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

Graphing y = x^2*cos(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
        2       
f(x) = x *cos(x)
f(x)=x2cos⁡(x)f{\left(x \right)} = x^{2} \cos{\left(x \right)} 
f = x^2*cos(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:
x2cos⁡(x)=0x^{2} \cos{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}
Numerical solution
x1=48.6946861306418x_{1} = 48.6946861306418
x2=92.6769832808989x_{2} = 92.6769832808989
x3=86.3937979737193x_{3} = 86.3937979737193
x4=−7.85398163397448x_{4} = -7.85398163397448
x5=−86.3937979737193x_{5} = -86.3937979737193
x6=1.5707963267949x_{6} = 1.5707963267949
x7=−64.4026493985908x_{7} = -64.4026493985908
x8=−58.1194640914112x_{8} = -58.1194640914112
x9=−83.2522053201295x_{9} = -83.2522053201295
x10=−54.9778714378214x_{10} = -54.9778714378214
x11=54.9778714378214x_{11} = 54.9778714378214
x12=89.5353906273091x_{12} = 89.5353906273091
x13=−20.4203522483337x_{13} = -20.4203522483337
x14=32.9867228626928x_{14} = 32.9867228626928
x15=−17.2787595947439x_{15} = -17.2787595947439
x16=23.5619449019235x_{16} = 23.5619449019235
x17=−45.553093477052x_{17} = -45.553093477052
x18=64.4026493985908x_{18} = 64.4026493985908
x19=45.553093477052x_{19} = 45.553093477052
x20=83.2522053201295x_{20} = 83.2522053201295
x21=−29.845130209103x_{21} = -29.845130209103
x22=−51.8362787842316x_{22} = -51.8362787842316
x23=80.1106126665397x_{23} = 80.1106126665397
x24=−39.2699081698724x_{24} = -39.2699081698724
x25=−92.6769832808989x_{25} = -92.6769832808989
x26=4.71238898038469x_{26} = 4.71238898038469
x27=70.6858347057703x_{27} = 70.6858347057703
x28=36.1283155162826x_{28} = 36.1283155162826
x29=−70.6858347057703x_{29} = -70.6858347057703
x30=−48.6946861306418x_{30} = -48.6946861306418
x31=42.4115008234622x_{31} = 42.4115008234622
x32=−42.4115008234622x_{32} = -42.4115008234622
x33=−67.5442420521806x_{33} = -67.5442420521806
x34=10.9955742875643x_{34} = 10.9955742875643
x35=98.9601685880785x_{35} = 98.9601685880785
x36=−23.5619449019235x_{36} = -23.5619449019235
x37=20.4203522483337x_{37} = 20.4203522483337
x38=−61.261056745001x_{38} = -61.261056745001
x39=−10.9955742875643x_{39} = -10.9955742875643
x40=17.2787595947439x_{40} = 17.2787595947439
x41=−95.8185759344887x_{41} = -95.8185759344887
x42=−36.1283155162826x_{42} = -36.1283155162826
x43=61.261056745001x_{43} = 61.261056745001
x44=73.8274273593601x_{44} = 73.8274273593601
x45=14.1371669411541x_{45} = 14.1371669411541
x46=−26.7035375555132x_{46} = -26.7035375555132
x47=51.8362787842316x_{47} = 51.8362787842316
x48=−89.5353906273091x_{48} = -89.5353906273091
x49=39.2699081698724x_{49} = 39.2699081698724
x50=−32.9867228626928x_{50} = -32.9867228626928
x51=−14.1371669411541x_{51} = -14.1371669411541
x52=−4.71238898038469x_{52} = -4.71238898038469
x53=−76.9690200129499x_{53} = -76.9690200129499
x54=95.8185759344887x_{54} = 95.8185759344887
x55=76.9690200129499x_{55} = 76.9690200129499
x56=58.1194640914112x_{56} = 58.1194640914112
x57=−80.1106126665397x_{57} = -80.1106126665397
x58=−73.8274273593601x_{58} = -73.8274273593601
x59=7.85398163397448x_{59} = 7.85398163397448
x60=−1.5707963267949x_{60} = -1.5707963267949
x61=29.845130209103x_{61} = 29.845130209103
x62=0x_{62} = 0
x63=67.5442420521806x_{63} = 67.5442420521806
x64=26.7035375555132x_{64} = 26.7035375555132
x65=−98.9601685880785x_{65} = -98.9601685880785
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x^2*cos(x).
02cos⁡(0)0^{2} \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
−x2sin⁡(x)+2xcos⁡(x)=0- x^{2} \sin{\left(x \right)} + 2 x \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−87.9873209346887x_{1} = -87.9873209346887
x2=69.1439554764926x_{2} = 69.1439554764926
x3=−44.0276918992479x_{3} = -44.0276918992479
x4=−100.550852725424x_{4} = -100.550852725424
x5=−9.62956034329743x_{5} = -9.62956034329743
x6=31.479374920314x_{6} = 31.479374920314
x7=−40.8895777660408x_{7} = -40.8895777660408
x8=−47.1662676027767x_{8} = -47.1662676027767
x9=97.4099011706723x_{9} = 97.4099011706723
x10=−6.57833373272234x_{10} = -6.57833373272234
x11=28.3447768697864x_{11} = 28.3447768697864
x12=100.550852725424x_{12} = 100.550852725424
x13=−59.7237354324305x_{13} = -59.7237354324305
x14=18.954681766529x_{14} = 18.954681766529
x15=59.7237354324305x_{15} = 59.7237354324305
x16=−37.7520396346102x_{16} = -37.7520396346102
x17=−1.0768739863118x_{17} = -1.0768739863118
x18=50.3052188363296x_{18} = 50.3052188363296
x19=−56.5839987378634x_{19} = -56.5839987378634
x20=−34.6152330552306x_{20} = -34.6152330552306
x21=40.8895777660408x_{21} = 40.8895777660408
x22=84.8465692433091x_{22} = 84.8465692433091
x23=−15.8336114149477x_{23} = -15.8336114149477
x24=−31.479374920314x_{24} = -31.479374920314
x25=62.863657228703x_{25} = 62.863657228703
x26=9.62956034329743x_{26} = 9.62956034329743
x27=47.1662676027767x_{27} = 47.1662676027767
x28=53.4444796697636x_{28} = 53.4444796697636
x29=−50.3052188363296x_{29} = -50.3052188363296
x30=−78.5652673845995x_{30} = -78.5652673845995
x31=34.6152330552306x_{31} = 34.6152330552306
x32=12.7222987717666x_{32} = 12.7222987717666
x33=75.4247339745236x_{33} = 75.4247339745236
x34=−3.6435971674254x_{34} = -3.6435971674254
x35=−69.1439554764926x_{35} = -69.1439554764926
x36=66.0037377708277x_{36} = 66.0037377708277
x37=25.2119030642106x_{37} = 25.2119030642106
x38=6.57833373272234x_{38} = 6.57833373272234
x39=91.1281305511393x_{39} = 91.1281305511393
x40=81.7058821480364x_{40} = 81.7058821480364
x41=44.0276918992479x_{41} = 44.0276918992479
x42=56.5839987378634x_{42} = 56.5839987378634
x43=37.7520396346102x_{43} = 37.7520396346102
x44=22.0814757672807x_{44} = 22.0814757672807
x45=−18.954681766529x_{45} = -18.954681766529
x46=−81.7058821480364x_{46} = -81.7058821480364
x47=1.0768739863118x_{47} = 1.0768739863118
x48=15.8336114149477x_{48} = 15.8336114149477
x49=87.9873209346887x_{49} = 87.9873209346887
x50=−72.2842925036825x_{50} = -72.2842925036825
x51=−75.4247339745236x_{51} = -75.4247339745236
x52=−12.7222987717666x_{52} = -12.7222987717666
x53=−94.2689923093066x_{53} = -94.2689923093066
x54=−66.0037377708277x_{54} = -66.0037377708277
x55=−62.863657228703x_{55} = -62.863657228703
x56=−28.3447768697864x_{56} = -28.3447768697864
x57=−97.4099011706723x_{57} = -97.4099011706723
x58=72.2842925036825x_{58} = 72.2842925036825
x59=−22.0814757672807x_{59} = -22.0814757672807
x60=0x_{60} = 0
x61=−53.4444796697636x_{61} = -53.4444796697636
x62=−25.2119030642106x_{62} = -25.2119030642106
x63=−84.8465692433091x_{63} = -84.8465692433091
x64=78.5652673845995x_{64} = 78.5652673845995
x65=−91.1281305511393x_{65} = -91.1281305511393
x66=94.2689923093066x_{66} = 94.2689923093066
x67=3.6435971674254x_{67} = 3.6435971674254
The values of the extrema at the points:
(-87.9873209346887, 7739.76941994707)

(69.1439554764926, 4778.88783305817)

(-44.02769189924788, 1936.44074393829)

(-100.55085272542402, 10108.4745770583)

(-9.62956034329743, -90.7908960221418)

(31.479374920314047, 988.957079867956)

(-40.889577766040844, -1669.96115135336)

(-47.1662676027767, -2222.65949258718)

(97.40990117067226, -9486.68947818992)

(-6.578333732722339, 41.4032422108864)

(28.344776869786372, -801.433812963046)

(100.55085272542402, 10108.4745770583)

(-59.72373543243046, -3564.92625455388)

(18.954681766529042, 357.296507493256)

(59.72373543243046, -3564.92625455388)

(-37.75203963461023, 1423.22069663783)

(-1.0768739863118038, 0.549774025605498)

(50.30521883632959, 2528.6174100174)

(-56.58399873786344, 3199.75078519348)

(-34.61523305523058, -1196.21935302944)

(40.889577766040844, -1669.96115135336)

(84.84656924330915, -7196.94114542993)

(-15.833611414947718, -248.726869289185)

(-31.479374920314047, 988.957079867956)

(62.863657228703005, 3949.84091716867)

(9.62956034329743, -90.7908960221418)

(47.1662676027767, -2222.65949258718)

(53.44447966976355, -2854.3145053339)

(-50.30521883632959, 2528.6174100174)

(-78.56526738459954, -6170.50221074225)

(34.61523305523058, -1196.21935302944)

(12.722298771766635, 159.893208545431)

(75.4247339745236, 5686.89154919726)

(-3.643597167425401, -11.6378292117556)

(-69.1439554764926, 4778.88783305817)

(66.00373777082767, -4354.4947759217)

(25.21190306421058, 633.649446194351)

(6.578333732722339, 41.4032422108864)

(91.1281305511393, -8302.3368999697)

(81.70588214803641, 6673.85207590208)

(44.02769189924788, 1936.44074393829)

(56.58399873786344, 3199.75078519348)

(37.75203963461023, 1423.22069663783)

(22.081475767280747, -485.603793917741)

(-18.954681766529042, 357.296507493256)

(-81.70588214803641, 6673.85207590208)

(1.0768739863118038, 0.549774025605498)

(15.833611414947718, -248.726869289185)

(87.9873209346887, 7739.76941994707)

(-72.2842925036825, -5223.02009034704)

(-75.4247339745236, 5686.89154919726)

(-12.722298771766635, 159.893208545431)

(-94.26899230930657, 8884.64358592955)

(-66.00373777082767, -4354.4947759217)

(-62.863657228703005, 3949.84091716867)

(-28.344776869786372, -801.433812963046)

(-97.40990117067226, -9486.68947818992)

(72.2842925036825, -5223.02009034704)

(-22.081475767280747, -485.603793917741)

(0, 0)

(-53.44447966976355, -2854.3145053339)

(-25.21190306421058, 633.649446194351)

(-84.84656924330915, -7196.94114542993)

(78.56526738459954, -6170.50221074225)

(-91.1281305511393, -8302.3368999697)

(94.26899230930657, 8884.64358592955)

(3.643597167425401, -11.6378292117556)


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=−9.62956034329743x_{1} = -9.62956034329743
x2=−40.8895777660408x_{2} = -40.8895777660408
x3=−47.1662676027767x_{3} = -47.1662676027767
x4=97.4099011706723x_{4} = 97.4099011706723
x5=28.3447768697864x_{5} = 28.3447768697864
x6=−59.7237354324305x_{6} = -59.7237354324305
x7=59.7237354324305x_{7} = 59.7237354324305
x8=−34.6152330552306x_{8} = -34.6152330552306
x9=40.8895777660408x_{9} = 40.8895777660408
x10=84.8465692433091x_{10} = 84.8465692433091
x11=−15.8336114149477x_{11} = -15.8336114149477
x12=9.62956034329743x_{12} = 9.62956034329743
x13=47.1662676027767x_{13} = 47.1662676027767
x14=53.4444796697636x_{14} = 53.4444796697636
x15=−78.5652673845995x_{15} = -78.5652673845995
x16=34.6152330552306x_{16} = 34.6152330552306
x17=−3.6435971674254x_{17} = -3.6435971674254
x18=66.0037377708277x_{18} = 66.0037377708277
x19=91.1281305511393x_{19} = 91.1281305511393
x20=22.0814757672807x_{20} = 22.0814757672807
x21=15.8336114149477x_{21} = 15.8336114149477
x22=−72.2842925036825x_{22} = -72.2842925036825
x23=−66.0037377708277x_{23} = -66.0037377708277
x24=−28.3447768697864x_{24} = -28.3447768697864
x25=−97.4099011706723x_{25} = -97.4099011706723
x26=72.2842925036825x_{26} = 72.2842925036825
x27=−22.0814757672807x_{27} = -22.0814757672807
x28=0x_{28} = 0
x29=−53.4444796697636x_{29} = -53.4444796697636
x30=−84.8465692433091x_{30} = -84.8465692433091
x31=78.5652673845995x_{31} = 78.5652673845995
x32=−91.1281305511393x_{32} = -91.1281305511393
x33=3.6435971674254x_{33} = 3.6435971674254
Maxima of the function at points:
x33=−87.9873209346887x_{33} = -87.9873209346887
x33=69.1439554764926x_{33} = 69.1439554764926
x33=−44.0276918992479x_{33} = -44.0276918992479
x33=−100.550852725424x_{33} = -100.550852725424
x33=31.479374920314x_{33} = 31.479374920314
x33=−6.57833373272234x_{33} = -6.57833373272234
x33=100.550852725424x_{33} = 100.550852725424
x33=18.954681766529x_{33} = 18.954681766529
x33=−37.7520396346102x_{33} = -37.7520396346102
x33=−1.0768739863118x_{33} = -1.0768739863118
x33=50.3052188363296x_{33} = 50.3052188363296
x33=−56.5839987378634x_{33} = -56.5839987378634
x33=−31.479374920314x_{33} = -31.479374920314
x33=62.863657228703x_{33} = 62.863657228703
x33=−50.3052188363296x_{33} = -50.3052188363296
x33=12.7222987717666x_{33} = 12.7222987717666
x33=75.4247339745236x_{33} = 75.4247339745236
x33=−69.1439554764926x_{33} = -69.1439554764926
x33=25.2119030642106x_{33} = 25.2119030642106
x33=6.57833373272234x_{33} = 6.57833373272234
x33=81.7058821480364x_{33} = 81.7058821480364
x33=44.0276918992479x_{33} = 44.0276918992479
x33=56.5839987378634x_{33} = 56.5839987378634
x33=37.7520396346102x_{33} = 37.7520396346102
x33=−18.954681766529x_{33} = -18.954681766529
x33=−81.7058821480364x_{33} = -81.7058821480364
x33=1.0768739863118x_{33} = 1.0768739863118
x33=87.9873209346887x_{33} = 87.9873209346887
x33=−75.4247339745236x_{33} = -75.4247339745236
x33=−12.7222987717666x_{33} = -12.7222987717666
x33=−94.2689923093066x_{33} = -94.2689923093066
x33=−62.863657228703x_{33} = -62.863657228703
x33=−25.2119030642106x_{33} = -25.2119030642106
x33=94.2689923093066x_{33} = 94.2689923093066
Decreasing at intervals
[97.4099011706723,∞)\left[97.4099011706723, \infty\right)
Increasing at intervals
(−∞,−97.4099011706723]\left(-\infty, -97.4099011706723\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
−x2cos⁡(x)−4xsin⁡(x)+2cos⁡(x)=0- x^{2} \cos{\left(x \right)} - 4 x \sin{\left(x \right)} + 2 \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−14.4104316197583x_{1} = -14.4104316197583
x2=−26.8518211351302x_{2} = -26.8518211351302
x3=77.0209248041715x_{3} = 77.0209248041715
x4=51.9132353685519x_{4} = 51.9132353685519
x5=29.978069966608x_{5} = 29.978069966608
x6=99.0005586689926x_{6} = 99.0005586689926
x7=14.4104316197583x_{7} = 14.4104316197583
x8=−11.3396400694003x_{8} = -11.3396400694003
x9=70.7423403203845x_{9} = 70.7423403203845
x10=−2.68896749918069x_{10} = -2.68896749918069
x11=−33.1071774728736x_{11} = -33.1071774728736
x12=55.0504523208123x_{12} = 55.0504523208123
x13=−20.6129075025746x_{13} = -20.6129075025746
x14=86.4400521937386x_{14} = 86.4400521937386
x15=0.599741421027824x_{15} = 0.599741421027824
x16=95.86028820339x_{16} = 95.86028820339
x17=8.31398471094493x_{17} = 8.31398471094493
x18=−99.0005586689926x_{18} = -99.0005586689926
x19=−17.5048370447411x_{19} = -17.5048370447411
x20=80.1604867055981x_{20} = 80.1604867055981
x21=−42.5054334829582x_{21} = -42.5054334829582
x22=64.4646492226716x_{22} = 64.4646492226716
x23=−45.6405949760108x_{23} = -45.6405949760108
x24=−23.7295255625192x_{24} = -23.7295255625192
x25=−70.7423403203845x_{25} = -70.7423403203845
x26=20.6129075025746x_{26} = 20.6129075025746
x27=89.5800249069094x_{27} = 89.5800249069094
x28=48.7765781423002x_{28} = 48.7765781423002
x29=−58.1881390858586x_{29} = -58.1881390858586
x30=−5.38572965402801x_{30} = -5.38572965402801
x31=92.7201071560939x_{31} = 92.7201071560939
x32=−89.5800249069094x_{32} = -89.5800249069094
x33=11.3396400694003x_{33} = 11.3396400694003
x34=−61.3262239938688x_{34} = -61.3262239938688
x35=−36.2384169134664x_{35} = -36.2384169134664
x36=23.7295255625192x_{36} = 23.7295255625192
x37=−55.0504523208123x_{37} = -55.0504523208123
x38=−29.978069966608x_{38} = -29.978069966608
x39=2.68896749918069x_{39} = 2.68896749918069
x40=−8.31398471094493x_{40} = -8.31398471094493
x41=61.3262239938688x_{41} = 61.3262239938688
x42=−39.3712875521166x_{42} = -39.3712875521166
x43=5.38572965402801x_{43} = 5.38572965402801
x44=−83.3002013673394x_{44} = -83.3002013673394
x45=58.1881390858586x_{45} = 58.1881390858586
x46=73.8815350660339x_{46} = 73.8815350660339
x47=−0.599741421027824x_{47} = -0.599741421027824
x48=−73.8815350660339x_{48} = -73.8815350660339
x49=−64.4646492226716x_{49} = -64.4646492226716
x50=83.3002013673394x_{50} = 83.3002013673394
x51=39.3712875521166x_{51} = 39.3712875521166
x52=−95.86028820339x_{52} = -95.86028820339
x53=67.6033676125647x_{53} = 67.6033676125647
x54=−48.7765781423002x_{54} = -48.7765781423002
x55=−80.1604867055981x_{55} = -80.1604867055981
x56=45.6405949760108x_{56} = 45.6405949760108
x57=26.8518211351302x_{57} = 26.8518211351302
x58=42.5054334829582x_{58} = 42.5054334829582
x59=36.2384169134664x_{59} = 36.2384169134664
x60=33.1071774728736x_{60} = 33.1071774728736
x61=−86.4400521937386x_{61} = -86.4400521937386
x62=17.5048370447411x_{62} = 17.5048370447411
x63=−51.9132353685519x_{63} = -51.9132353685519
x64=−67.6033676125647x_{64} = -67.6033676125647
x65=−77.0209248041715x_{65} = -77.0209248041715
x66=−92.7201071560939x_{66} = -92.7201071560939

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
[95.86028820339,∞)\left[95.86028820339, \infty\right)
Convex at the intervals
(−∞,−99.0005586689926]\left(-\infty, -99.0005586689926\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(x2cos⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x^{2} \cos{\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→∞(x2cos⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(x^{2} \cos{\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*cos(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(xcos⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x \cos{\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→∞(xcos⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(x \cos{\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:
x2cos⁡(x)=x2cos⁡(x)x^{2} \cos{\left(x \right)} = x^{2} \cos{\left(x \right)}
- Yes
x2cos⁡(x)=−x2cos⁡(x)x^{2} \cos{\left(x \right)} = - x^{2} \cos{\left(x \right)}
- No
so, the function
is
even
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