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Plotting a function graph/ x^3*sin(x)

Graphing y = x^3*sin(x)

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
0-60-50-40-30-20-10102030405060-500000500000
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=43.9822971502571x_{1} = 43.9822971502571
x2=−97.3893722612836x_{2} = -97.3893722612836
x3=−43.9822971502571x_{3} = -43.9822971502571
x4=−72.2566310325652x_{4} = -72.2566310325652
x5=−59.6902604182061x_{5} = -59.6902604182061
x6=81.6814089933346x_{6} = 81.6814089933346
x7=−31.4159265358979x_{7} = -31.4159265358979
x8=−78.5398163397448x_{8} = -78.5398163397448
x9=97.3893722612836x_{9} = 97.3893722612836
x10=9.42477796076938x_{10} = 9.42477796076938
x11=−25.1327412287183x_{11} = -25.1327412287183
x12=84.8230016469244x_{12} = 84.8230016469244
x13=−21.9911485751286x_{13} = -21.9911485751286
x14=−94.2477796076938x_{14} = -94.2477796076938
x15=6.28318530717959x_{15} = 6.28318530717959
x16=3.14159265358979x_{16} = 3.14159265358979
x17=−50.2654824574367x_{17} = -50.2654824574367
x18=28.2743338823081x_{18} = 28.2743338823081
x19=−75.398223686155x_{19} = -75.398223686155
x20=−28.2743338823081x_{20} = -28.2743338823081
x21=−56.5486677646163x_{21} = -56.5486677646163
x22=−65.9734457253857x_{22} = -65.9734457253857
x23=−40.8407044966673x_{23} = -40.8407044966673
x24=−91.106186954104x_{24} = -91.106186954104
x25=50.2654824574367x_{25} = 50.2654824574367
x26=−69.1150383789755x_{26} = -69.1150383789755
x27=−100.530964914873x_{27} = -100.530964914873
x28=56.5486677646163x_{28} = 56.5486677646163
x29=−62.8318530717959x_{29} = -62.8318530717959
x30=−87.9645943005142x_{30} = -87.9645943005142
x31=40.8407044966673x_{31} = 40.8407044966673
x32=100.530964914873x_{32} = 100.530964914873
x33=18.8495559215388x_{33} = 18.8495559215388
x34=62.8318530717959x_{34} = 62.8318530717959
x35=−53.4070751110265x_{35} = -53.4070751110265
x36=94.2477796076938x_{36} = 94.2477796076938
x37=−3.14159265358979x_{37} = -3.14159265358979
x38=21.9911485751286x_{38} = 21.9911485751286
x39=12.5663706143592x_{39} = 12.5663706143592
x40=−84.8230016469244x_{40} = -84.8230016469244
x41=34.5575191894877x_{41} = 34.5575191894877
x42=−125.663706143592x_{42} = -125.663706143592
x43=47.1238898038469x_{43} = 47.1238898038469
x44=−15.707963267949x_{44} = -15.707963267949
x45=53.4070751110265x_{45} = 53.4070751110265
x46=65.9734457253857x_{46} = 65.9734457253857
x47=87.9645943005142x_{47} = 87.9645943005142
x48=91.106186954104x_{48} = 91.106186954104
x49=59.6902604182061x_{49} = 59.6902604182061
x50=69.1150383789755x_{50} = 69.1150383789755
x51=−6.28318530717959x_{51} = -6.28318530717959
x52=75.398223686155x_{52} = 75.398223686155
x53=−37.6991118430775x_{53} = -37.6991118430775
x54=−12.5663706143592x_{54} = -12.5663706143592
x55=−18.8495559215388x_{55} = -18.8495559215388
x56=31.4159265358979x_{56} = 31.4159265358979
x57=−81.6814089933346x_{57} = -81.6814089933346
x58=78.5398163397448x_{58} = 78.5398163397448
x59=−135.088484104361x_{59} = -135.088484104361
x60=15.707963267949x_{60} = 15.707963267949
x61=72.2566310325652x_{61} = 72.2566310325652
x62=37.6991118430775x_{62} = 37.6991118430775
x63=25.1327412287183x_{63} = 25.1327412287183
x64=−47.1238898038469x_{64} = -47.1238898038469
x65=0x_{65} = 0
x66=−9.42477796076938x_{66} = -9.42477796076938
x67=−34.5575191894877x_{67} = -34.5575191894877
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=51.894024636399x_{1} = 51.894024636399
x2=95.8498646688189x_{2} = 95.8498646688189
x3=89.5688718899173x_{3} = 89.5688718899173
x4=−73.8680180276454x_{4} = -73.8680180276454
x5=61.3099494475655x_{5} = 61.3099494475655
x6=−70.7282251775385x_{6} = -70.7282251775385
x7=48.75613936684x_{7} = 48.75613936684
x8=−45.6187613383417x_{8} = -45.6187613383417
x9=−48.75613936684x_{9} = -48.75613936684
x10=45.6187613383417x_{10} = 45.6187613383417
x11=17.4490243427188x_{11} = 17.4490243427188
x12=92.7093311956205x_{12} = 92.7093311956205
x13=−39.3460075465194x_{13} = -39.3460075465194
x14=67.5885991217338x_{14} = 67.5885991217338
x15=−61.3099494475655x_{15} = -61.3099494475655
x16=29.9449807735163x_{16} = 29.9449807735163
x17=−86.4284948180722x_{17} = -86.4284948180722
x18=−33.0771723843072x_{18} = -33.0771723843072
x19=58.170990540028x_{19} = 58.170990540028
x20=−98.9904652640992x_{20} = -98.9904652640992
x21=−5.23293845351241x_{21} = -5.23293845351241
x22=−51.894024636399x_{22} = -51.894024636399
x23=70.7282251775385x_{23} = 70.7282251775385
x24=−64.4491641378738x_{24} = -64.4491641378738
x25=−8.20453136258127x_{25} = -8.20453136258127
x26=98.9904652640992x_{26} = 98.9904652640992
x27=5.23293845351241x_{27} = 5.23293845351241
x28=−67.5885991217338x_{28} = -67.5885991217338
x29=33.0771723843072x_{29} = 33.0771723843072
x30=−29.9449807735163x_{30} = -29.9449807735163
x31=−36.2109745555852x_{31} = -36.2109745555852
x32=−14.3433507883915x_{32} = -14.3433507883915
x33=36.2109745555852x_{33} = 36.2109745555852
x34=−77.0079573362515x_{34} = -77.0079573362515
x35=0x_{35} = 0
x36=−83.2882092591146x_{36} = -83.2882092591146
x37=2.45564386287944x_{37} = 2.45564386287944
x38=−20.5652079398333x_{38} = -20.5652079398333
x39=−26.814952130975x_{39} = -26.814952130975
x40=14.3433507883915x_{40} = 14.3433507883915
x41=42.4820019253669x_{41} = 42.4820019253669
x42=83.2882092591146x_{42} = 83.2882092591146
x43=−17.4490243427188x_{43} = -17.4490243427188
x44=8.20453136258127x_{44} = 8.20453136258127
x45=77.0079573362515x_{45} = 77.0079573362515
x46=55.0323309441547x_{46} = 55.0323309441547
x47=−42.4820019253669x_{47} = -42.4820019253669
x48=23.6879210560017x_{48} = 23.6879210560017
x49=73.8680180276454x_{49} = 73.8680180276454
x50=−80.1480259413025x_{50} = -80.1480259413025
x51=−89.5688718899173x_{51} = -89.5688718899173
x52=−92.7093311956205x_{52} = -92.7093311956205
x53=−2.45564386287944x_{53} = -2.45564386287944
x54=−55.0323309441547x_{54} = -55.0323309441547
x55=26.814952130975x_{55} = 26.814952130975
x56=−95.8498646688189x_{56} = -95.8498646688189
x57=−23.6879210560017x_{57} = -23.6879210560017
x58=80.1480259413025x_{58} = 80.1480259413025
x59=86.4284948180722x_{59} = 86.4284948180722
x60=11.2560430143535x_{60} = 11.2560430143535
x61=20.5652079398333x_{61} = 20.5652079398333
x62=−11.2560430143535x_{62} = -11.2560430143535
x63=−58.170990540028x_{63} = -58.170990540028
x64=64.4491641378738x_{64} = 64.4491641378738
x65=39.3460075465194x_{65} = 39.3460075465194
The values of the extrema at the points:
(51.894024636399, 139517.139252855)

(95.84986466881885, 880160.538929613)

(89.56887188991735, 718170.970965642)

(-73.86801802764536, -402727.669491498)

(61.309949447565465, -230183.175698878)

(-70.72822517753846, 353498.813601871)

(48.756139366839975, -115682.417566907)

(-45.6187613383417, 94731.2779158677)

(-48.756139366839975, -115682.417566907)

(45.6187613383417, 94731.2779158677)

(17.449024342718843, -5235.85577950966)

(92.70933119562048, -796421.699586266)

(-39.34600754651944, 60735.5924841558)

(67.5885991217338, -308455.804503574)

(-61.309949447565465, -230183.175698878)

(29.944980773516342, -26717.9738988985)

(-86.42849481807224, -645222.315380553)

(-33.07717238430719, 36041.7770225777)

(58.17099054002796, 196581.480455827)

(-98.99046526409923, -969573.526679447)

(-5.232938453512406, -124.316680634702)

(-51.894024636399, 139517.139252855)

(70.72822517753846, 353498.813601871)

(-64.44916413787378, 267412.604455205)

(-8.204531362581267, 518.694993552911)

(98.99046526409923, -969573.526679447)

(5.232938453512406, -124.316680634702)

(-67.5885991217338, -308455.804503574)

(33.07717238430719, 36041.7770225777)

(-29.944980773516342, -26717.9738988985)

(-36.21097455558523, -47318.9702503321)

(-14.34335078839151, 2888.3803804149)

(36.21097455558523, -47318.9702503321)

(-77.00795733625147, 456328.409900699)

(0, 0)

(-83.28820925911458, 577389.695139745)

(2.45564386287944, 9.37949248744233)

(-20.56520793983334, 8606.50554943)

(-26.81495213097502, 19161.5214252829)

(14.34335078839151, 2888.3803804149)

(42.48200192536688, -76477.6822699254)

(83.28820925911458, 577389.695139745)

(-17.449024342718843, -5235.85577950966)

(8.204531362581267, 518.694993552911)

(77.00795733625147, 456328.409900699)

(55.032330944154715, -166421.48092055)

(-42.48200192536688, -76477.6822699254)

(23.687921056001688, -13186.37925766)

(73.86801802764536, -402727.669491498)

(-80.14802594130248, -514487.072547109)

(-89.56887188991735, 718170.970965642)

(-92.70933119562048, -796421.699586266)

(-2.45564386287944, 9.37949248744233)

(-55.032330944154715, -166421.48092055)

(26.81495213097502, 19161.5214252829)

(-95.84986466881885, 880160.538929613)

(-23.687921056001688, -13186.37925766)

(80.14802594130248, -514487.072547109)

(86.42849481807224, -645222.315380553)

(11.256043014353493, -1378.01976203725)

(20.56520793983334, 8606.50554943)

(-11.256043014353493, -1378.01976203725)

(-58.17099054002796, 196581.480455827)

(64.44916413787378, 267412.604455205)

(39.34600754651944, 60735.5924841558)


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=−73.8680180276454x_{1} = -73.8680180276454
x2=61.3099494475655x_{2} = 61.3099494475655
x3=48.75613936684x_{3} = 48.75613936684
x4=−48.75613936684x_{4} = -48.75613936684
x5=17.4490243427188x_{5} = 17.4490243427188
x6=92.7093311956205x_{6} = 92.7093311956205
x7=67.5885991217338x_{7} = 67.5885991217338
x8=−61.3099494475655x_{8} = -61.3099494475655
x9=29.9449807735163x_{9} = 29.9449807735163
x10=−86.4284948180722x_{10} = -86.4284948180722
x11=−98.9904652640992x_{11} = -98.9904652640992
x12=−5.23293845351241x_{12} = -5.23293845351241
x13=98.9904652640992x_{13} = 98.9904652640992
x14=5.23293845351241x_{14} = 5.23293845351241
x15=−67.5885991217338x_{15} = -67.5885991217338
x16=−29.9449807735163x_{16} = -29.9449807735163
x17=−36.2109745555852x_{17} = -36.2109745555852
x18=36.2109745555852x_{18} = 36.2109745555852
x19=0x_{19} = 0
x20=42.4820019253669x_{20} = 42.4820019253669
x21=−17.4490243427188x_{21} = -17.4490243427188
x22=55.0323309441547x_{22} = 55.0323309441547
x23=−42.4820019253669x_{23} = -42.4820019253669
x24=23.6879210560017x_{24} = 23.6879210560017
x25=73.8680180276454x_{25} = 73.8680180276454
x26=−80.1480259413025x_{26} = -80.1480259413025
x27=−92.7093311956205x_{27} = -92.7093311956205
x28=−55.0323309441547x_{28} = -55.0323309441547
x29=−23.6879210560017x_{29} = -23.6879210560017
x30=80.1480259413025x_{30} = 80.1480259413025
x31=86.4284948180722x_{31} = 86.4284948180722
x32=11.2560430143535x_{32} = 11.2560430143535
x33=−11.2560430143535x_{33} = -11.2560430143535
Maxima of the function at points:
x33=51.894024636399x_{33} = 51.894024636399
x33=95.8498646688189x_{33} = 95.8498646688189
x33=89.5688718899173x_{33} = 89.5688718899173
x33=−70.7282251775385x_{33} = -70.7282251775385
x33=−45.6187613383417x_{33} = -45.6187613383417
x33=45.6187613383417x_{33} = 45.6187613383417
x33=−39.3460075465194x_{33} = -39.3460075465194
x33=−33.0771723843072x_{33} = -33.0771723843072
x33=58.170990540028x_{33} = 58.170990540028
x33=−51.894024636399x_{33} = -51.894024636399
x33=70.7282251775385x_{33} = 70.7282251775385
x33=−64.4491641378738x_{33} = -64.4491641378738
x33=−8.20453136258127x_{33} = -8.20453136258127
x33=33.0771723843072x_{33} = 33.0771723843072
x33=−14.3433507883915x_{33} = -14.3433507883915
x33=−77.0079573362515x_{33} = -77.0079573362515
x33=−83.2882092591146x_{33} = -83.2882092591146
x33=2.45564386287944x_{33} = 2.45564386287944
x33=−20.5652079398333x_{33} = -20.5652079398333
x33=−26.814952130975x_{33} = -26.814952130975
x33=14.3433507883915x_{33} = 14.3433507883915
x33=83.2882092591146x_{33} = 83.2882092591146
x33=8.20453136258127x_{33} = 8.20453136258127
x33=77.0079573362515x_{33} = 77.0079573362515
x33=−89.5688718899173x_{33} = -89.5688718899173
x33=−2.45564386287944x_{33} = -2.45564386287944
x33=26.814952130975x_{33} = 26.814952130975
x33=−95.8498646688189x_{33} = -95.8498646688189
x33=20.5652079398333x_{33} = 20.5652079398333
x33=−58.170990540028x_{33} = -58.170990540028
x33=64.4491641378738x_{33} = 64.4491641378738
x33=39.3460075465194x_{33} = 39.3460075465194
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=37.85694574059x_{1} = 37.85694574059
x2=25.3671067403905x_{2} = 25.3671067403905
x3=−97.4509028811532x_{3} = -97.4509028811532
x4=97.4509028811532x_{4} = 97.4509028811532
x5=69.2016332025015x_{5} = 69.2016332025015
x6=−37.85694574059x_{6} = -37.85694574059
x7=−100.590577325313x_{7} = -100.590577325313
x8=84.893619603402x_{8} = 84.893619603402
x9=7.05208859318106x_{9} = 7.05208859318106
x10=13.012298102066x_{10} = 13.012298102066
x11=−25.3671067403905x_{11} = -25.3671067403905
x12=−84.893619603402x_{12} = -84.893619603402
x13=53.5189511635138x_{13} = 53.5189511635138
x14=1.81453497473617x_{14} = 1.81453497473617
x15=−22.2575395263247x_{15} = -22.2575395263247
x16=34.7294331656689x_{16} = 34.7294331656689
x17=4.26739380712901x_{17} = 4.26739380712901
x18=−28.4834495364194x_{18} = -28.4834495364194
x19=28.4834495364194x_{19} = 28.4834495364194
x20=31.6046464917248x_{20} = 31.6046464917248
x21=62.9270575685711x_{21} = 62.9270575685711
x22=−94.3113558182004x_{22} = -94.3113558182004
x23=−53.5189511635138x_{23} = -53.5189511635138
x24=−78.6160626870951x_{24} = -78.6160626870951
x25=81.7547334875609x_{25} = 81.7547334875609
x26=−19.1578008101567x_{26} = -19.1578008101567
x27=−50.3842871966612x_{27} = -50.3842871966612
x28=−66.064142073588x_{28} = -66.064142073588
x29=72.3394784349932x_{29} = 72.3394784349932
x30=−9.99322916735933x_{30} = -9.99322916735933
x31=−7.05208859318106x_{31} = -7.05208859318106
x32=−91.1719492416891x_{32} = -91.1719492416891
x33=−13.012298102066x_{33} = -13.012298102066
x34=44.1178800161513x_{34} = 44.1178800161513
x35=100.590577325313x_{35} = 100.590577325313
x36=−69.2016332025015x_{36} = -69.2016332025015
x37=−62.9270575685711x_{37} = -62.9270575685711
x38=−81.7547334875609x_{38} = -81.7547334875609
x39=0x_{39} = 0
x40=50.3842871966612x_{40} = 50.3842871966612
x41=75.4776339039269x_{41} = 75.4776339039269
x42=−88.0326981155368x_{42} = -88.0326981155368
x43=−31.6046464917248x_{43} = -31.6046464917248
x44=−72.3394784349932x_{44} = -72.3394784349932
x45=40.9865751123733x_{45} = 40.9865751123733
x46=−56.6543759268167x_{46} = -56.6543759268167
x47=19.1578008101567x_{47} = 19.1578008101567
x48=78.6160626870951x_{48} = 78.6160626870951
x49=−40.9865751123733x_{49} = -40.9865751123733
x50=9.99322916735933x_{50} = 9.99322916735933
x51=22.2575395263247x_{51} = 22.2575395263247
x52=88.0326981155368x_{52} = 88.0326981155368
x53=−34.7294331656689x_{53} = -34.7294331656689
x54=−47.2505332434495x_{54} = -47.2505332434495
x55=91.1719492416891x_{55} = 91.1719492416891
x56=−1.81453497473617x_{56} = -1.81453497473617
x57=59.7904430977076x_{57} = 59.7904430977076
x58=94.3113558182004x_{58} = 94.3113558182004
x59=−59.7904430977076x_{59} = -59.7904430977076
x60=−75.4776339039269x_{60} = -75.4776339039269
x61=16.0730074093467x_{61} = 16.0730074093467
x62=66.064142073588x_{62} = 66.064142073588
x63=56.6543759268167x_{63} = 56.6543759268167
x64=−44.1178800161513x_{64} = -44.1178800161513
x65=−16.0730074093467x_{65} = -16.0730074093467
x66=47.2505332434495x_{66} = 47.2505332434495
x67=−4.26739380712901x_{67} = -4.26739380712901

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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