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

Graphing y = x^2*sin(2*x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
        2         
f(x) = x *sin(2*x)
f(x)=x2sin⁡(2x)f{\left(x \right)} = x^{2} \sin{\left(2 x \right)} 
f = x^2*sin(2*x)
The graph of the function
0-50-40-30-20-101020304050607080-200200
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⁡(2x)=0x^{2} \sin{\left(2 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}
Numerical solution
x1=−29.845130209103x_{1} = -29.845130209103
x2=−59.6902604182061x_{2} = -59.6902604182061
x3=86.3937979737193x_{3} = 86.3937979737193
x4=−86.3937979737193x_{4} = -86.3937979737193
x5=−67.5442420521806x_{5} = -67.5442420521806
x6=−87.9645943005142x_{6} = -87.9645943005142
x7=−14.1371669411541x_{7} = -14.1371669411541
x8=−20.4203522483337x_{8} = -20.4203522483337
x9=6.28318530717959x_{9} = 6.28318530717959
x10=100.530964914873x_{10} = 100.530964914873
x11=−42.4115008234622x_{11} = -42.4115008234622
x12=26.7035375555132x_{12} = 26.7035375555132
x13=−37.6991118430775x_{13} = -37.6991118430775
x14=−61.261056745001x_{14} = -61.261056745001
x15=45.553093477052x_{15} = 45.553093477052
x16=−4.71238898038469x_{16} = -4.71238898038469
x17=89.5353906273091x_{17} = 89.5353906273091
x18=3.14159265358979x_{18} = 3.14159265358979
x19=78.5398163397448x_{19} = 78.5398163397448
x20=67.5442420521806x_{20} = 67.5442420521806
x21=0x_{21} = 0
x22=36.1283155162826x_{22} = 36.1283155162826
x23=−1.5707963267949x_{23} = -1.5707963267949
x24=−31.4159265358979x_{24} = -31.4159265358979
x25=87.9645943005142x_{25} = 87.9645943005142
x26=−39.2699081698724x_{26} = -39.2699081698724
x27=−45.553093477052x_{27} = -45.553093477052
x28=20.4203522483337x_{28} = 20.4203522483337
x29=−89.5353906273091x_{29} = -89.5353906273091
x30=−65.9734457253857x_{30} = -65.9734457253857
x31=−53.4070751110265x_{31} = -53.4070751110265
x32=−75.398223686155x_{32} = -75.398223686155
x33=−72.2566310325652x_{33} = -72.2566310325652
x34=56.5486677646163x_{34} = 56.5486677646163
x35=65.9734457253857x_{35} = 65.9734457253857
x36=−28.2743338823081x_{36} = -28.2743338823081
x37=48.6946861306418x_{37} = 48.6946861306418
x38=28.2743338823081x_{38} = 28.2743338823081
x39=−51.8362787842316x_{39} = -51.8362787842316
x40=15.707963267949x_{40} = 15.707963267949
x41=−15.707963267949x_{41} = -15.707963267949
x42=−36.1283155162826x_{42} = -36.1283155162826
x43=−95.8185759344887x_{43} = -95.8185759344887
x44=12.5663706143592x_{44} = 12.5663706143592
x45=14.1371669411541x_{45} = 14.1371669411541
x46=−50.2654824574367x_{46} = -50.2654824574367
x47=95.8185759344887x_{47} = 95.8185759344887
x48=−7.85398163397448x_{48} = -7.85398163397448
x49=−21.9911485751286x_{49} = -21.9911485751286
x50=23.5619449019235x_{50} = 23.5619449019235
x51=37.6991118430775x_{51} = 37.6991118430775
x52=64.4026493985908x_{52} = 64.4026493985908
x53=21.9911485751286x_{53} = 21.9911485751286
x54=58.1194640914112x_{54} = 58.1194640914112
x55=43.9822971502571x_{55} = 43.9822971502571
x56=70.6858347057703x_{56} = 70.6858347057703
x57=42.4115008234622x_{57} = 42.4115008234622
x58=29.845130209103x_{58} = 29.845130209103
x59=94.2477796076938x_{59} = 94.2477796076938
x60=34.5575191894877x_{60} = 34.5575191894877
x61=1.5707963267949x_{61} = 1.5707963267949
x62=−9.42477796076938x_{62} = -9.42477796076938
x63=−97.3893722612836x_{63} = -97.3893722612836
x64=−73.8274273593601x_{64} = -73.8274273593601
x65=−81.6814089933346x_{65} = -81.6814089933346
x66=92.6769832808989x_{66} = 92.6769832808989
x67=80.1106126665397x_{67} = 80.1106126665397
x68=81.6814089933346x_{68} = 81.6814089933346
x69=−17.2787595947439x_{69} = -17.2787595947439
x70=−23.5619449019235x_{70} = -23.5619449019235
x71=4.71238898038469x_{71} = 4.71238898038469
x72=−64.4026493985908x_{72} = -64.4026493985908
x73=−80.1106126665397x_{73} = -80.1106126665397
x74=−83.2522053201295x_{74} = -83.2522053201295
x75=−6.28318530717959x_{75} = -6.28318530717959
x76=−3.14159265358979x_{76} = -3.14159265358979
x77=7.85398163397448x_{77} = 7.85398163397448
x78=51.8362787842316x_{78} = 51.8362787842316
x79=−58.1194640914112x_{79} = -58.1194640914112
x80=59.6902604182061x_{80} = 59.6902604182061
x81=−94.2477796076938x_{81} = -94.2477796076938
x82=−43.9822971502571x_{82} = -43.9822971502571
x83=72.2566310325652x_{83} = 72.2566310325652
x84=−103.672557568463x_{84} = -103.672557568463
x85=73.8274273593601x_{85} = 73.8274273593601
x86=50.2654824574367x_{86} = 50.2654824574367
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(2*x).
02sin⁡(2⋅0)0^{2} \sin{\left(2 \cdot 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
2x2cos⁡(2x)+2xsin⁡(2x)=02 x^{2} \cos{\left(2 x \right)} + 2 x \sin{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=16.5235843473527x_{1} = 16.5235843473527
x2=60.4839244878466x_{2} = 60.4839244878466
x3=40.0677825970372x_{3} = 40.0677825970372
x4=−11.8231619098018x_{4} = -11.8231619098018
x5=25.9374070267134x_{5} = 25.9374070267134
x6=82.4728694594266x_{6} = 82.4728694594266
x7=38.4974949445838x_{7} = 38.4974949445838
x8=−41.6381085824888x_{8} = -41.6381085824888
x9=−63.6251091208926x_{9} = -63.6251091208926
x10=27.5071048394191x_{10} = 27.5071048394191
x11=77.760847792972x_{11} = 77.760847792972
x12=−98.1798629425939x_{12} = -98.1798629425939
x13=−62.0545116429054x_{13} = -62.0545116429054
x14=−99.7505790857949x_{14} = -99.7505790857949
x15=−27.5071048394191x_{15} = -27.5071048394191
x16=74.6195257807054x_{16} = 74.6195257807054
x17=10.2587614549708x_{17} = 10.2587614549708
x18=0x_{18} = 0
x19=33.7869153354295x_{19} = 33.7869153354295
x20=8.69662198229738x_{20} = 8.69662198229738
x21=−21.2292853858495x_{21} = -21.2292853858495
x22=85.6142396947314x_{22} = 85.6142396947314
x23=54.2016970313842x_{23} = 54.2016970313842
x24=−77.760847792972x_{24} = -77.760847792972
x25=11.8231619098018x_{25} = 11.8231619098018
x26=−18.0917665453763x_{26} = -18.0917665453763
x27=52.6311758774383x_{27} = 52.6311758774383
x28=63.6251091208926x_{28} = 63.6251091208926
x29=−69.9075883539626x_{29} = -69.9075883539626
x30=−55.7722336752062x_{30} = -55.7722336752062
x31=−71.4782275499213x_{31} = -71.4782275499213
x32=18.0917665453763x_{32} = 18.0917665453763
x33=−4.04808180161146x_{33} = -4.04808180161146
x34=35.3570550332742x_{34} = 35.3570550332742
x35=−76.1901839979235x_{35} = -76.1901839979235
x36=−49.4901859325761x_{36} = -49.4901859325761
x37=−79.3315168346756x_{37} = -79.3315168346756
x38=−5.58635293416499x_{38} = -5.58635293416499
x39=32.2168395518658x_{39} = 32.2168395518658
x40=24.3678503974527x_{40} = 24.3678503974527
x41=−10.2587614549708x_{41} = -10.2587614549708
x42=84.0435524991391x_{42} = 84.0435524991391
x43=−46.3492776216985x_{43} = -46.3492776216985
x44=−33.7869153354295x_{44} = -33.7869153354295
x45=69.9075883539626x_{45} = 69.9075883539626
x46=−38.4974949445838x_{46} = -38.4974949445838
x47=−40.0677825970372x_{47} = -40.0677825970372
x48=62.0545116429054x_{48} = 62.0545116429054
x49=68.3369563786298x_{49} = 68.3369563786298
x50=88.7556256712795x_{50} = 88.7556256712795
x51=−25.9374070267134x_{51} = -25.9374070267134
x52=−93.4677306800165x_{52} = -93.4677306800165
x53=−57.3427845371101x_{53} = -57.3427845371101
x54=−3.42962943093331⋅10−7x_{54} = -3.42962943093331 \cdot 10^{-7}
x55=−54.2016970313842x_{55} = -54.2016970313842
x56=71.4782275499213x_{56} = 71.4782275499213
x57=30.6468374831214x_{57} = 30.6468374831214
x58=−24.3678503974527x_{58} = -24.3678503974527
x59=41.6381085824888x_{59} = 41.6381085824888
x60=−84.0435524991391x_{60} = -84.0435524991391
x61=4.04808180161146x_{61} = 4.04808180161146
x62=−60.4839244878466x_{62} = -60.4839244878466
x63=−68.3369563786298x_{63} = -68.3369563786298
x64=46.3492776216985x_{64} = 46.3492776216985
x65=−47.9197205706165x_{65} = -47.9197205706165
x66=−32.2168395518658x_{66} = -32.2168395518658
x67=65.1957161761796x_{67} = 65.1957161761796
x68=47.9197205706165x_{68} = 47.9197205706165
x69=98.1798629425939x_{69} = 98.1798629425939
x70=−91.8970257752571x_{70} = -91.8970257752571
x71=55.7722336752062x_{71} = 55.7722336752062
x72=91.8970257752571x_{72} = 91.8970257752571
x73=−44.7788594413622x_{73} = -44.7788594413622
x74=−2.54349254705114x_{74} = -2.54349254705114
x75=−3.16473361148914⋅10−7x_{75} = -3.16473361148914 \cdot 10^{-7}
x76=−13.3890435377793x_{76} = -13.3890435377793
x77=−82.4728694594266x_{77} = -82.4728694594266
x78=−85.6142396947314x_{78} = -85.6142396947314
x79=19.6603640661261x_{79} = 19.6603640661261
x80=2.54349254705114x_{80} = 2.54349254705114
x81=76.1901839979235x_{81} = 76.1901839979235
x82=49.4901859325761x_{82} = 49.4901859325761
x83=−19.6603640661261x_{83} = -19.6603640661261
x84=90.3263240494369x_{84} = 90.3263240494369
x85=−35.3570550332742x_{85} = -35.3570550332742
x86=−90.3263240494369x_{86} = -90.3263240494369
x87=7.13817645916824x_{87} = 7.13817645916824
x88=96.6091494063022x_{88} = 96.6091494063022
x89=95.0384386061415x_{89} = 95.0384386061415
x90=5.58635293416499x_{90} = 5.58635293416499
x91=99.7505790857949x_{91} = 99.7505790857949
The values of the extrema at the points:
(16.5235843473527, 272.530208986636)

(60.4839244878466, 3657.80522393468)

(40.0677825970372, -1604.92743570495)

(-11.8231619098018, 139.289824302256)

(25.9374070267134, 672.24963999419)

(82.4728694594266, 6801.27425199754)

(38.4974949445838, 1481.55736989275)

(-41.6381085824888, -1733.23230251961)

(-63.6251091208926, -4047.65460326123)

(27.5071048394191, -756.141311713221)

(77.760847792972, -6046.24951149001)

(-98.1798629425939, -9638.78552632646)

(-62.0545116429054, 3850.26251260173)

(-99.7505790857949, 9949.67806563604)

(-27.5071048394191, 756.141311713221)

(74.6195257807054, -5567.57369507552)

(10.2587614549708, 104.745721818108)

(0, 0)

(33.7869153354295, -1141.05597614296)

(8.69662198229738, -75.1361381644989)

(-21.2292853858495, 450.183388529538)

(85.6142396947314, 7329.29808966213)

(54.2016970313842, 2937.32408869126)

(-77.760847792972, 6046.24951149001)

(11.8231619098018, -139.289824302256)

(-18.0917665453763, 326.813159519034)

(52.6311758774383, -2769.54080957821)

(63.6251091208926, 4047.65460326123)

(-69.9075883539626, -4886.57098618708)

(-55.7722336752062, 3110.04216964728)

(-71.4782275499213, 5108.63708706427)

(18.0917665453763, -326.813159519034)

(-4.04808180161146, -15.9087454878886)

(35.3570550332742, 1249.62164039704)

(-76.1901839979235, -5804.44420222827)

(-49.4901859325761, 2448.7786566952)

(-79.3315168346756, -6292.98962286791)

(-5.58635293416499, 30.719043378479)

(32.2168395518658, 1037.4251117187)

(24.3678503974527, -593.292763641772)

(-10.2587614549708, -104.745721818108)

(84.0435524991391, -7062.81876976048)

(-46.3492776216985, 2147.75571054583)

(-33.7869153354295, 1141.05597614296)

(69.9075883539626, 4886.57098618708)

(-38.4974949445838, -1481.55736989275)

(-40.0677825970372, 1604.92743570495)

(62.0545116429054, -3850.26251260173)

(68.3369563786298, -4669.43968738125)

(88.7556256712795, 7877.06113589882)

(-25.9374070267134, -672.24963999419)

(-93.4677306800165, 8735.71672139277)

(-57.3427845371101, -3287.69505248487)

(-3.42962943093331e-7, -8.06810585778905e-20)

(-54.2016970313842, -2937.32408869126)

(71.4782275499213, -5108.63708706427)

(30.6468374831214, -938.729046626741)

(-24.3678503974527, 593.292763641772)

(41.6381085824888, 1733.23230251961)

(-84.0435524991391, 7062.81876976048)

(4.04808180161146, 15.9087454878886)

(-60.4839244878466, -3657.80522393468)

(-68.3369563786298, 4669.43968738125)

(46.3492776216985, -2147.75571054583)

(-47.9197205706165, -2295.79978281294)

(-32.2168395518658, -1037.4251117187)

(65.1957161761796, -4249.98149593298)

(47.9197205706165, 2295.79978281294)

(98.1798629425939, 9638.78552632646)

(-91.8970257752571, -8444.56339073853)

(55.7722336752062, -3110.04216964728)

(91.8970257752571, 8444.56339073853)

(-44.7788594413622, -2004.64643981036)

(-2.54349254705114, 6.02074005576708)

(-3.16473361148914e-7, -6.33930247556382e-20)

(-13.3890435377793, -178.768569037428)

(-82.4728694594266, -6801.27425199754)

(-85.6142396947314, -7329.29808966213)

(19.6603640661261, 386.030883296424)

(2.54349254705114, -6.02074005576708)

(76.1901839979235, 5804.44420222827)

(49.4901859325761, -2448.7786566952)

(-19.6603640661261, -386.030883296424)

(90.3263240494369, -8158.34486224158)

(-35.3570550332742, -1249.62164039704)

(-90.3263240494369, 8158.34486224158)

(7.13817645916824, 50.4608044704652)

(96.6091494063022, -9332.82778918424)

(95.0384386061415, 9031.80485420714)

(5.58635293416499, -30.719043378479)

(99.7505790857949, -9949.67806563604)


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=40.0677825970372x_{1} = 40.0677825970372
x2=−41.6381085824888x_{2} = -41.6381085824888
x3=−63.6251091208926x_{3} = -63.6251091208926
x4=27.5071048394191x_{4} = 27.5071048394191
x5=77.760847792972x_{5} = 77.760847792972
x6=−98.1798629425939x_{6} = -98.1798629425939
x7=74.6195257807054x_{7} = 74.6195257807054
x8=33.7869153354295x_{8} = 33.7869153354295
x9=8.69662198229738x_{9} = 8.69662198229738
x10=11.8231619098018x_{10} = 11.8231619098018
x11=52.6311758774383x_{11} = 52.6311758774383
x12=−69.9075883539626x_{12} = -69.9075883539626
x13=18.0917665453763x_{13} = 18.0917665453763
x14=−4.04808180161146x_{14} = -4.04808180161146
x15=−76.1901839979235x_{15} = -76.1901839979235
x16=−79.3315168346756x_{16} = -79.3315168346756
x17=24.3678503974527x_{17} = 24.3678503974527
x18=−10.2587614549708x_{18} = -10.2587614549708
x19=84.0435524991391x_{19} = 84.0435524991391
x20=−38.4974949445838x_{20} = -38.4974949445838
x21=62.0545116429054x_{21} = 62.0545116429054
x22=68.3369563786298x_{22} = 68.3369563786298
x23=−25.9374070267134x_{23} = -25.9374070267134
x24=−57.3427845371101x_{24} = -57.3427845371101
x25=−54.2016970313842x_{25} = -54.2016970313842
x26=71.4782275499213x_{26} = 71.4782275499213
x27=30.6468374831214x_{27} = 30.6468374831214
x28=−60.4839244878466x_{28} = -60.4839244878466
x29=46.3492776216985x_{29} = 46.3492776216985
x30=−47.9197205706165x_{30} = -47.9197205706165
x31=−32.2168395518658x_{31} = -32.2168395518658
x32=65.1957161761796x_{32} = 65.1957161761796
x33=−91.8970257752571x_{33} = -91.8970257752571
x34=55.7722336752062x_{34} = 55.7722336752062
x35=−44.7788594413622x_{35} = -44.7788594413622
x36=−13.3890435377793x_{36} = -13.3890435377793
x37=−82.4728694594266x_{37} = -82.4728694594266
x38=−85.6142396947314x_{38} = -85.6142396947314
x39=2.54349254705114x_{39} = 2.54349254705114
x40=49.4901859325761x_{40} = 49.4901859325761
x41=−19.6603640661261x_{41} = -19.6603640661261
x42=90.3263240494369x_{42} = 90.3263240494369
x43=−35.3570550332742x_{43} = -35.3570550332742
x44=96.6091494063022x_{44} = 96.6091494063022
x45=5.58635293416499x_{45} = 5.58635293416499
x46=99.7505790857949x_{46} = 99.7505790857949
Maxima of the function at points:
x46=16.5235843473527x_{46} = 16.5235843473527
x46=60.4839244878466x_{46} = 60.4839244878466
x46=−11.8231619098018x_{46} = -11.8231619098018
x46=25.9374070267134x_{46} = 25.9374070267134
x46=82.4728694594266x_{46} = 82.4728694594266
x46=38.4974949445838x_{46} = 38.4974949445838
x46=−62.0545116429054x_{46} = -62.0545116429054
x46=−99.7505790857949x_{46} = -99.7505790857949
x46=−27.5071048394191x_{46} = -27.5071048394191
x46=10.2587614549708x_{46} = 10.2587614549708
x46=−21.2292853858495x_{46} = -21.2292853858495
x46=85.6142396947314x_{46} = 85.6142396947314
x46=54.2016970313842x_{46} = 54.2016970313842
x46=−77.760847792972x_{46} = -77.760847792972
x46=−18.0917665453763x_{46} = -18.0917665453763
x46=63.6251091208926x_{46} = 63.6251091208926
x46=−55.7722336752062x_{46} = -55.7722336752062
x46=−71.4782275499213x_{46} = -71.4782275499213
x46=35.3570550332742x_{46} = 35.3570550332742
x46=−49.4901859325761x_{46} = -49.4901859325761
x46=−5.58635293416499x_{46} = -5.58635293416499
x46=32.2168395518658x_{46} = 32.2168395518658
x46=−46.3492776216985x_{46} = -46.3492776216985
x46=−33.7869153354295x_{46} = -33.7869153354295
x46=69.9075883539626x_{46} = 69.9075883539626
x46=−40.0677825970372x_{46} = -40.0677825970372
x46=88.7556256712795x_{46} = 88.7556256712795
x46=−93.4677306800165x_{46} = -93.4677306800165
x46=−24.3678503974527x_{46} = -24.3678503974527
x46=41.6381085824888x_{46} = 41.6381085824888
x46=−84.0435524991391x_{46} = -84.0435524991391
x46=4.04808180161146x_{46} = 4.04808180161146
x46=−68.3369563786298x_{46} = -68.3369563786298
x46=47.9197205706165x_{46} = 47.9197205706165
x46=98.1798629425939x_{46} = 98.1798629425939
x46=91.8970257752571x_{46} = 91.8970257752571
x46=−2.54349254705114x_{46} = -2.54349254705114
x46=19.6603640661261x_{46} = 19.6603640661261
x46=76.1901839979235x_{46} = 76.1901839979235
x46=−90.3263240494369x_{46} = -90.3263240494369
x46=7.13817645916824x_{46} = 7.13817645916824
x46=95.0384386061415x_{46} = 95.0384386061415
Decreasing at intervals
[99.7505790857949,∞)\left[99.7505790857949, \infty\right)
Increasing at intervals
(−∞,−98.1798629425939]\left(-\infty, -98.1798629425939\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
2(−2x2sin⁡(2x)+4xcos⁡(2x)+sin⁡(2x))=02 \left(- 2 x^{2} \sin{\left(2 x \right)} + 4 x \cos{\left(2 x \right)} + \sin{\left(2 x \right)}\right) = 0
Solve this equation
The roots of this equation
x1=−81.6936483190068x_{1} = -81.6936483190068
x2=20.4691095857577x_{2} = 20.4691095857577
x3=26.7408899940119x_{3} = 26.7408899940119
x4=−6.43557028921917x_{4} = -6.43557028921917
x5=44.0050120637787x_{5} = 44.0050120637787
x6=15.7711591859629x_{6} = 15.7711591859629
x7=−59.7070049585797x_{7} = -59.7070049585797
x8=94.258387748226x_{8} = 94.258387748226
x9=−3.41607287346559x_{9} = -3.41607287346559
x10=45.5750265225894x_{10} = 45.5750265225894
x11=−89.5465568422098x_{11} = -89.5465568422098
x12=−83.2642138383723x_{12} = -83.2642138383723
x13=−1.99722235787071x_{13} = -1.99722235787071
x14=67.5590412185413x_{14} = 67.5590412185413
x15=70.6999766247443x_{15} = 70.6999766247443
x16=28.3096209125642x_{16} = 28.3096209125642
x17=58.1366607029306x_{17} = 58.1366607029306
x18=−58.1366607029306x_{18} = -58.1366607029306
x19=7.97773271487555x_{19} = 7.97773271487555
x20=59.7070049585797x_{20} = 59.7070049585797
x21=−37.725603538235x_{21} = -37.725603538235
x22=−67.5590412185413x_{22} = -67.5590412185413
x23=−51.8555571480304x_{23} = -51.8555571480304
x24=0x_{24} = 0
x25=−20.4691095857577x_{25} = -20.4691095857577
x26=−44.0050120637787x_{26} = -44.0050120637787
x27=1.99722235787071x_{27} = 1.99722235787071
x28=92.6877711443551x_{28} = 92.6877711443551
x29=−23.6042469916597x_{29} = -23.6042469916597
x30=23.6042469916597x_{30} = 23.6042469916597
x31=−75.4114823236753x_{31} = -75.4114823236753
x32=42.4350553508244x_{32} = 42.4350553508244
x33=73.8409679079427x_{33} = 73.8409679079427
x34=−17.3362830681118x_{34} = -17.3362830681118
x35=81.6936483190068x_{35} = 81.6936483190068
x36=3.41607287346559x_{36} = 3.41607287346559
x37=−64.418169852971x_{37} = -64.418169852971
x38=56.5663415203821x_{38} = 56.5663415203821
x39=−26.7408899940119x_{39} = -26.7408899940119
x40=89.5465568422098x_{40} = 89.5465568422098
x41=−53.4257872025938x_{41} = -53.4257872025938
x42=−94.258387748226x_{42} = -94.258387748226
x43=−15.7711591859629x_{43} = -15.7711591859629
x44=−22.0364503381404x_{44} = -22.0364503381404
x45=−102.111553670193x_{45} = -102.111553670193
x46=37.725603538235x_{46} = 37.725603538235
x47=12.6450452480401x_{47} = 12.6450452480401
x48=−39.2953427597448x_{48} = -39.2953427597448
x49=−28.3096209125642x_{49} = -28.3096209125642
x50=65.9885969598317x_{50} = 65.9885969598317
x51=−7.97773271487555x_{51} = -7.97773271487555
x52=72.2704657365879x_{52} = 72.2704657365879
x53=−87.9759598185177x_{53} = -87.9759598185177
x54=50.2853624109229x_{54} = 50.2853624109229
x55=−31.4476986173355x_{55} = -31.4476986173355
x56=22.0364503381404x_{56} = 22.0364503381404
x57=−14.2072653485813x_{57} = -14.2072653485813
x58=−97.3996383381479x_{58} = -97.3996383381479
x59=78.5525449532803x_{59} = 78.5525449532803
x60=−29.8785678341332x_{60} = -29.8785678341332
x61=−42.4350553508244x_{61} = -42.4350553508244
x62=−72.2704657365879x_{62} = -72.2704657365879
x63=−86.4053700369695x_{63} = -86.4053700369695
x64=64.418169852971x_{64} = 64.418169852971
x65=9.52877807686926x_{65} = 9.52877807686926
x66=−76.9820082350865x_{66} = -76.9820082350865
x67=6.43557028921917x_{67} = 6.43557028921917
x68=−95.829010241065x_{68} = -95.829010241065
x69=−50.2853624109229x_{69} = -50.2853624109229
x70=−73.8409679079427x_{70} = -73.8409679079427
x71=−9.52877807686926x_{71} = -9.52877807686926
x72=36.1559558691412x_{72} = 36.1559558691412
x73=14.2072653485813x_{73} = 14.2072653485813
x74=−61.2773723625442x_{74} = -61.2773723625442
x75=29.8785678341332x_{75} = 29.8785678341332
x76=80.1230918433114x_{76} = 80.1230918433114
x77=48.7152063990254x_{77} = 48.7152063990254
x78=95.829010241065x_{78} = 95.829010241065
x79=34.5864121653729x_{79} = 34.5864121653729
x80=−36.1559558691412x_{80} = -36.1559558691412
x81=−80.1230918433114x_{81} = -80.1230918433114
x82=86.4053700369695x_{82} = 86.4053700369695
x83=87.9759598185177x_{83} = 87.9759598185177
x84=−65.9885969598317x_{84} = -65.9885969598317
x85=51.8555571480304x_{85} = 51.8555571480304
x86=−45.5750265225894x_{86} = -45.5750265225894
x87=100.540910295039x_{87} = 100.540910295039

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
[95.829010241065,∞)\left[95.829010241065, \infty\right)
Convex at the intervals
(−∞,−102.111553670193]\left(-\infty, -102.111553670193\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⁡(2x))=∞sign⁡(⟨−1,1⟩)\lim_{x \to -\infty}\left(x^{2} \sin{\left(2 x \right)}\right) = \infty \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=∞sign⁡(⟨−1,1⟩)y = \infty \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}
lim⁡x→∞(x2sin⁡(2x))=∞sign⁡(⟨−1,1⟩)\lim_{x \to \infty}\left(x^{2} \sin{\left(2 x \right)}\right) = \infty \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=∞sign⁡(⟨−1,1⟩)y = \infty \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x^2*sin(2*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(xsin⁡(2x))=−∞sign⁡(⟨−1,1⟩)\lim_{x \to -\infty}\left(x \sin{\left(2 x \right)}\right) = - \infty \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}
Let's take the limit
so,
inclined asymptote equation on the left:
y=−∞xsign⁡(⟨−1,1⟩)y = - \infty x \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}
lim⁡x→∞(xsin⁡(2x))=∞sign⁡(⟨−1,1⟩)\lim_{x \to \infty}\left(x \sin{\left(2 x \right)}\right) = \infty \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}
Let's take the limit
so,
inclined asymptote equation on the right:
y=∞xsign⁡(⟨−1,1⟩)y = \infty x \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \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:
x2sin⁡(2x)=−x2sin⁡(2x)x^{2} \sin{\left(2 x \right)} = - x^{2} \sin{\left(2 x \right)}
- No
x2sin⁡(2x)=x2sin⁡(2x)x^{2} \sin{\left(2 x \right)} = x^{2} \sin{\left(2 x \right)}
- Yes
so, the function
is
odd
The graph
Graphing y = x^2*sin(2*x)
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