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Plotting a function graph/ sinx(x)

Graphing y = sinx(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = sin(x)*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)=0x \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=25.1327412287183x_{8} = 25.1327412287183
x9=−3.14159265358979x_{9} = -3.14159265358979
x10=−6.28318530717959x_{10} = -6.28318530717959
x11=−40.8407044966673x_{11} = -40.8407044966673
x12=−18.8495559215388x_{12} = -18.8495559215388
x13=78.5398163397448x_{13} = 78.5398163397448
x14=−75.398223686155x_{14} = -75.398223686155
x15=−9.42477796076938x_{15} = -9.42477796076938
x16=72.2566310325652x_{16} = 72.2566310325652
x17=−43.9822971502571x_{17} = -43.9822971502571
x18=31.4159265358979x_{18} = 31.4159265358979
x19=9.42477796076938x_{19} = 9.42477796076938
x20=40.8407044966673x_{20} = 40.8407044966673
x21=−69.1150383789755x_{21} = -69.1150383789755
x22=12.5663706143592x_{22} = 12.5663706143592
x23=87.9645943005142x_{23} = 87.9645943005142
x24=59.6902604182061x_{24} = 59.6902604182061
x25=−37.6991118430775x_{25} = -37.6991118430775
x26=−100.530964914873x_{26} = -100.530964914873
x27=697.433569096934x_{27} = 697.433569096934
x28=97.3893722612836x_{28} = 97.3893722612836
x29=0x_{29} = 0
x30=−12.5663706143592x_{30} = -12.5663706143592
x31=−78.5398163397448x_{31} = -78.5398163397448
x32=18.8495559215388x_{32} = 18.8495559215388
x33=34.5575191894877x_{33} = 34.5575191894877
x34=−94.2477796076938x_{34} = -94.2477796076938
x35=−91.106186954104x_{35} = -91.106186954104
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=−97.3893722612836x_{53} = -97.3893722612836
x54=21.9911485751286x_{54} = 21.9911485751286
x55=65.9734457253857x_{55} = 65.9734457253857
x56=37.6991118430775x_{56} = 37.6991118430775
x57=−87.9645943005142x_{57} = -87.9645943005142
x58=−72.2566310325652x_{58} = -72.2566310325652
x59=−34.5575191894877x_{59} = -34.5575191894877
x60=−25.1327412287183x_{60} = -25.1327412287183
x61=−28.2743338823081x_{61} = -28.2743338823081
x62=81.6814089933346x_{62} = 81.6814089933346
x63=6.28318530717959x_{63} = 6.28318530717959
x64=100.530964914873x_{64} = 100.530964914873
x65=53.4070751110265x_{65} = 53.4070751110265
x66=−47.1238898038469x_{66} = -47.1238898038469
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x)*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)=0x \cos{\left(x \right)} + \sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−64.4181717218392x_{1} = -64.4181717218392
x2=51.855560729152x_{2} = 51.855560729152
x3=−58.1366632448992x_{3} = -58.1366632448992
x4=26.7409160147873x_{4} = 26.7409160147873
x5=−17.3363779239834x_{5} = -17.3363779239834
x6=−61.2773745335697x_{6} = -61.2773745335697
x7=−42.4350618814099x_{7} = -42.4350618814099
x8=102.111554139654x_{8} = 102.111554139654
x9=29.8785865061074x_{9} = 29.8785865061074
x10=64.4181717218392x_{10} = 64.4181717218392
x11=−2.02875783811043x_{11} = -2.02875783811043
x12=−33.0170010333572x_{12} = -33.0170010333572
x13=73.8409691490209x_{13} = 73.8409691490209
x14=−80.1230928148503x_{14} = -80.1230928148503
x15=89.5465575382492x_{15} = 89.5465575382492
x16=−20.469167402741x_{16} = -20.469167402741
x17=−26.7409160147873x_{17} = -26.7409160147873
x18=−36.1559664195367x_{18} = -36.1559664195367
x19=33.0170010333572x_{19} = 33.0170010333572
x20=20.469167402741x_{20} = 20.469167402741
x21=54.9960525574964x_{21} = 54.9960525574964
x22=7.97866571241324x_{22} = 7.97866571241324
x23=−14.2074367251912x_{23} = -14.2074367251912
x24=39.295350981473x_{24} = 39.295350981473
x25=36.1559664195367x_{25} = 36.1559664195367
x26=83.2642147040886x_{26} = 83.2642147040886
x27=86.4053708116885x_{27} = 86.4053708116885
x28=−92.687771772017x_{28} = -92.687771772017
x29=−29.8785865061074x_{29} = -29.8785865061074
x30=−67.5590428388084x_{30} = -67.5590428388084
x31=76.9820093304187x_{31} = 76.9820093304187
x32=−11.085538406497x_{32} = -11.085538406497
x33=70.69997803861x_{33} = 70.69997803861
x34=−51.855560729152x_{34} = -51.855560729152
x35=48.7152107175577x_{35} = 48.7152107175577
x36=17.3363779239834x_{36} = 17.3363779239834
x37=−4.91318043943488x_{37} = -4.91318043943488
x38=−86.4053708116885x_{38} = -86.4053708116885
x39=92.687771772017x_{39} = 92.687771772017
x40=−39.295350981473x_{40} = -39.295350981473
x41=−73.8409691490209x_{41} = -73.8409691490209
x42=80.1230928148503x_{42} = 80.1230928148503
x43=58.1366632448992x_{43} = 58.1366632448992
x44=−45.57503179559x_{44} = -45.57503179559
x45=67.5590428388084x_{45} = 67.5590428388084
x46=−89.5465575382492x_{46} = -89.5465575382492
x47=−70.69997803861x_{47} = -70.69997803861
x48=95.8290108090195x_{48} = 95.8290108090195
x49=11.085538406497x_{49} = 11.085538406497
x50=−95.8290108090195x_{50} = -95.8290108090195
x51=0x_{51} = 0
x52=98.9702722883957x_{52} = 98.9702722883957
x53=2.02875783811043x_{53} = 2.02875783811043
x54=−83.2642147040886x_{54} = -83.2642147040886
x55=4.91318043943488x_{55} = 4.91318043943488
x56=−23.6042847729804x_{56} = -23.6042847729804
x57=−48.7152107175577x_{57} = -48.7152107175577
x58=−76.9820093304187x_{58} = -76.9820093304187
x59=61.2773745335697x_{59} = 61.2773745335697
x60=42.4350618814099x_{60} = 42.4350618814099
x61=−54.9960525574964x_{61} = -54.9960525574964
x62=−7.97866571241324x_{62} = -7.97866571241324
x63=−98.9702722883957x_{63} = -98.9702722883957
x64=23.6042847729804x_{64} = 23.6042847729804
x65=45.57503179559x_{65} = 45.57503179559
x66=14.2074367251912x_{66} = 14.2074367251912
The values of the extrema at the points:
(-64.41817172183916, 64.4104113393753)

(51.85556072915197, 51.8459212502015)

(-58.13666324489916, 58.1280647280857)

(26.74091601478731, 26.7222376646974)

(-17.33637792398336, -17.3076086078585)

(-61.277374533569656, -61.2692165444766)

(-42.43506188140989, -42.4232840772591)

(102.11155413965392, 102.106657886316)

(29.878586506107393, -29.8618661591868)

(64.41817172183916, 64.4104113393753)

(-2.028757838110434, 1.81970574115965)

(-33.017001033357246, 33.0018677308454)

(73.8409691490209, -73.8341987715416)

(-80.12309281485025, -80.1168531456592)

(89.54655753824919, 89.5409743728852)

(-20.46916740274095, 20.4447840582523)

(-26.74091601478731, 26.7222376646974)

(-36.15596641953672, -36.1421453722421)

(33.017001033357246, 33.0018677308454)

(20.46916740274095, 20.4447840582523)

(54.99605255749639, -54.9869632496976)

(7.978665712413241, 7.91672737158778)

(-14.207436725191188, 14.1723741137743)

(39.295350981472986, 39.2826330068918)

(36.15596641953672, -36.1421453722421)

(83.26421470408864, 83.2582103729533)

(86.40537081168854, -86.3995847156108)

(-92.687771772017, -92.6823777880592)

(-29.878586506107393, -29.8618661591868)

(-67.5590428388084, -67.5516431209725)

(76.98200933041872, 76.9755151282637)

(-11.085538406497022, -11.04070801593)

(70.69997803861, 70.6929069615931)

(-51.85556072915197, 51.8459212502015)

(48.715210717557724, -48.7049502253679)

(17.33637792398336, -17.3076086078585)

(-4.913180439434884, -4.81446988971227)

(-86.40537081168854, -86.3995847156108)

(92.687771772017, -92.6823777880592)

(-39.295350981472986, 39.2826330068918)

(-73.8409691490209, -73.8341987715416)

(80.12309281485025, -80.1168531456592)

(58.13666324489916, 58.1280647280857)

(-45.57503179559002, 45.5640648360268)

(67.5590428388084, -67.5516431209725)

(-89.54655753824919, 89.5409743728852)

(-70.69997803861, 70.6929069615931)

(95.82901080901948, 95.8237936084657)

(11.085538406497022, -11.04070801593)

(-95.82901080901948, 95.8237936084657)

(0, 0)

(98.9702722883957, -98.9652206531187)

(2.028757838110434, 1.81970574115965)

(-83.26421470408864, 83.2582103729533)

(4.913180439434884, -4.81446988971227)

(-23.604284772980407, -23.5831306496334)

(-48.715210717557724, -48.7049502253679)

(-76.98200933041872, 76.9755151282637)

(61.277374533569656, -61.2692165444766)

(42.43506188140989, -42.4232840772591)

(-54.99605255749639, -54.9869632496976)

(-7.978665712413241, 7.91672737158778)

(-98.9702722883957, -98.9652206531187)

(23.604284772980407, -23.5831306496334)

(45.57503179559002, 45.5640648360268)

(14.207436725191188, 14.1723741137743)


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

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
[97.4099011706723,∞)\left[97.4099011706723, \infty\right)
Convex at the intervals
(−∞,−100.550852725424]\left(-\infty, -100.550852725424\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 sin(x)*x, divided by x at x->+oo and x ->-oo
lim⁡x→−∞sin⁡(x)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(x \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} \sin{\left(x \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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