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Plotting a function graph/ -x*sin(x)

Graphing y = -x*sin(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = -x*sin(x)
f(x)=−xsin⁡(x)f{\left(x \right)} = - x \sin{\left(x \right)} 
f = (-x)*sin(x)
The graph of the function
02468-8-6-4-2-1010-1010
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:
−xsin⁡(x)=0- x \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=−75.398223686155x_{1} = -75.398223686155
x2=47.1238898038469x_{2} = 47.1238898038469
x3=−31.4159265358979x_{3} = -31.4159265358979
x4=9.42477796076938x_{4} = 9.42477796076938
x5=−34.5575191894877x_{5} = -34.5575191894877
x6=−97.3893722612836x_{6} = -97.3893722612836
x7=−62.8318530717959x_{7} = -62.8318530717959
x8=87.9645943005142x_{8} = 87.9645943005142
x9=−87.9645943005142x_{9} = -87.9645943005142
x10=−3.14159265358979x_{10} = -3.14159265358979
x11=6.28318530717959x_{11} = 6.28318530717959
x12=59.6902604182061x_{12} = 59.6902604182061
x13=−47.1238898038469x_{13} = -47.1238898038469
x14=−40.8407044966673x_{14} = -40.8407044966673
x15=100.530964914873x_{15} = 100.530964914873
x16=62.8318530717959x_{16} = 62.8318530717959
x17=3.14159265358979x_{17} = 3.14159265358979
x18=28.2743338823081x_{18} = 28.2743338823081
x19=−69.1150383789755x_{19} = -69.1150383789755
x20=97.3893722612836x_{20} = 97.3893722612836
x21=12.5663706143592x_{21} = 12.5663706143592
x22=94.2477796076938x_{22} = 94.2477796076938
x23=31.4159265358979x_{23} = 31.4159265358979
x24=25.1327412287183x_{24} = 25.1327412287183
x25=−37.6991118430775x_{25} = -37.6991118430775
x26=−59.6902604182061x_{26} = -59.6902604182061
x27=−94.2477796076938x_{27} = -94.2477796076938
x28=−56.5486677646163x_{28} = -56.5486677646163
x29=81.6814089933346x_{29} = 81.6814089933346
x30=43.9822971502571x_{30} = 43.9822971502571
x31=−91.106186954104x_{31} = -91.106186954104
x32=15.707963267949x_{32} = 15.707963267949
x33=34.5575191894877x_{33} = 34.5575191894877
x34=21.9911485751286x_{34} = 21.9911485751286
x35=40.8407044966673x_{35} = 40.8407044966673
x36=69.1150383789755x_{36} = 69.1150383789755
x37=65.9734457253857x_{37} = 65.9734457253857
x38=−72.2566310325652x_{38} = -72.2566310325652
x39=−21.9911485751286x_{39} = -21.9911485751286
x40=91.106186954104x_{40} = 91.106186954104
x41=53.4070751110265x_{41} = 53.4070751110265
x42=−28.2743338823081x_{42} = -28.2743338823081
x43=56.5486677646163x_{43} = 56.5486677646163
x44=−65.9734457253857x_{44} = -65.9734457253857
x45=−18.8495559215388x_{45} = -18.8495559215388
x46=−100.530964914873x_{46} = -100.530964914873
x47=−53.4070751110265x_{47} = -53.4070751110265
x48=697.433569096934x_{48} = 697.433569096934
x49=−15.707963267949x_{49} = -15.707963267949
x50=84.8230016469244x_{50} = 84.8230016469244
x51=72.2566310325652x_{51} = 72.2566310325652
x52=18.8495559215388x_{52} = 18.8495559215388
x53=0x_{53} = 0
x54=−43.9822971502571x_{54} = -43.9822971502571
x55=−84.8230016469244x_{55} = -84.8230016469244
x56=−78.5398163397448x_{56} = -78.5398163397448
x57=−12.5663706143592x_{57} = -12.5663706143592
x58=75.398223686155x_{58} = 75.398223686155
x59=−6.28318530717959x_{59} = -6.28318530717959
x60=78.5398163397448x_{60} = 78.5398163397448
x61=−50.2654824574367x_{61} = -50.2654824574367
x62=−81.6814089933346x_{62} = -81.6814089933346
x63=50.2654824574367x_{63} = 50.2654824574367
x64=−9.42477796076938x_{64} = -9.42477796076938
x65=37.6991118430775x_{65} = 37.6991118430775
x66=−25.1327412287183x_{66} = -25.1327412287183
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (-x)*sin(x).
−0sin⁡(0)- 0 \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
−xcos⁡(x)−sin⁡(x)=0- x \cos{\left(x \right)} - \sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−80.1230928148503x_{1} = -80.1230928148503
x2=92.687771772017x_{2} = 92.687771772017
x3=−70.69997803861x_{3} = -70.69997803861
x4=54.9960525574964x_{4} = 54.9960525574964
x5=−20.469167402741x_{5} = -20.469167402741
x6=−42.4350618814099x_{6} = -42.4350618814099
x7=0x_{7} = 0
x8=−17.3363779239834x_{8} = -17.3363779239834
x9=−45.57503179559x_{9} = -45.57503179559
x10=−76.9820093304187x_{10} = -76.9820093304187
x11=29.8785865061074x_{11} = 29.8785865061074
x12=45.57503179559x_{12} = 45.57503179559
x13=89.5465575382492x_{13} = 89.5465575382492
x14=36.1559664195367x_{14} = 36.1559664195367
x15=−29.8785865061074x_{15} = -29.8785865061074
x16=−48.7152107175577x_{16} = -48.7152107175577
x17=−23.6042847729804x_{17} = -23.6042847729804
x18=95.8290108090195x_{18} = 95.8290108090195
x19=−58.1366632448992x_{19} = -58.1366632448992
x20=−61.2773745335697x_{20} = -61.2773745335697
x21=−4.91318043943488x_{21} = -4.91318043943488
x22=−33.0170010333572x_{22} = -33.0170010333572
x23=76.9820093304187x_{23} = 76.9820093304187
x24=−14.2074367251912x_{24} = -14.2074367251912
x25=42.4350618814099x_{25} = 42.4350618814099
x26=23.6042847729804x_{26} = 23.6042847729804
x27=14.2074367251912x_{27} = 14.2074367251912
x28=−86.4053708116885x_{28} = -86.4053708116885
x29=70.69997803861x_{29} = 70.69997803861
x30=−39.295350981473x_{30} = -39.295350981473
x31=20.469167402741x_{31} = 20.469167402741
x32=−26.7409160147873x_{32} = -26.7409160147873
x33=83.2642147040886x_{33} = 83.2642147040886
x34=−64.4181717218392x_{34} = -64.4181717218392
x35=80.1230928148503x_{35} = 80.1230928148503
x36=−89.5465575382492x_{36} = -89.5465575382492
x37=−7.97866571241324x_{37} = -7.97866571241324
x38=33.0170010333572x_{38} = 33.0170010333572
x39=73.8409691490209x_{39} = 73.8409691490209
x40=−83.2642147040886x_{40} = -83.2642147040886
x41=11.085538406497x_{41} = 11.085538406497
x42=86.4053708116885x_{42} = 86.4053708116885
x43=51.855560729152x_{43} = 51.855560729152
x44=−51.855560729152x_{44} = -51.855560729152
x45=4.91318043943488x_{45} = 4.91318043943488
x46=−2.02875783811043x_{46} = -2.02875783811043
x47=26.7409160147873x_{47} = 26.7409160147873
x48=48.7152107175577x_{48} = 48.7152107175577
x49=39.295350981473x_{49} = 39.295350981473
x50=67.5590428388084x_{50} = 67.5590428388084
x51=−54.9960525574964x_{51} = -54.9960525574964
x52=64.4181717218392x_{52} = 64.4181717218392
x53=2.02875783811043x_{53} = 2.02875783811043
x54=98.9702722883957x_{54} = 98.9702722883957
x55=−73.8409691490209x_{55} = -73.8409691490209
x56=61.2773745335697x_{56} = 61.2773745335697
x57=7.97866571241324x_{57} = 7.97866571241324
x58=−67.5590428388084x_{58} = -67.5590428388084
x59=58.1366632448992x_{59} = 58.1366632448992
x60=−92.687771772017x_{60} = -92.687771772017
x61=−98.9702722883957x_{61} = -98.9702722883957
x62=102.111554139654x_{62} = 102.111554139654
x63=−11.085538406497x_{63} = -11.085538406497
x64=−36.1559664195367x_{64} = -36.1559664195367
x65=−95.8290108090195x_{65} = -95.8290108090195
x66=17.3363779239834x_{66} = 17.3363779239834
The values of the extrema at the points:
(-80.12309281485025, 80.1168531456592)

(92.687771772017, 92.6823777880592)

(-70.69997803861, -70.6929069615931)

(54.99605255749639, 54.9869632496976)

(-20.46916740274095, -20.4447840582523)

(-42.43506188140989, 42.4232840772591)

(0, 0)

(-17.33637792398336, 17.3076086078585)

(-45.57503179559002, -45.5640648360268)

(-76.98200933041872, -76.9755151282637)

(29.878586506107393, 29.8618661591868)

(45.57503179559002, -45.5640648360268)

(89.54655753824919, -89.5409743728852)

(36.15596641953672, 36.1421453722421)

(-29.878586506107393, 29.8618661591868)

(-48.715210717557724, 48.7049502253679)

(-23.604284772980407, 23.5831306496334)

(95.82901080901948, -95.8237936084657)

(-58.13666324489916, -58.1280647280857)

(-61.277374533569656, 61.2692165444766)

(-4.913180439434884, 4.81446988971227)

(-33.017001033357246, -33.0018677308454)

(76.98200933041872, -76.9755151282637)

(-14.207436725191188, -14.1723741137743)

(42.43506188140989, 42.4232840772591)

(23.604284772980407, 23.5831306496334)

(14.207436725191188, -14.1723741137743)

(-86.40537081168854, 86.3995847156108)

(70.69997803861, -70.6929069615931)

(-39.295350981472986, -39.2826330068918)

(20.46916740274095, -20.4447840582523)

(-26.74091601478731, -26.7222376646974)

(83.26421470408864, -83.2582103729533)

(-64.41817172183916, -64.4104113393753)

(80.12309281485025, 80.1168531456592)

(-89.54655753824919, -89.5409743728852)

(-7.978665712413241, -7.91672737158778)

(33.017001033357246, -33.0018677308454)

(73.8409691490209, 73.8341987715416)

(-83.26421470408864, -83.2582103729533)

(11.085538406497022, 11.04070801593)

(86.40537081168854, 86.3995847156108)

(51.85556072915197, -51.8459212502015)

(-51.85556072915197, -51.8459212502015)

(4.913180439434884, 4.81446988971227)

(-2.028757838110434, -1.81970574115965)

(26.74091601478731, -26.7222376646974)

(48.715210717557724, 48.7049502253679)

(39.295350981472986, -39.2826330068918)

(67.5590428388084, 67.5516431209725)

(-54.99605255749639, 54.9869632496976)

(64.41817172183916, -64.4104113393753)

(2.028757838110434, -1.81970574115965)

(98.9702722883957, 98.9652206531187)

(-73.8409691490209, 73.8341987715416)

(61.277374533569656, 61.2692165444766)

(7.978665712413241, -7.91672737158778)

(-67.5590428388084, 67.5516431209725)

(58.13666324489916, -58.1280647280857)

(-92.687771772017, 92.6823777880592)

(-98.9702722883957, 98.9652206531187)

(102.11155413965392, -102.106657886316)

(-11.085538406497022, 11.04070801593)

(-36.15596641953672, 36.1421453722421)

(-95.82901080901948, -95.8237936084657)

(17.33637792398336, 17.3076086078585)


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=−70.69997803861x_{1} = -70.69997803861
x2=−20.469167402741x_{2} = -20.469167402741
x3=−45.57503179559x_{3} = -45.57503179559
x4=−76.9820093304187x_{4} = -76.9820093304187
x5=45.57503179559x_{5} = 45.57503179559
x6=89.5465575382492x_{6} = 89.5465575382492
x7=95.8290108090195x_{7} = 95.8290108090195
x8=−58.1366632448992x_{8} = -58.1366632448992
x9=−33.0170010333572x_{9} = -33.0170010333572
x10=76.9820093304187x_{10} = 76.9820093304187
x11=−14.2074367251912x_{11} = -14.2074367251912
x12=14.2074367251912x_{12} = 14.2074367251912
x13=70.69997803861x_{13} = 70.69997803861
x14=−39.295350981473x_{14} = -39.295350981473
x15=20.469167402741x_{15} = 20.469167402741
x16=−26.7409160147873x_{16} = -26.7409160147873
x17=83.2642147040886x_{17} = 83.2642147040886
x18=−64.4181717218392x_{18} = -64.4181717218392
x19=−89.5465575382492x_{19} = -89.5465575382492
x20=−7.97866571241324x_{20} = -7.97866571241324
x21=33.0170010333572x_{21} = 33.0170010333572
x22=−83.2642147040886x_{22} = -83.2642147040886
x23=51.855560729152x_{23} = 51.855560729152
x24=−51.855560729152x_{24} = -51.855560729152
x25=−2.02875783811043x_{25} = -2.02875783811043
x26=26.7409160147873x_{26} = 26.7409160147873
x27=39.295350981473x_{27} = 39.295350981473
x28=64.4181717218392x_{28} = 64.4181717218392
x29=2.02875783811043x_{29} = 2.02875783811043
x30=7.97866571241324x_{30} = 7.97866571241324
x31=58.1366632448992x_{31} = 58.1366632448992
x32=102.111554139654x_{32} = 102.111554139654
x33=−95.8290108090195x_{33} = -95.8290108090195
Maxima of the function at points:
x33=−80.1230928148503x_{33} = -80.1230928148503
x33=92.687771772017x_{33} = 92.687771772017
x33=54.9960525574964x_{33} = 54.9960525574964
x33=−42.4350618814099x_{33} = -42.4350618814099
x33=0x_{33} = 0
x33=−17.3363779239834x_{33} = -17.3363779239834
x33=29.8785865061074x_{33} = 29.8785865061074
x33=36.1559664195367x_{33} = 36.1559664195367
x33=−29.8785865061074x_{33} = -29.8785865061074
x33=−48.7152107175577x_{33} = -48.7152107175577
x33=−23.6042847729804x_{33} = -23.6042847729804
x33=−61.2773745335697x_{33} = -61.2773745335697
x33=−4.91318043943488x_{33} = -4.91318043943488
x33=42.4350618814099x_{33} = 42.4350618814099
x33=23.6042847729804x_{33} = 23.6042847729804
x33=−86.4053708116885x_{33} = -86.4053708116885
x33=80.1230928148503x_{33} = 80.1230928148503
x33=73.8409691490209x_{33} = 73.8409691490209
x33=11.085538406497x_{33} = 11.085538406497
x33=86.4053708116885x_{33} = 86.4053708116885
x33=4.91318043943488x_{33} = 4.91318043943488
x33=48.7152107175577x_{33} = 48.7152107175577
x33=67.5590428388084x_{33} = 67.5590428388084
x33=−54.9960525574964x_{33} = -54.9960525574964
x33=98.9702722883957x_{33} = 98.9702722883957
x33=−73.8409691490209x_{33} = -73.8409691490209
x33=61.2773745335697x_{33} = 61.2773745335697
x33=−67.5590428388084x_{33} = -67.5590428388084
x33=−92.687771772017x_{33} = -92.687771772017
x33=−98.9702722883957x_{33} = -98.9702722883957
x33=−11.085538406497x_{33} = -11.085538406497
x33=−36.1559664195367x_{33} = -36.1559664195367
x33=17.3363779239834x_{33} = 17.3363779239834
Decreasing at intervals
[102.111554139654,∞)\left[102.111554139654, \infty\right)
Increasing at intervals
(−∞,−95.8290108090195]\left(-\infty, -95.8290108090195\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
xsin⁡(x)−2cos⁡(x)=0x \sin{\left(x \right)} - 2 \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−59.7237354324305x_{1} = -59.7237354324305
x2=−81.7058821480364x_{2} = -81.7058821480364
x3=40.8895777660408x_{3} = 40.8895777660408
x4=100.550852725424x_{4} = 100.550852725424
x5=97.4099011706723x_{5} = 97.4099011706723
x6=56.5839987378634x_{6} = 56.5839987378634
x7=91.1281305511393x_{7} = 91.1281305511393
x8=−44.0276918992479x_{8} = -44.0276918992479
x9=−53.4444796697636x_{9} = -53.4444796697636
x10=−9.62956034329743x_{10} = -9.62956034329743
x11=−3.6435971674254x_{11} = -3.6435971674254
x12=31.479374920314x_{12} = 31.479374920314
x13=75.4247339745236x_{13} = 75.4247339745236
x14=−128.820822990274x_{14} = -128.820822990274
x15=3.6435971674254x_{15} = 3.6435971674254
x16=59.7237354324305x_{16} = 59.7237354324305
x17=53.4444796697636x_{17} = 53.4444796697636
x18=−50.3052188363296x_{18} = -50.3052188363296
x19=−78.5652673845995x_{19} = -78.5652673845995
x20=9.62956034329743x_{20} = 9.62956034329743
x21=−87.9873209346887x_{21} = -87.9873209346887
x22=−100.550852725424x_{22} = -100.550852725424
x23=−47.1662676027767x_{23} = -47.1662676027767
x24=−6.57833373272234x_{24} = -6.57833373272234
x25=−15.8336114149477x_{25} = -15.8336114149477
x26=−28.3447768697864x_{26} = -28.3447768697864
x27=−18.954681766529x_{27} = -18.954681766529
x28=−75.4247339745236x_{28} = -75.4247339745236
x29=−62.863657228703x_{29} = -62.863657228703
x30=−91.1281305511393x_{30} = -91.1281305511393
x31=50.3052188363296x_{31} = 50.3052188363296
x32=72.2842925036825x_{32} = 72.2842925036825
x33=81.7058821480364x_{33} = 81.7058821480364
x34=66.0037377708277x_{34} = 66.0037377708277
x35=18.954681766529x_{35} = 18.954681766529
x36=−72.2842925036825x_{36} = -72.2842925036825
x37=62.863657228703x_{37} = 62.863657228703
x38=−69.1439554764926x_{38} = -69.1439554764926
x39=−22.0814757672807x_{39} = -22.0814757672807
x40=69.1439554764926x_{40} = 69.1439554764926
x41=−40.8895777660408x_{41} = -40.8895777660408
x42=15.8336114149477x_{42} = 15.8336114149477
x43=−31.479374920314x_{43} = -31.479374920314
x44=−56.5839987378634x_{44} = -56.5839987378634
x45=25.2119030642106x_{45} = 25.2119030642106
x46=28.3447768697864x_{46} = 28.3447768697864
x47=1.0768739863118x_{47} = 1.0768739863118
x48=37.7520396346102x_{48} = 37.7520396346102
x49=78.5652673845995x_{49} = 78.5652673845995
x50=12.7222987717666x_{50} = 12.7222987717666
x51=87.9873209346887x_{51} = 87.9873209346887
x52=6.57833373272234x_{52} = 6.57833373272234
x53=−34.6152330552306x_{53} = -34.6152330552306
x54=34.6152330552306x_{54} = 34.6152330552306
x55=−94.2689923093066x_{55} = -94.2689923093066
x56=−37.7520396346102x_{56} = -37.7520396346102
x57=22.0814757672807x_{57} = 22.0814757672807
x58=−25.2119030642106x_{58} = -25.2119030642106
x59=−1.0768739863118x_{59} = -1.0768739863118
x60=−66.0037377708277x_{60} = -66.0037377708277
x61=94.2689923093066x_{61} = 94.2689923093066
x62=−12.7222987717666x_{62} = -12.7222987717666
x63=44.0276918992479x_{63} = 44.0276918992479
x64=−97.4099011706723x_{64} = -97.4099011706723
x65=47.1662676027767x_{65} = 47.1662676027767
x66=−84.8465692433091x_{66} = -84.8465692433091
x67=84.8465692433091x_{67} = 84.8465692433091

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
[100.550852725424,∞)\left[100.550852725424, \infty\right)
Convex at the intervals
(−∞,−128.820822990274]\left(-\infty, -128.820822990274\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function 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,
equation of the horizontal asymptote on the left:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
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,
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)*sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(−sin⁡(x))=⟨−1,1⟩\lim_{x \to -\infty}\left(- \sin{\left(x \right)}\right) = \left\langle -1, 1\right\rangle
Let's take the limit
so,
inclined asymptote equation on the left:
y=⟨−1,1⟩xy = \left\langle -1, 1\right\rangle x
lim⁡x→∞(−sin⁡(x))=⟨−1,1⟩\lim_{x \to \infty}\left(- \sin{\left(x \right)}\right) = \left\langle -1, 1\right\rangle
Let's take the limit
so,
inclined asymptote equation on the right:
y=⟨−1,1⟩xy = \left\langle -1, 1\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:
−xsin⁡(x)=−xsin⁡(x)- x \sin{\left(x \right)} = - x \sin{\left(x \right)}
- No
−xsin⁡(x)=xsin⁡(x)- x \sin{\left(x \right)} = x \sin{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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