
x33=−51.894024636399x_{33} = -51.894024636399
x33=26.814952130975x_{33} = 26.814952130975
x33=−70.7282251775385x_{33} = -70.7282251775385
Decreasing at intervals
[98.9904652640992,∞)\left[98.9904652640992, \infty\right)
Increasing at intervals
(−∞,−98.9904652640992]\left(-\infty, -98.9904652640992\right]
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
x(−x2sin⁡(x)+6xcos⁡(x)+6sin⁡(x))=0x \left(- x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)} + 6 \sin{\left(x \right)}\right) = 0
Solve this equation
The roots of this equation
x1=44.1178800161513x_{1} = 44.1178800161513
x2=−37.85694574059x_{2} = -37.85694574059
x3=7.05208859318106x_{3} = 7.05208859318106
x4=28.4834495364194x_{4} = 28.4834495364194
x5=53.5189511635138x_{5} = 53.5189511635138
x6=81.7547334875609x_{6} = 81.7547334875609
x7=−22.2575395263247x_{7} = -22.2575395263247
x8=−44.1178800161513x_{8} = -44.1178800161513
x9=−50.3842871966612x_{9} = -50.3842871966612
x10=−40.9865751123733x_{10} = -40.9865751123733
x11=75.4776339039269x_{11} = 75.4776339039269
x12=47.2505332434495x_{12} = 47.2505332434495
x13=−13.012298102066x_{13} = -13.012298102066
x14=16.0730074093467x_{14} = 16.0730074093467
x15=−72.3394784349932x_{15} = -72.3394784349932
x16=0x_{16} = 0
x17=97.4509028811532x_{17} = 97.4509028811532
x18=−88.0326981155368x_{18} = -88.0326981155368
x19=−91.1719492416891x_{19} = -91.1719492416891
x20=−4.26739380712901x_{20} = -4.26739380712901
x21=−28.4834495364194x_{21} = -28.4834495364194
x22=−16.0730074093467x_{22} = -16.0730074093467
x23=56.6543759268167x_{23} = 56.6543759268167
x24=31.6046464917248x_{24} = 31.6046464917248
x25=−84.893619603402x_{25} = -84.893619603402
x26=91.1719492416891x_{26} = 91.1719492416891
x27=−1.81453497473617x_{27} = -1.81453497473617
x28=22.2575395263247x_{28} = 22.2575395263247
x29=72.3394784349932x_{29} = 72.3394784349932
x30=−47.2505332434495x_{30} = -47.2505332434495
x31=19.1578008101567x_{31} = 19.1578008101567
x32=−56.6543759268167x_{32} = -56.6543759268167
x33=−62.9270575685711x_{33} = -62.9270575685711
x34=−59.7904430977076x_{34} = -59.7904430977076
x35=84.893619603402x_{35} = 84.893619603402
x36=69.2016332025015x_{36} = 69.2016332025015
x37=−31.6046464917248x_{37} = -31.6046464917248
x38=40.9865751123733x_{38} = 40.9865751123733
x39=59.7904430977076x_{39} = 59.7904430977076
x40=−66.064142073588x_{40} = -66.064142073588
x41=25.3671067403905x_{41} = 25.3671067403905
x42=−9.99322916735933x_{42} = -9.99322916735933
x43=−81.7547334875609x_{43} = -81.7547334875609
x44=−53.5189511635138x_{44} = -53.5189511635138
x45=−78.6160626870951x_{45} = -78.6160626870951
x46=−94.3113558182004x_{46} = -94.3113558182004
x47=94.3113558182004x_{47} = 94.3113558182004
x48=−69.2016332025015x_{48} = -69.2016332025015
x49=−25.3671067403905x_{49} = -25.3671067403905
x50=−97.4509028811532x_{50} = -97.4509028811532
x51=4.26739380712901x_{51} = 4.26739380712901
x52=−100.590577325313x_{52} = -100.590577325313
x53=88.0326981155368x_{53} = 88.0326981155368
x54=50.3842871966612x_{54} = 50.3842871966612
x55=−34.7294331656689x_{55} = -34.7294331656689
x56=34.7294331656689x_{56} = 34.7294331656689
x57=78.6160626870951x_{57} = 78.6160626870951
x58=−19.1578008101567x_{58} = -19.1578008101567
x59=37.85694574059x_{59} = 37.85694574059
x60=1.81453497473617x_{60} = 1.81453497473617
x61=100.590577325313x_{61} = 100.590577325313
x62=13.012298102066x_{62} = 13.012298102066
x63=−75.4776339039269x_{63} = -75.4776339039269
x64=9.99322916735933x_{64} = 9.99322916735933
x65=−7.05208859318106x_{65} = -7.05208859318106
x66=66.064142073588x_{66} = 66.064142073588
x67=62.9270575685711x_{67} = 62.9270575685711

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
[97.4509028811532,∞)\left[97.4509028811532, \infty\right)
Convex at the intervals
(−∞,−100.590577325313]\left(-\infty, -100.590577325313\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(x3sin⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x^{3} \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(x3sin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(x^{3} \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x^3*sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(x2sin⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x^{2} \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
inclined asymptote equation on the left:
y=⟨−∞,∞⟩xy = \left\langle -\infty, \infty\right\rangle x
lim⁡x→∞(x2sin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(x^{2} \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
inclined asymptote equation on the right:
y=⟨−∞,∞⟩xy = \left\langle -\infty, \infty\right\rangle x
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
x3sin⁡(x)=x3sin⁡(x)x^{3} \sin{\left(x \right)} = x^{3} \sin{\left(x \right)}
- Yes
x3sin⁡(x)=−x3sin⁡(x)x^{3} \sin{\left(x \right)} = - x^{3} \sin{\left(x \right)}
- No
so, the function
is
even
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