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Plotting a function graph/ sin(x)+x*cos(x)

Graphing y = sin(x)+x*cos(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = sin(x) + x*cos(x)
f(x)=xcos⁡(x)+sin⁡(x)f{\left(x \right)} = x \cos{\left(x \right)} + \sin{\left(x \right)} 
f = x*cos(x) + sin(x)
The graph of the function
02468-8-6-4-2-1010-2020
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:
xcos⁡(x)+sin⁡(x)=0x \cos{\left(x \right)} + \sin{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
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 points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x) + x*cos(x).
sin⁡(0)+0cos⁡(0)\sin{\left(0 \right)} + 0 \cos{\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
−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=78.5652673845995x_{1} = 78.5652673845995
x2=15.8336114149477x_{2} = 15.8336114149477
x3=18.954681766529x_{3} = 18.954681766529
x4=−40.8895777660408x_{4} = -40.8895777660408
x5=81.7058821480364x_{5} = 81.7058821480364
x6=−59.7237354324305x_{6} = -59.7237354324305
x7=−28.3447768697864x_{7} = -28.3447768697864
x8=87.9873209346887x_{8} = 87.9873209346887
x9=−1.0768739863118x_{9} = -1.0768739863118
x10=−53.4444796697636x_{10} = -53.4444796697636
x11=−44.0276918992479x_{11} = -44.0276918992479
x12=84.8465692433091x_{12} = 84.8465692433091
x13=−94.2689923093066x_{13} = -94.2689923093066
x14=12.7222987717666x_{14} = 12.7222987717666
x15=−100.550852725424x_{15} = -100.550852725424
x16=59.7237354324305x_{16} = 59.7237354324305
x17=94.2689923093066x_{17} = 94.2689923093066
x18=−50.3052188363296x_{18} = -50.3052188363296
x19=−128.820822990274x_{19} = -128.820822990274
x20=44.0276918992479x_{20} = 44.0276918992479
x21=28.3447768697864x_{21} = 28.3447768697864
x22=31.479374920314x_{22} = 31.479374920314
x23=53.4444796697636x_{23} = 53.4444796697636
x24=1.0768739863118x_{24} = 1.0768739863118
x25=−37.7520396346102x_{25} = -37.7520396346102
x26=−69.1439554764926x_{26} = -69.1439554764926
x27=−66.0037377708277x_{27} = -66.0037377708277
x28=22.0814757672807x_{28} = 22.0814757672807
x29=−75.4247339745236x_{29} = -75.4247339745236
x30=100.550852725424x_{30} = 100.550852725424
x31=97.4099011706723x_{31} = 97.4099011706723
x32=66.0037377708277x_{32} = 66.0037377708277
x33=−72.2842925036825x_{33} = -72.2842925036825
x34=9.62956034329743x_{34} = 9.62956034329743
x35=37.7520396346102x_{35} = 37.7520396346102
x36=−78.5652673845995x_{36} = -78.5652673845995
x37=−81.7058821480364x_{37} = -81.7058821480364
x38=−15.8336114149477x_{38} = -15.8336114149477
x39=−62.863657228703x_{39} = -62.863657228703
x40=−12.7222987717666x_{40} = -12.7222987717666
x41=3.6435971674254x_{41} = 3.6435971674254
x42=56.5839987378634x_{42} = 56.5839987378634
x43=25.2119030642106x_{43} = 25.2119030642106
x44=−9.62956034329743x_{44} = -9.62956034329743
x45=−84.8465692433091x_{45} = -84.8465692433091
x46=−3.6435971674254x_{46} = -3.6435971674254
x47=50.3052188363296x_{47} = 50.3052188363296
x48=−56.5839987378634x_{48} = -56.5839987378634
x49=−91.1281305511393x_{49} = -91.1281305511393
x50=−34.6152330552306x_{50} = -34.6152330552306
x51=−6.57833373272234x_{51} = -6.57833373272234
x52=62.863657228703x_{52} = 62.863657228703
x53=75.4247339745236x_{53} = 75.4247339745236
x54=47.1662676027767x_{54} = 47.1662676027767
x55=69.1439554764926x_{55} = 69.1439554764926
x56=−97.4099011706723x_{56} = -97.4099011706723
x57=6.57833373272234x_{57} = 6.57833373272234
x58=40.8895777660408x_{58} = 40.8895777660408
x59=34.6152330552306x_{59} = 34.6152330552306
x60=−87.9873209346887x_{60} = -87.9873209346887
x61=−31.479374920314x_{61} = -31.479374920314
x62=−47.1662676027767x_{62} = -47.1662676027767
x63=72.2842925036825x_{63} = 72.2842925036825
x64=−25.2119030642106x_{64} = -25.2119030642106
x65=91.1281305511393x_{65} = 91.1281305511393
x66=−22.0814757672807x_{66} = -22.0814757672807
x67=−18.954681766529x_{67} = -18.954681766529
The values of the extrema at the points:
(78.56526738459954, -78.5652715061143)

(15.833611414947718, -15.834107331638)

(18.954681766529042, 18.9549722147554)

(-40.889577766040844, 40.8896069506711)

(81.70588214803641, 81.7058858124955)

(-59.72373543243046, 59.7237448102597)

(-28.344776869786372, 28.3448642580985)

(87.9873209346887, 87.9873238692648)

(-1.0768739863118038, -1.39100784545588)

(-53.44447966976355, 53.4444927529527)

(-44.02769189924788, -44.0277152852979)

(84.84656924330915, -84.8465725158561)

(-94.26899230930657, -94.2689946956226)

(12.722298771766635, 12.7232465674385)

(-100.55085272542402, -100.550854691956)

(59.72373543243046, -59.7237448102597)

(94.26899230930657, 94.2689946956226)

(-50.30521883632959, -50.3052345220647)

(-128.8208229902735, 128.820823925608)

(44.02769189924788, 44.0277152852979)

(28.344776869786372, -28.3448642580985)

(31.479374920314047, 31.4794387763188)

(53.44447966976355, -53.4444927529527)

(1.0768739863118038, 1.39100784545588)

(-37.75203963461023, -37.7520767019434)

(-69.1439554764926, -69.1439615216012)

(-66.00373777082767, 66.0037447198836)

(22.081475767280747, -22.0816600122592)

(-75.4247339745236, -75.4247386323507)

(100.55085272542402, 100.550854691956)

(97.40990117067226, -97.4099033335782)

(66.00373777082767, -66.0037447198836)

(-72.2842925036825, 72.2842977950245)

(9.62956034329743, -9.63170728857969)

(37.75203963461023, 37.7520767019434)

(-78.56526738459954, 78.5652715061143)

(-81.70588214803641, -81.7058858124955)

(-15.833611414947718, 15.834107331638)

(-62.863657228703005, -62.8636652712142)

(-12.722298771766635, -12.7232465674385)

(3.643597167425401, -3.67523306366032)

(56.58399873786344, 56.5840097635798)

(25.21190306421058, 25.2120270830452)

(-9.62956034329743, 9.63170728857969)

(-84.84656924330915, 84.8465725158561)

(-3.643597167425401, 3.67523306366032)

(50.30521883632959, 50.3052345220647)

(-56.58399873786344, -56.5840097635798)

(-91.1281305511393, 91.1281331927175)

(-34.61523305523058, 34.6152811148717)

(-6.578333732722339, -6.58476172355643)

(62.863657228703005, 62.8636652712142)

(75.4247339745236, 75.4247386323507)

(47.1662676027767, -47.1662866291145)

(69.1439554764926, 69.1439615216012)

(-97.40990117067226, 97.4099033335782)

(6.578333732722339, 6.58476172355643)

(40.889577766040844, -40.8896069506711)

(34.61523305523058, -34.6152811148717)

(-87.9873209346887, -87.9873238692648)

(-31.479374920314047, -31.4794387763188)

(-47.1662676027767, 47.1662866291145)

(72.2842925036825, -72.2842977950245)

(-25.21190306421058, -25.2120270830452)

(91.1281305511393, -91.1281331927175)

(-22.081475767280747, 22.0816600122592)

(-18.954681766529042, -18.9549722147554)


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=78.5652673845995x_{1} = 78.5652673845995
x2=15.8336114149477x_{2} = 15.8336114149477
x3=−1.0768739863118x_{3} = -1.0768739863118
x4=−44.0276918992479x_{4} = -44.0276918992479
x5=84.8465692433091x_{5} = 84.8465692433091
x6=−94.2689923093066x_{6} = -94.2689923093066
x7=−100.550852725424x_{7} = -100.550852725424
x8=59.7237354324305x_{8} = 59.7237354324305
x9=−50.3052188363296x_{9} = -50.3052188363296
x10=28.3447768697864x_{10} = 28.3447768697864
x11=53.4444796697636x_{11} = 53.4444796697636
x12=−37.7520396346102x_{12} = -37.7520396346102
x13=−69.1439554764926x_{13} = -69.1439554764926
x14=22.0814757672807x_{14} = 22.0814757672807
x15=−75.4247339745236x_{15} = -75.4247339745236
x16=97.4099011706723x_{16} = 97.4099011706723
x17=66.0037377708277x_{17} = 66.0037377708277
x18=9.62956034329743x_{18} = 9.62956034329743
x19=−81.7058821480364x_{19} = -81.7058821480364
x20=−62.863657228703x_{20} = -62.863657228703
x21=−12.7222987717666x_{21} = -12.7222987717666
x22=3.6435971674254x_{22} = 3.6435971674254
x23=−56.5839987378634x_{23} = -56.5839987378634
x24=−6.57833373272234x_{24} = -6.57833373272234
x25=47.1662676027767x_{25} = 47.1662676027767
x26=40.8895777660408x_{26} = 40.8895777660408
x27=34.6152330552306x_{27} = 34.6152330552306
x28=−87.9873209346887x_{28} = -87.9873209346887
x29=−31.479374920314x_{29} = -31.479374920314
x30=72.2842925036825x_{30} = 72.2842925036825
x31=−25.2119030642106x_{31} = -25.2119030642106
x32=91.1281305511393x_{32} = 91.1281305511393
x33=−18.954681766529x_{33} = -18.954681766529
Maxima of the function at points:
x33=18.954681766529x_{33} = 18.954681766529
x33=−40.8895777660408x_{33} = -40.8895777660408
x33=81.7058821480364x_{33} = 81.7058821480364
x33=−59.7237354324305x_{33} = -59.7237354324305
x33=−28.3447768697864x_{33} = -28.3447768697864
x33=87.9873209346887x_{33} = 87.9873209346887
x33=−53.4444796697636x_{33} = -53.4444796697636
x33=12.7222987717666x_{33} = 12.7222987717666
x33=94.2689923093066x_{33} = 94.2689923093066
x33=−128.820822990274x_{33} = -128.820822990274
x33=44.0276918992479x_{33} = 44.0276918992479
x33=31.479374920314x_{33} = 31.479374920314
x33=1.0768739863118x_{33} = 1.0768739863118
x33=−66.0037377708277x_{33} = -66.0037377708277
x33=100.550852725424x_{33} = 100.550852725424
x33=−72.2842925036825x_{33} = -72.2842925036825
x33=37.7520396346102x_{33} = 37.7520396346102
x33=−78.5652673845995x_{33} = -78.5652673845995
x33=−15.8336114149477x_{33} = -15.8336114149477
x33=56.5839987378634x_{33} = 56.5839987378634
x33=25.2119030642106x_{33} = 25.2119030642106
x33=−9.62956034329743x_{33} = -9.62956034329743
x33=−84.8465692433091x_{33} = -84.8465692433091
x33=−3.6435971674254x_{33} = -3.6435971674254
x33=50.3052188363296x_{33} = 50.3052188363296
x33=−91.1281305511393x_{33} = -91.1281305511393
x33=−34.6152330552306x_{33} = -34.6152330552306
x33=62.863657228703x_{33} = 62.863657228703
x33=75.4247339745236x_{33} = 75.4247339745236
x33=69.1439554764926x_{33} = 69.1439554764926
x33=−97.4099011706723x_{33} = -97.4099011706723
x33=6.57833373272234x_{33} = 6.57833373272234
x33=−47.1662676027767x_{33} = -47.1662676027767
x33=−22.0814757672807x_{33} = -22.0814757672807
Decreasing at intervals
[97.4099011706723,∞)\left[97.4099011706723, \infty\right)
Increasing at intervals
(−∞,−100.550852725424]\left(-\infty, -100.550852725424\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
−(xcos⁡(x)+3sin⁡(x))=0- (x \cos{\left(x \right)} + 3 \sin{\left(x \right)}) = 0
Solve this equation
The roots of this equation
x1=48.75613936684x_{1} = 48.75613936684
x2=14.3433507883915x_{2} = 14.3433507883915
x3=42.4820019253669x_{3} = 42.4820019253669
x4=−58.170990540028x_{4} = -58.170990540028
x5=92.7093311956205x_{5} = 92.7093311956205
x6=−67.5885991217338x_{6} = -67.5885991217338
x7=−61.3099494475655x_{7} = -61.3099494475655
x8=39.3460075465194x_{8} = 39.3460075465194
x9=−17.4490243427188x_{9} = -17.4490243427188
x10=−36.2109745555852x_{10} = -36.2109745555852
x11=80.1480259413025x_{11} = 80.1480259413025
x12=61.3099494475655x_{12} = 61.3099494475655
x13=−89.5688718899173x_{13} = -89.5688718899173
x14=−45.6187613383417x_{14} = -45.6187613383417
x15=86.4284948180722x_{15} = 86.4284948180722
x16=95.8498646688189x_{16} = 95.8498646688189
x17=23.6879210560017x_{17} = 23.6879210560017
x18=−80.1480259413025x_{18} = -80.1480259413025
x19=−8.20453136258127x_{19} = -8.20453136258127
x20=−42.4820019253669x_{20} = -42.4820019253669
x21=45.6187613383417x_{21} = 45.6187613383417
x22=−5.23293845351241x_{22} = -5.23293845351241
x23=−73.8680180276454x_{23} = -73.8680180276454
x24=89.5688718899173x_{24} = 89.5688718899173
x25=29.9449807735163x_{25} = 29.9449807735163
x26=51.894024636399x_{26} = 51.894024636399
x27=33.0771723843072x_{27} = 33.0771723843072
x28=−29.9449807735163x_{28} = -29.9449807735163
x29=73.8680180276454x_{29} = 73.8680180276454
x30=−92.7093311956205x_{30} = -92.7093311956205
x31=70.7282251775385x_{31} = 70.7282251775385
x32=17.4490243427188x_{32} = 17.4490243427188
x33=−39.3460075465194x_{33} = -39.3460075465194
x34=−14.3433507883915x_{34} = -14.3433507883915
x35=−98.9904652640992x_{35} = -98.9904652640992
x36=26.814952130975x_{36} = 26.814952130975
x37=−20.5652079398333x_{37} = -20.5652079398333
x38=8.20453136258127x_{38} = 8.20453136258127
x39=−83.2882092591146x_{39} = -83.2882092591146
x40=−86.4284948180722x_{40} = -86.4284948180722
x41=−26.814952130975x_{41} = -26.814952130975
x42=36.2109745555852x_{42} = 36.2109745555852
x43=−55.0323309441547x_{43} = -55.0323309441547
x44=−51.894024636399x_{44} = -51.894024636399
x45=83.2882092591146x_{45} = 83.2882092591146
x46=20.5652079398333x_{46} = 20.5652079398333
x47=−23.6879210560017x_{47} = -23.6879210560017
x48=64.4491641378738x_{48} = 64.4491641378738
x49=−95.8498646688189x_{49} = -95.8498646688189
x50=58.170990540028x_{50} = 58.170990540028
x51=−11.2560430143535x_{51} = -11.2560430143535
x52=−64.4491641378738x_{52} = -64.4491641378738
x53=77.0079573362515x_{53} = 77.0079573362515
x54=98.9904652640992x_{54} = 98.9904652640992
x55=−70.7282251775385x_{55} = -70.7282251775385
x56=−33.0771723843072x_{56} = -33.0771723843072
x57=−48.75613936684x_{57} = -48.75613936684
x58=5.23293845351241x_{58} = 5.23293845351241
x59=67.5885991217338x_{59} = 67.5885991217338
x60=−2.45564386287944x_{60} = -2.45564386287944
x61=0x_{61} = 0
x62=−77.0079573362515x_{62} = -77.0079573362515
x63=2.45564386287944x_{63} = 2.45564386287944
x64=55.0323309441547x_{64} = 55.0323309441547
x65=11.2560430143535x_{65} = 11.2560430143535

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
[95.8498646688189,∞)\left[95.8498646688189, \infty\right)
Convex at the intervals
(−∞,−95.8498646688189]\left(-\infty, -95.8498646688189\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(xcos⁡(x)+sin⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x \cos{\left(x \right)} + \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→∞(xcos⁡(x)+sin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(x \cos{\left(x \right)} + \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*cos(x), divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(xcos⁡(x)+sin⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{x \cos{\left(x \right)} + \sin{\left(x \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(xcos⁡(x)+sin⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{x \cos{\left(x \right)} + \sin{\left(x \right)}}{x}\right)
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:
xcos⁡(x)+sin⁡(x)=−xcos⁡(x)−sin⁡(x)x \cos{\left(x \right)} + \sin{\left(x \right)} = - x \cos{\left(x \right)} - \sin{\left(x \right)}
- No
xcos⁡(x)+sin⁡(x)=xcos⁡(x)+sin⁡(x)x \cos{\left(x \right)} + \sin{\left(x \right)} = x \cos{\left(x \right)} + \sin{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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