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Plotting a function graph/ x^2*sin(x)

Graphing y = x^2*sin(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
        2       
f(x) = x *sin(x)
f(x)=x2sin⁡(x)f{\left(x \right)} = x^{2} \sin{\left(x \right)} 
f = x^2*sin(x)
The graph of the function
02468-8-6-4-2-1010-100100
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:
x2sin⁡(x)=0x^{2} \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=40.8407044966673x_{1} = 40.8407044966673
x2=0x_{2} = 0
x3=−18.8495559215388x_{3} = -18.8495559215388
x4=−56.5486677646163x_{4} = -56.5486677646163
x5=97.3893722612836x_{5} = 97.3893722612836
x6=34.5575191894877x_{6} = 34.5575191894877
x7=53.4070751110265x_{7} = 53.4070751110265
x8=47.1238898038469x_{8} = 47.1238898038469
x9=−97.3893722612836x_{9} = -97.3893722612836
x10=62.8318530717959x_{10} = 62.8318530717959
x11=87.9645943005142x_{11} = 87.9645943005142
x12=43.9822971502571x_{12} = 43.9822971502571
x13=37.6991118430775x_{13} = 37.6991118430775
x14=−21.9911485751286x_{14} = -21.9911485751286
x15=3.14159265358979x_{15} = 3.14159265358979
x16=65.9734457253857x_{16} = 65.9734457253857
x17=69.1150383789755x_{17} = 69.1150383789755
x18=−50.2654824574367x_{18} = -50.2654824574367
x19=−94.2477796076938x_{19} = -94.2477796076938
x20=−75.398223686155x_{20} = -75.398223686155
x21=−53.4070751110265x_{21} = -53.4070751110265
x22=12.5663706143592x_{22} = 12.5663706143592
x23=−9.42477796076938x_{23} = -9.42477796076938
x24=−34.5575191894877x_{24} = -34.5575191894877
x25=21.9911485751286x_{25} = 21.9911485751286
x26=−47.1238898038469x_{26} = -47.1238898038469
x27=−43.9822971502571x_{27} = -43.9822971502571
x28=28.2743338823081x_{28} = 28.2743338823081
x29=−31.4159265358979x_{29} = -31.4159265358979
x30=−3.14159265358979x_{30} = -3.14159265358979
x31=−6.28318530717959x_{31} = -6.28318530717959
x32=−25.1327412287183x_{32} = -25.1327412287183
x33=−62.8318530717959x_{33} = -62.8318530717959
x34=31.4159265358979x_{34} = 31.4159265358979
x35=−65.9734457253857x_{35} = -65.9734457253857
x36=72.2566310325652x_{36} = 72.2566310325652
x37=−59.6902604182061x_{37} = -59.6902604182061
x38=−12.5663706143592x_{38} = -12.5663706143592
x39=94.2477796076938x_{39} = 94.2477796076938
x40=81.6814089933346x_{40} = 81.6814089933346
x41=−91.106186954104x_{41} = -91.106186954104
x42=−100.530964914873x_{42} = -100.530964914873
x43=59.6902604182061x_{43} = 59.6902604182061
x44=−40.8407044966673x_{44} = -40.8407044966673
x45=91.106186954104x_{45} = 91.106186954104
x46=78.5398163397448x_{46} = 78.5398163397448
x47=−106.814150222053x_{47} = -106.814150222053
x48=56.5486677646163x_{48} = 56.5486677646163
x49=84.8230016469244x_{49} = 84.8230016469244
x50=100.530964914873x_{50} = 100.530964914873
x51=−69.1150383789755x_{51} = -69.1150383789755
x52=9.42477796076938x_{52} = 9.42477796076938
x53=−84.8230016469244x_{53} = -84.8230016469244
x54=−78.5398163397448x_{54} = -78.5398163397448
x55=−87.9645943005142x_{55} = -87.9645943005142
x56=−81.6814089933346x_{56} = -81.6814089933346
x57=15.707963267949x_{57} = 15.707963267949
x58=−28.2743338823081x_{58} = -28.2743338823081
x59=−15.707963267949x_{59} = -15.707963267949
x60=−37.6991118430775x_{60} = -37.6991118430775
x61=18.8495559215388x_{61} = 18.8495559215388
x62=25.1327412287183x_{62} = 25.1327412287183
x63=50.2654824574367x_{63} = 50.2654824574367
x64=−72.2566310325652x_{64} = -72.2566310325652
x65=75.398223686155x_{65} = 75.398223686155
x66=6.28318530717959x_{66} = 6.28318530717959
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x^2*sin(x).
02sin⁡(0)0^{2} \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
x2cos⁡(x)+2xsin⁡(x)=0x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=29.9118938695518x_{1} = 29.9118938695518
x2=48.7357007949054x_{2} = 48.7357007949054
x3=86.4169374541167x_{3} = 86.4169374541167
x4=95.839441141233x_{4} = 95.839441141233
x5=83.2762171649775x_{5} = 83.2762171649775
x6=−51.8748140534268x_{6} = -51.8748140534268
x7=36.1835330907526x_{7} = 36.1835330907526
x8=0x_{8} = 0
x9=51.8748140534268x_{9} = 51.8748140534268
x10=67.573830670859x_{10} = 67.573830670859
x11=33.0471686947054x_{11} = 33.0471686947054
x12=8.09616360322292x_{12} = 8.09616360322292
x13=−26.7780870755585x_{13} = -26.7780870755585
x14=5.08698509410227x_{14} = 5.08698509410227
x15=−55.0142096788381x_{15} = -55.0142096788381
x16=−92.6985552433969x_{16} = -92.6985552433969
x17=61.2936749662429x_{17} = 61.2936749662429
x18=11.17270586833x_{18} = 11.17270586833
x19=20.5175229099417x_{19} = 20.5175229099417
x20=−36.1835330907526x_{20} = -36.1835330907526
x21=−23.6463238196036x_{21} = -23.6463238196036
x22=−58.153842078645x_{22} = -58.153842078645
x23=−20.5175229099417x_{23} = -20.5175229099417
x24=70.7141100665485x_{24} = 70.7141100665485
x25=45.5969279840735x_{25} = 45.5969279840735
x26=14.2763529183365x_{26} = 14.2763529183365
x27=42.458570771699x_{27} = 42.458570771699
x28=−5.08698509410227x_{28} = -5.08698509410227
x29=−29.9118938695518x_{29} = -29.9118938695518
x30=98.9803718651523x_{30} = 98.9803718651523
x31=−42.458570771699x_{31} = -42.458570771699
x32=−80.1355651940744x_{32} = -80.1355651940744
x33=−89.5577188827244x_{33} = -89.5577188827244
x34=−11.17270586833x_{34} = -11.17270586833
x35=2.2889297281034x_{35} = 2.2889297281034
x36=−48.7357007949054x_{36} = -48.7357007949054
x37=−17.3932439645948x_{37} = -17.3932439645948
x38=−120.967848975693x_{38} = -120.967848975693
x39=92.6985552433969x_{39} = 92.6985552433969
x40=39.3207281322521x_{40} = 39.3207281322521
x41=−39.3207281322521x_{41} = -39.3207281322521
x42=−83.2762171649775x_{42} = -83.2762171649775
x43=73.8545010149048x_{43} = 73.8545010149048
x44=58.153842078645x_{44} = 58.153842078645
x45=−8.09616360322292x_{45} = -8.09616360322292
x46=−76.9949898891676x_{46} = -76.9949898891676
x47=64.4336791037316x_{47} = 64.4336791037316
x48=−64.4336791037316x_{48} = -64.4336791037316
x49=89.5577188827244x_{49} = 89.5577188827244
x50=55.0142096788381x_{50} = 55.0142096788381
x51=−33.0471686947054x_{51} = -33.0471686947054
x52=−67.573830670859x_{52} = -67.573830670859
x53=80.1355651940744x_{53} = 80.1355651940744
x54=76.9949898891676x_{54} = 76.9949898891676
x55=−70.7141100665485x_{55} = -70.7141100665485
x56=−61.2936749662429x_{56} = -61.2936749662429
x57=17.3932439645948x_{57} = 17.3932439645948
x58=26.7780870755585x_{58} = 26.7780870755585
x59=−14.2763529183365x_{59} = -14.2763529183365
x60=−98.9803718651523x_{60} = -98.9803718651523
x61=23.6463238196036x_{61} = 23.6463238196036
x62=−86.4169374541167x_{62} = -86.4169374541167
x63=−73.8545010149048x_{63} = -73.8545010149048
x64=−45.5969279840735x_{64} = -45.5969279840735
x65=3.95930141892882⋅10−7x_{65} = 3.95930141892882 \cdot 10^{-7}
x66=−2.2889297281034x_{66} = -2.2889297281034
x67=−95.839441141233x_{67} = -95.839441141233
The values of the extrema at the points:
(29.911893869551772, -892.728075975236)

(48.73570079490539, -2373.17105456709)

(86.4169374541167, -7465.88788203037)

(95.83944114123304, 9183.19913125177)

(83.27621716497754, 6932.92921007843)

(-51.874814053426775, -2688.99855997676)

(36.18353309075258, -1307.25263807613)

(0, 0)

(51.874814053426775, 2688.99855997676)

(67.573830670859, -4564.22390457183)

(33.04716869470536, 1090.12083594654)

(8.096163603222921, 63.6349819515545)

(-26.778087075558506, -715.074276149712)

(5.08698509410227, -24.0829602230683)

(-55.01420967883812, 3024.56524685288)

(-92.69855524339692, 8591.02284218332)

(61.2936749662429, -3754.91618650696)

(11.172705868329984, -122.876173513916)

(20.51752290994169, 418.982887272434)

(-36.18353309075258, 1307.25263807613)

(-23.64632381960362, 557.159297209023)

(-58.153842078645, -3379.87112092779)

(-20.51752290994169, -418.982887272434)

(70.7141100665485, 4998.48656158818)

(45.59692798407349, 2077.08272285774)

(14.276352918336478, 201.843217881861)

(42.458570771699044, -1800.73355411815)

(-5.08698509410227, 24.0829602230683)

(-29.911893869551772, 892.728075975236)

(98.98037186515228, -9795.11462678079)

(-42.458570771699044, 1800.73355411815)

(-80.13556519407445, 6419.70974281978)

(-89.55771888272442, -8018.58575924144)

(-11.172705868329984, 122.876173513916)

(2.2889297281034042, 3.94530162528433)

(-48.73570079490539, 2373.17105456709)

(-17.393243964594753, 300.544552657996)

(-120.96784897569329, -14631.2208957387)

(92.69855524339692, -8591.02284218332)

(39.32072813225213, 1544.1235331857)

(-39.32072813225213, -1544.1235331857)

(-83.27621716497754, -6932.92921007843)

(73.85450101490484, -5452.4884195005)

(58.153842078645, 3379.87112092779)

(-8.096163603222921, -63.6349819515545)

(-76.9949898891676, -5926.22947957101)

(64.43367910373156, 4149.70044687478)

(-64.43367910373156, -4149.70044687478)

(89.55771888272442, 8018.58575924144)

(55.01420967883812, -3024.56524685288)

(-33.04716869470536, -1090.12083594654)

(-67.573830670859, 4564.22390457183)

(80.13556519407445, -6419.70974281978)

(76.9949898891676, 5926.22947957101)

(-70.7141100665485, -4998.48656158818)

(-61.2936749662429, 3754.91618650696)

(17.393243964594753, -300.544552657996)

(26.778087075558506, 715.074276149712)

(-14.276352918336478, -201.843217881861)

(-98.98037186515228, 9795.11462678079)

(23.64632381960362, -557.159297209023)

(-86.4169374541167, 7465.88788203037)

(-73.85450101490484, 5452.4884195005)

(-45.59692798407349, -2077.08272285774)

(3.9593014189288195e-07, 6.20662771905043e-20)

(-2.2889297281034042, -3.94530162528433)

(-95.83944114123304, -9183.19913125177)


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=29.9118938695518x_{1} = 29.9118938695518
x2=48.7357007949054x_{2} = 48.7357007949054
x3=86.4169374541167x_{3} = 86.4169374541167
x4=−51.8748140534268x_{4} = -51.8748140534268
x5=36.1835330907526x_{5} = 36.1835330907526
x6=67.573830670859x_{6} = 67.573830670859
x7=−26.7780870755585x_{7} = -26.7780870755585
x8=5.08698509410227x_{8} = 5.08698509410227
x9=61.2936749662429x_{9} = 61.2936749662429
x10=11.17270586833x_{10} = 11.17270586833
x11=−58.153842078645x_{11} = -58.153842078645
x12=−20.5175229099417x_{12} = -20.5175229099417
x13=42.458570771699x_{13} = 42.458570771699
x14=98.9803718651523x_{14} = 98.9803718651523
x15=−89.5577188827244x_{15} = -89.5577188827244
x16=−120.967848975693x_{16} = -120.967848975693
x17=92.6985552433969x_{17} = 92.6985552433969
x18=−39.3207281322521x_{18} = -39.3207281322521
x19=−83.2762171649775x_{19} = -83.2762171649775
x20=73.8545010149048x_{20} = 73.8545010149048
x21=−8.09616360322292x_{21} = -8.09616360322292
x22=−76.9949898891676x_{22} = -76.9949898891676
x23=−64.4336791037316x_{23} = -64.4336791037316
x24=55.0142096788381x_{24} = 55.0142096788381
x25=−33.0471686947054x_{25} = -33.0471686947054
x26=80.1355651940744x_{26} = 80.1355651940744
x27=−70.7141100665485x_{27} = -70.7141100665485
x28=17.3932439645948x_{28} = 17.3932439645948
x29=−14.2763529183365x_{29} = -14.2763529183365
x30=23.6463238196036x_{30} = 23.6463238196036
x31=−45.5969279840735x_{31} = -45.5969279840735
x32=−2.2889297281034x_{32} = -2.2889297281034
x33=−95.839441141233x_{33} = -95.839441141233
Maxima of the function at points:
x33=95.839441141233x_{33} = 95.839441141233
x33=83.2762171649775x_{33} = 83.2762171649775
x33=51.8748140534268x_{33} = 51.8748140534268
x33=33.0471686947054x_{33} = 33.0471686947054
x33=8.09616360322292x_{33} = 8.09616360322292
x33=−55.0142096788381x_{33} = -55.0142096788381
x33=−92.6985552433969x_{33} = -92.6985552433969
x33=20.5175229099417x_{33} = 20.5175229099417
x33=−36.1835330907526x_{33} = -36.1835330907526
x33=−23.6463238196036x_{33} = -23.6463238196036
x33=70.7141100665485x_{33} = 70.7141100665485
x33=45.5969279840735x_{33} = 45.5969279840735
x33=14.2763529183365x_{33} = 14.2763529183365
x33=−5.08698509410227x_{33} = -5.08698509410227
x33=−29.9118938695518x_{33} = -29.9118938695518
x33=−42.458570771699x_{33} = -42.458570771699
x33=−80.1355651940744x_{33} = -80.1355651940744
x33=−11.17270586833x_{33} = -11.17270586833
x33=2.2889297281034x_{33} = 2.2889297281034
x33=−48.7357007949054x_{33} = -48.7357007949054
x33=−17.3932439645948x_{33} = -17.3932439645948
x33=39.3207281322521x_{33} = 39.3207281322521
x33=58.153842078645x_{33} = 58.153842078645
x33=64.4336791037316x_{33} = 64.4336791037316
x33=89.5577188827244x_{33} = 89.5577188827244
x33=−67.573830670859x_{33} = -67.573830670859
x33=76.9949898891676x_{33} = 76.9949898891676
x33=−61.2936749662429x_{33} = -61.2936749662429
x33=26.7780870755585x_{33} = 26.7780870755585
x33=−98.9803718651523x_{33} = -98.9803718651523
x33=−86.4169374541167x_{33} = -86.4169374541167
x33=−73.8545010149048x_{33} = -73.8545010149048
Decreasing at intervals
[98.9803718651523,∞)\left[98.9803718651523, \infty\right)
Increasing at intervals
(−∞,−120.967848975693]\left(-\infty, -120.967848975693\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
−x2sin⁡(x)+4xcos⁡(x)+2sin⁡(x)=0- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 2 \sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−62.895397234671x_{1} = -62.895397234671
x2=−1.51985529843113x_{2} = -1.51985529843113
x3=−75.4512070764701x_{3} = -75.4512070764701
x4=−100.570724821846x_{4} = -100.570724821846
x5=−56.6192418251285x_{5} = -56.6192418251285
x6=−22.1703631077661x_{6} = -22.1703631077661
x7=−40.9382191715155x_{7} = -40.9382191715155
x8=0x_{8} = 0
x9=1.51985529843113x_{9} = 1.51985529843113
x10=56.6192418251285x_{10} = 56.6192418251285
x11=−6.83214574693118x_{11} = -6.83214574693118
x12=−47.2084939833195x_{12} = -47.2084939833195
x13=97.4304127980508x_{13} = 97.4304127980508
x14=−97.4304127980508x_{14} = -97.4304127980508
x15=9.81900340196872x_{15} = 9.81900340196872
x16=−44.0729006762809x_{16} = -44.0729006762809
x17=69.1728243307457x_{17} = 69.1728243307457
x18=81.7303260381702x_{18} = 81.7303260381702
x19=47.2084939833195x_{19} = 47.2084939833195
x20=−31.5423183719258x_{20} = -31.5423183719258
x21=−3.99444471574142x_{21} = -3.99444471574142
x22=91.1500530451789x_{22} = 91.1500530451789
x23=75.4512070764701x_{23} = 75.4512070764701
x24=−59.7571356682663x_{24} = -59.7571356682663
x25=−66.0339743721325x_{25} = -66.0339743721325
x26=−84.8701107016488x_{26} = -84.8701107016488
x27=53.4817799880237x_{27} = 53.4817799880237
x28=34.6725661362236x_{28} = 34.6725661362236
x29=94.290185945407x_{29} = 94.290185945407
x30=3.99444471574142x_{30} = 3.99444471574142
x31=−53.4817799880237x_{31} = -53.4817799880237
x32=−28.4145306971625x_{32} = -28.4145306971625
x33=6.83214574693118x_{33} = 6.83214574693118
x34=−91.1500530451789x_{34} = -91.1500530451789
x35=59.7571356682663x_{35} = 59.7571356682663
x36=−69.1728243307457x_{36} = -69.1728243307457
x37=−94.290185945407x_{37} = -94.290185945407
x38=−19.0575561537385x_{38} = -19.0575561537385
x39=−72.3119117382824x_{39} = -72.3119117382824
x40=−12.8711405784383x_{40} = -12.8711405784383
x41=−50.3448303040845x_{41} = -50.3448303040845
x42=28.4145306971625x_{42} = 28.4145306971625
x43=40.9382191715155x_{43} = 40.9382191715155
x44=37.8046732869526x_{44} = 37.8046732869526
x45=−25.2900904960802x_{45} = -25.2900904960802
x46=84.8701107016488x_{46} = 84.8701107016488
x47=−37.8046732869526x_{47} = -37.8046732869526
x48=22.1703631077661x_{48} = 22.1703631077661
x49=88.0100241275575x_{49} = 88.0100241275575
x50=−9.81900340196872x_{50} = -9.81900340196872
x51=−15.9554654297511x_{51} = -15.9554654297511
x52=62.895397234671x_{52} = 62.895397234671
x53=−81.7303260381702x_{53} = -81.7303260381702
x54=−78.5906855194896x_{54} = -78.5906855194896
x55=19.0575561537385x_{55} = 19.0575561537385
x56=100.570724821846x_{56} = 100.570724821846
x57=25.2900904960802x_{57} = 25.2900904960802
x58=31.5423183719258x_{58} = 31.5423183719258
x59=−88.0100241275575x_{59} = -88.0100241275575
x60=78.5906855194896x_{60} = 78.5906855194896
x61=66.0339743721325x_{61} = 66.0339743721325
x62=50.3448303040845x_{62} = 50.3448303040845
x63=44.0729006762809x_{63} = 44.0729006762809
x64=15.9554654297511x_{64} = 15.9554654297511
x65=−34.6725661362236x_{65} = -34.6725661362236
x66=72.3119117382824x_{66} = 72.3119117382824
x67=12.8711405784383x_{67} = 12.8711405784383

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
[97.4304127980508,∞)\left[97.4304127980508, \infty\right)
Convex at the intervals
(−∞,−97.4304127980508]\left(-\infty, -97.4304127980508\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function 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,
equation of the horizontal asymptote on the left:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
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,
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^2*sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(xsin⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x \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→∞(xsin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(x \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:
x2sin⁡(x)=−x2sin⁡(x)x^{2} \sin{\left(x \right)} = - x^{2} \sin{\left(x \right)}
- No
x2sin⁡(x)=x2sin⁡(x)x^{2} \sin{\left(x \right)} = x^{2} \sin{\left(x \right)}
- Yes
so, the function
is
odd
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