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Plotting a function graph/ xsin^2x

Graphing y = xsin^2x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
            2   
f(x) = x*sin (x)
f(x)=xsin⁡2(x)f{\left(x \right)} = x \sin^{2}{\left(x \right)} 
f = x*sin(x)^2
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:
xsin⁡2(x)=0x \sin^{2}{\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=−12.5663703832991x_{1} = -12.5663703832991
x2=25.1327410420105x_{2} = 25.1327410420105
x3=−18.8495561496377x_{3} = -18.8495561496377
x4=−37.6991118772194x_{4} = -37.6991118772194
x5=3.1415923353488x_{5} = 3.1415923353488
x6=−87.964594358858x_{6} = -87.964594358858
x7=94.2477796093525x_{7} = 94.2477796093525
x8=−3.14159271719906x_{8} = -3.14159271719906
x9=−31.4159267074656x_{9} = -31.4159267074656
x10=−9.09618852922⋅10−5x_{10} = -9.09618852922 \cdot 10^{-5}
x11=−28.2743337190252x_{11} = -28.2743337190252
x12=37.6991120212708x_{12} = 37.6991120212708
x13=3.14159299901751x_{13} = 3.14159299901751
x14=−15.7079632966406x_{14} = -15.7079632966406
x15=−3.14159295661011x_{15} = -3.14159295661011
x16=−65.97344576507x_{16} = -65.97344576507
x17=−9.42477813384597x_{17} = -9.42477813384597
x18=28.2743338652241x_{18} = 28.2743338652241
x19=34.5575190322918x_{19} = 34.5575190322918
x20=−81.6814090380603x_{20} = -81.6814090380603
x21=−6.28318514963244x_{21} = -6.28318514963244
x22=65.9734457529229x_{22} = 65.9734457529229
x23=−18.8495557219808x_{23} = -18.8495557219808
x24=0x_{24} = 0
x25=43.9822971694455x_{25} = 43.9822971694455
x26=6.28318528443896x_{26} = 6.28318528443896
x27=87.964594335789x_{27} = 87.964594335789
x28=3.69946911765974⋅10−5x_{28} = 3.69946911765974 \cdot 10^{-5}
x29=−21.9911485864645x_{29} = -21.9911485864645
x30=−59.6902604576978x_{30} = -59.6902604576978
x31=50.2654824463527x_{31} = 50.2654824463527
x32=21.9911485852065x_{32} = 21.9911485852065
x33=9.42477823217446x_{33} = 9.42477823217446
x34=3.1415931516833x_{34} = 3.1415931516833
x35=72.2566310277197x_{35} = 72.2566310277197
x36=18.8495556906624x_{36} = 18.8495556906624
x37=−43.9822971746086x_{37} = -43.9822971746086
x38=15.7079634453651x_{38} = 15.7079634453651
x39=12.5663704571697x_{39} = 12.5663704571697
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)^2.
0sin⁡2(0)0 \sin^{2}{\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
2xsin⁡(x)cos⁡(x)+sin⁡2(x)=02 x \sin{\left(x \right)} \cos{\left(x \right)} + \sin^{2}{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−97.3893722612836x_{1} = -97.3893722612836
x2=−43.9822971502571x_{2} = -43.9822971502571
x3=278.032748190065x_{3} = 278.032748190065
x4=−45.5640665961997x_{4} = -45.5640665961997
x5=56.5486677646163x_{5} = 56.5486677646163
x6=−14.1724320747999x_{6} = -14.1724320747999
x7=−73.8341991854591x_{7} = -73.8341991854591
x8=28.2743338823081x_{8} = 28.2743338823081
x9=−81.6814089933346x_{9} = -81.6814089933346
x10=−15.707963267949x_{10} = -15.707963267949
x11=72.2566310325652x_{11} = 72.2566310325652
x12=48.7049516666752x_{12} = 48.7049516666752
x13=−80.1168534696549x_{13} = -80.1168534696549
x14=95.8237937978449x_{14} = 95.8237937978449
x15=−29.861872403816x_{15} = -29.861872403816
x16=89.5409746049841x_{16} = 89.5409746049841
x17=−53.4070751110265x_{17} = -53.4070751110265
x18=25.1327412287183x_{18} = 25.1327412287183
x19=100.530964914873x_{19} = 100.530964914873
x20=−9.42477796076938x_{20} = -9.42477796076938
x21=81.6814089933346x_{21} = 81.6814089933346
x22=−83.2582106616487x_{22} = -83.2582106616487
x23=−21.9911485751286x_{23} = -21.9911485751286
x24=1.83659720315213x_{24} = 1.83659720315213
x25=80.1168534696549x_{25} = 80.1168534696549
x26=−20.4448034666183x_{26} = -20.4448034666183
x27=−58.1280655761511x_{27} = -58.1280655761511
x28=−72.2566310325652x_{28} = -72.2566310325652
x29=15.707963267949x_{29} = 15.707963267949
x30=50.2654824574367x_{30} = 50.2654824574367
x31=−28.2743338823081x_{31} = -28.2743338823081
x32=−6.28318530717959x_{32} = -6.28318530717959
x33=23.5831433102848x_{33} = 23.5831433102848
x34=3.14159265358979x_{34} = 3.14159265358979
x35=−306.306916073247x_{35} = -306.306916073247
x36=−84.8230016469244x_{36} = -84.8230016469244
x37=−86.3995849739529x_{37} = -86.3995849739529
x38=42.4232862577008x_{38} = 42.4232862577008
x39=−51.8459224452234x_{39} = -51.8459224452234
x40=51.8459224452234x_{40} = 51.8459224452234
x41=−36.1421488970061x_{41} = -36.1421488970061
x42=−87.9645943005142x_{42} = -87.9645943005142
x43=−42.4232862577008x_{43} = -42.4232862577008
x44=58.1280655761511x_{44} = 58.1280655761511
x45=−61.2692172687226x_{45} = -61.2692172687226
x46=0x_{46} = 0
x47=−1.83659720315213x_{47} = -1.83659720315213
x48=−65.9734457253857x_{48} = -65.9734457253857
x49=−89.5409746049841x_{49} = -89.5409746049841
x50=−67.5516436614121x_{50} = -67.5516436614121
x51=6.28318530717959x_{51} = 6.28318530717959
x52=94.2477796076938x_{52} = 94.2477796076938
x53=65.9734457253857x_{53} = 65.9734457253857
x54=7.91705268466621x_{54} = 7.91705268466621
x55=21.9911485751286x_{55} = 21.9911485751286
x56=64.410411962776x_{56} = 64.410411962776
x57=78.5398163397448x_{57} = 78.5398163397448
x58=−64.410411962776x_{58} = -64.410411962776
x59=−39.2826357527234x_{59} = -39.2826357527234
x60=−23.5831433102848x_{60} = -23.5831433102848
x61=87.9645943005142x_{61} = 87.9645943005142
x62=12.5663706143592x_{62} = 12.5663706143592
x63=26.7222463741877x_{63} = 26.7222463741877
x64=−4.81584231784594x_{64} = -4.81584231784594
x65=73.8341991854591x_{65} = 73.8341991854591
x66=−37.6991118430775x_{66} = -37.6991118430775
x67=−95.8237937978449x_{67} = -95.8237937978449
x68=59.6902604182061x_{68} = 59.6902604182061
x69=43.9822971502571x_{69} = 43.9822971502571
x70=70.692907433161x_{70} = 70.692907433161
x71=−50.2654824574367x_{71} = -50.2654824574367
x72=37.6991118430775x_{72} = 37.6991118430775
x73=14.1724320747999x_{73} = 14.1724320747999
x74=92.682377997352x_{74} = 92.682377997352
x75=−59.6902604182061x_{75} = -59.6902604182061
x76=−105.248104538899x_{76} = -105.248104538899
x77=−75.398223686155x_{77} = -75.398223686155
x78=29.861872403816x_{78} = 29.861872403816
x79=67.5516436614121x_{79} = 67.5516436614121
x80=86.3995849739529x_{80} = 86.3995849739529
x81=34.5575191894877x_{81} = 34.5575191894877
x82=−17.3076405374146x_{82} = -17.3076405374146
x83=45.5640665961997x_{83} = 45.5640665961997
x84=−94.2477796076938x_{84} = -94.2477796076938
x85=−7.91705268466621x_{85} = -7.91705268466621
x86=20.4448034666183x_{86} = 20.4448034666183
x87=−31.4159265358979x_{87} = -31.4159265358979
x88=36.1421488970061x_{88} = 36.1421488970061
The values of the extrema at the points:
(-97.3893722612836, -4.58542475390885e-27)

(-43.982297150257104, -1.29287245613476e-28)

(278.0327481900649, 278.031849018319)

(-45.56406659619972, -45.5585804770373)

(56.548667764616276, 2.7478251327179e-28)

(-14.172432074799941, -14.1548141232633)

(-73.83419918545908, -73.8308133759219)

(28.274333882308138, 3.43478141589738e-29)

(-81.68140899333463, -1.25601110053315e-27)

(-15.707963267948966, -5.8895428941999e-30)

(72.25663103256524, 2.93139900017185e-27)

(48.70495166667517, 48.6998192592491)

(-80.11685346965491, -80.1137331491182)

(95.82379379784489, 95.8211849135206)

(-29.861872403816044, -29.853502870657)

(89.54097460498406, 89.5381826741839)

(-53.40707511102649, -1.15535214562331e-28)

(25.132741228718345, 2.41235676946428e-29)

(100.53096491487338, 1.54390833245714e-27)

(-9.42477796076938, -1.27214126514718e-30)

(81.68140899333463, 1.25601110053315e-27)

(-83.25821066164869, -83.255208063081)

(-21.991148575128552, -1.61609057016845e-29)

(1.8365972031521258, 1.70986852923209)

(80.11685346965491, 80.1137331491182)

(-20.4448034666183, -20.4325827297121)

(-58.12806557615112, -58.1237650459065)

(-72.25663103256524, -2.93139900017185e-27)

(15.707963267948966, 5.8895428941999e-30)

(50.26548245743669, 1.92988541557142e-28)

(-28.274333882308138, -3.43478141589738e-29)

(-6.283185307179586, -3.76930745228793e-31)

(23.583143310284843, 23.5725472811462)

(3.141592653589793, 4.71163431535992e-32)

(-306.30691607324667, -306.306099900576)

(-84.82300164692441, -3.99087542625273e-27)

(-86.3995849739529, -86.3966915384367)

(42.423286257700816, 42.4173940862181)

(-51.84592244522343, -51.8411009136761)

(51.84592244522343, 51.8411009136761)

(-36.142148897006074, -36.135233089007)

(-87.96459430051421, -1.03429796490781e-27)

(-42.423286257700816, -42.4173940862181)

(58.12806557615112, 58.1237650459065)

(-61.269217268722585, -61.2651371880071)

(0, 0)

(-1.8365972031521258, -1.70986852923209)

(-65.97344572538566, -6.34844983898999e-29)

(-89.54097460498406, -89.5381826741839)

(-67.5516436614121, -67.5479429919577)

(6.283185307179586, 3.76930745228793e-31)

(94.2477796076938, 1.10977728956951e-27)

(65.97344572538566, 6.34844983898999e-29)

(7.917052684666207, 7.88560072412753)

(21.991148575128552, 1.61609057016845e-29)

(64.41041196277601, 64.4065308365988)

(78.53981633974483, 1.8941914820334e-29)

(-64.41041196277601, -64.4065308365988)

(-39.282635752723394, -39.2762726485285)

(-23.583143310284843, -23.5725472811462)

(87.96459430051421, 1.03429796490781e-27)

(12.566370614359172, 3.01544596183035e-30)

(26.72224637418772, 26.7128941475173)

(-4.815842317845935, -4.76448393290203)

(73.83419918545908, 73.8308133759219)

(-37.69911184307752, -8.14170409694193e-29)

(-95.82379379784489, -95.8211849135206)

(59.69026041820607, 8.97021321364436e-29)

(43.982297150257104, 1.29287245613476e-28)

(70.692907433161, 70.6893711873986)

(-50.26548245743669, -1.92988541557142e-28)

(37.69911184307752, 8.14170409694193e-29)

(14.172432074799941, 14.1548141232633)

(92.68237799735202, 92.6796806914592)

(-59.69026041820607, -8.97021321364436e-29)

(-105.24810453889911, -105.245729252817)

(-75.39822368615503, -6.51336327755355e-28)

(29.861872403816044, 29.853502870657)

(67.5516436614121, 67.5479429919577)

(86.3995849739529, 86.3966915384367)

(34.55751918948773, 1.68111309202325e-28)

(-17.307640537414635, -17.2932080946897)

(45.56406659619972, 45.5585804770373)

(-94.2477796076938, -1.10977728956951e-27)

(-7.917052684666207, -7.88560072412753)

(20.4448034666183, 20.4325827297121)

(-31.41592653589793, -4.71163431535992e-29)

(36.142148897006074, 36.135233089007)


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=−45.5640665961997x_{1} = -45.5640665961997
x2=56.5486677646163x_{2} = 56.5486677646163
x3=−14.1724320747999x_{3} = -14.1724320747999
x4=−73.8341991854591x_{4} = -73.8341991854591
x5=28.2743338823081x_{5} = 28.2743338823081
x6=72.2566310325652x_{6} = 72.2566310325652
x7=−80.1168534696549x_{7} = -80.1168534696549
x8=−29.861872403816x_{8} = -29.861872403816
x9=25.1327412287183x_{9} = 25.1327412287183
x10=100.530964914873x_{10} = 100.530964914873
x11=81.6814089933346x_{11} = 81.6814089933346
x12=−83.2582106616487x_{12} = -83.2582106616487
x13=−20.4448034666183x_{13} = -20.4448034666183
x14=−58.1280655761511x_{14} = -58.1280655761511
x15=15.707963267949x_{15} = 15.707963267949
x16=50.2654824574367x_{16} = 50.2654824574367
x17=3.14159265358979x_{17} = 3.14159265358979
x18=−306.306916073247x_{18} = -306.306916073247
x19=−86.3995849739529x_{19} = -86.3995849739529
x20=−51.8459224452234x_{20} = -51.8459224452234
x21=−36.1421488970061x_{21} = -36.1421488970061
x22=−42.4232862577008x_{22} = -42.4232862577008
x23=−61.2692172687226x_{23} = -61.2692172687226
x24=−1.83659720315213x_{24} = -1.83659720315213
x25=−89.5409746049841x_{25} = -89.5409746049841
x26=−67.5516436614121x_{26} = -67.5516436614121
x27=6.28318530717959x_{27} = 6.28318530717959
x28=94.2477796076938x_{28} = 94.2477796076938
x29=65.9734457253857x_{29} = 65.9734457253857
x30=21.9911485751286x_{30} = 21.9911485751286
x31=78.5398163397448x_{31} = 78.5398163397448
x32=−64.410411962776x_{32} = -64.410411962776
x33=−39.2826357527234x_{33} = -39.2826357527234
x34=−23.5831433102848x_{34} = -23.5831433102848
x35=87.9645943005142x_{35} = 87.9645943005142
x36=12.5663706143592x_{36} = 12.5663706143592
x37=−4.81584231784594x_{37} = -4.81584231784594
x38=−95.8237937978449x_{38} = -95.8237937978449
x39=59.6902604182061x_{39} = 59.6902604182061
x40=43.9822971502571x_{40} = 43.9822971502571
x41=37.6991118430775x_{41} = 37.6991118430775
x42=−105.248104538899x_{42} = -105.248104538899
x43=34.5575191894877x_{43} = 34.5575191894877
x44=−17.3076405374146x_{44} = -17.3076405374146
x45=−7.91705268466621x_{45} = -7.91705268466621
Maxima of the function at points:
x45=−97.3893722612836x_{45} = -97.3893722612836
x45=−43.9822971502571x_{45} = -43.9822971502571
x45=278.032748190065x_{45} = 278.032748190065
x45=−81.6814089933346x_{45} = -81.6814089933346
x45=−15.707963267949x_{45} = -15.707963267949
x45=48.7049516666752x_{45} = 48.7049516666752
x45=95.8237937978449x_{45} = 95.8237937978449
x45=89.5409746049841x_{45} = 89.5409746049841
x45=−53.4070751110265x_{45} = -53.4070751110265
x45=−9.42477796076938x_{45} = -9.42477796076938
x45=−21.9911485751286x_{45} = -21.9911485751286
x45=1.83659720315213x_{45} = 1.83659720315213
x45=80.1168534696549x_{45} = 80.1168534696549
x45=−72.2566310325652x_{45} = -72.2566310325652
x45=−28.2743338823081x_{45} = -28.2743338823081
x45=−6.28318530717959x_{45} = -6.28318530717959
x45=23.5831433102848x_{45} = 23.5831433102848
x45=−84.8230016469244x_{45} = -84.8230016469244
x45=42.4232862577008x_{45} = 42.4232862577008
x45=51.8459224452234x_{45} = 51.8459224452234
x45=−87.9645943005142x_{45} = -87.9645943005142
x45=58.1280655761511x_{45} = 58.1280655761511
x45=−65.9734457253857x_{45} = -65.9734457253857
x45=7.91705268466621x_{45} = 7.91705268466621
x45=64.410411962776x_{45} = 64.410411962776
x45=26.7222463741877x_{45} = 26.7222463741877
x45=73.8341991854591x_{45} = 73.8341991854591
x45=−37.6991118430775x_{45} = -37.6991118430775
x45=70.692907433161x_{45} = 70.692907433161
x45=−50.2654824574367x_{45} = -50.2654824574367
x45=14.1724320747999x_{45} = 14.1724320747999
x45=92.682377997352x_{45} = 92.682377997352
x45=−59.6902604182061x_{45} = -59.6902604182061
x45=−75.398223686155x_{45} = -75.398223686155
x45=29.861872403816x_{45} = 29.861872403816
x45=67.5516436614121x_{45} = 67.5516436614121
x45=86.3995849739529x_{45} = 86.3995849739529
x45=45.5640665961997x_{45} = 45.5640665961997
x45=−94.2477796076938x_{45} = -94.2477796076938
x45=20.4448034666183x_{45} = 20.4448034666183
x45=−31.4159265358979x_{45} = -31.4159265358979
x45=36.1421488970061x_{45} = 36.1421488970061
Decreasing at intervals
[100.530964914873,∞)\left[100.530964914873, \infty\right)
Increasing at intervals
(−∞,−306.306916073247]\left(-\infty, -306.306916073247\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(−x(sin⁡2(x)−cos⁡2(x))+2sin⁡(x)cos⁡(x))=02 \left(- x \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) = 0
Solve this equation
The roots of this equation
x1=46.3492776216985x_{1} = 46.3492776216985
x2=27.5071048394191x_{2} = 27.5071048394191
x3=−77.760847792972x_{3} = -77.760847792972
x4=24.3678503974527x_{4} = 24.3678503974527
x5=−71.4782275499213x_{5} = -71.4782275499213
x6=11.8231619098018x_{6} = 11.8231619098018
x7=−99.7505790857949x_{7} = -99.7505790857949
x8=60.4839244878466x_{8} = 60.4839244878466
x9=−11.8231619098018x_{9} = -11.8231619098018
x10=−10.2587614549708x_{10} = -10.2587614549708
x11=−68.3369563786298x_{11} = -68.3369563786298
x12=−47.9197205706165x_{12} = -47.9197205706165
x13=41.6381085824888x_{13} = 41.6381085824888
x14=120.170079673253x_{14} = 120.170079673253
x15=47.9197205706165x_{15} = 47.9197205706165
x16=−69.9075883539626x_{16} = -69.9075883539626
x17=−32.2168395518658x_{17} = -32.2168395518658
x18=−82.4728694594266x_{18} = -82.4728694594266
x19=63.6251091208926x_{19} = 63.6251091208926
x20=74.6195257807054x_{20} = 74.6195257807054
x21=−63.6251091208926x_{21} = -63.6251091208926
x22=76.1901839979235x_{22} = 76.1901839979235
x23=0x_{23} = 0
x24=−40.0677825970372x_{24} = -40.0677825970372
x25=−21.2292853858495x_{25} = -21.2292853858495
x26=55.7722336752062x_{26} = 55.7722336752062
x27=−13.3890435377793x_{27} = -13.3890435377793
x28=66.766332133246x_{28} = 66.766332133246
x29=52.6311758774383x_{29} = 52.6311758774383
x30=−5.58635293416499x_{30} = -5.58635293416499
x31=8.69662198229738x_{31} = 8.69662198229738
x32=25.9374070267134x_{32} = 25.9374070267134
x33=10.2587614549708x_{33} = 10.2587614549708
x34=71.4782275499213x_{34} = 71.4782275499213
x35=2.54349254705114x_{35} = 2.54349254705114
x36=54.2016970313842x_{36} = 54.2016970313842
x37=96.6091494063022x_{37} = 96.6091494063022
x38=−19.6603640661261x_{38} = -19.6603640661261
x39=−79.3315168346756x_{39} = -79.3315168346756
x40=19.6603640661261x_{40} = 19.6603640661261
x41=−33.7869153354295x_{41} = -33.7869153354295
x42=5.58635293416499x_{42} = 5.58635293416499
x43=88.7556256712795x_{43} = 88.7556256712795
x44=91.8970257752571x_{44} = 91.8970257752571
x45=40.0677825970372x_{45} = 40.0677825970372
x46=−55.7722336752062x_{46} = -55.7722336752062
x47=77.760847792972x_{47} = 77.760847792972
x48=−46.3492776216985x_{48} = -46.3492776216985
x49=−91.8970257752571x_{49} = -91.8970257752571
x50=−93.4677306800165x_{50} = -93.4677306800165
x51=85.6142396947314x_{51} = 85.6142396947314
x52=−35.3570550332742x_{52} = -35.3570550332742
x53=−98.1798629425939x_{53} = -98.1798629425939
x54=30.6468374831214x_{54} = 30.6468374831214
x55=16.5235843473527x_{55} = 16.5235843473527
x56=38.4974949445838x_{56} = 38.4974949445838
x57=69.9075883539626x_{57} = 69.9075883539626
x58=82.4728694594266x_{58} = 82.4728694594266
x59=62.0545116429054x_{59} = 62.0545116429054
x60=18.0917665453763x_{60} = 18.0917665453763
x61=84.0435524991391x_{61} = 84.0435524991391
x62=−84.0435524991391x_{62} = -84.0435524991391
x63=49.4901859325761x_{63} = 49.4901859325761
x64=−85.6142396947314x_{64} = -85.6142396947314
x65=98.1798629425939x_{65} = 98.1798629425939
x66=−41.6381085824888x_{66} = -41.6381085824888
x67=−18.0917665453763x_{67} = -18.0917665453763
x68=68.3369563786298x_{68} = 68.3369563786298
x69=32.2168395518658x_{69} = 32.2168395518658
x70=−1.1444648640517x_{70} = -1.1444648640517
x71=−58.9133484807877x_{71} = -58.9133484807877
x72=33.7869153354295x_{72} = 33.7869153354295
x73=−54.2016970313842x_{73} = -54.2016970313842
x74=−90.3263240494369x_{74} = -90.3263240494369
x75=−49.4901859325761x_{75} = -49.4901859325761
x76=−25.9374070267134x_{76} = -25.9374070267134
x77=−66.766332133246x_{77} = -66.766332133246
x78=4.04808180161146x_{78} = 4.04808180161146
x79=90.3263240494369x_{79} = 90.3263240494369
x80=99.7505790857949x_{80} = 99.7505790857949
x81=−76.1901839979235x_{81} = -76.1901839979235
x82=−27.5071048394191x_{82} = -27.5071048394191
x83=−60.4839244878466x_{83} = -60.4839244878466
x84=−57.3427845371101x_{84} = -57.3427845371101
x85=−62.0545116429054x_{85} = -62.0545116429054
x86=−24.3678503974527x_{86} = -24.3678503974527
x87=−4.04808180161146x_{87} = -4.04808180161146
x88=−38.4974949445838x_{88} = -38.4974949445838

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
[99.7505790857949,∞)\left[99.7505790857949, \infty\right)
Convex at the intervals
(−∞,−99.7505790857949]\left(-\infty, -99.7505790857949\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⁡2(x))=⟨−∞,0⟩\lim_{x \to -\infty}\left(x \sin^{2}{\left(x \right)}\right) = \left\langle -\infty, 0\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−∞,0⟩y = \left\langle -\infty, 0\right\rangle
lim⁡x→∞(xsin⁡2(x))=⟨0,∞⟩\lim_{x \to \infty}\left(x \sin^{2}{\left(x \right)}\right) = \left\langle 0, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨0,∞⟩y = \left\langle 0, \infty\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x*sin(x)^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞sin⁡2(x)=⟨0,1⟩\lim_{x \to -\infty} \sin^{2}{\left(x \right)} = \left\langle 0, 1\right\rangle
Let's take the limit
so,
inclined asymptote equation on the left:
y=⟨0,1⟩xy = \left\langle 0, 1\right\rangle x
lim⁡x→∞sin⁡2(x)=⟨0,1⟩\lim_{x \to \infty} \sin^{2}{\left(x \right)} = \left\langle 0, 1\right\rangle
Let's take the limit
so,
inclined asymptote equation on the right:
y=⟨0,1⟩xy = \left\langle 0, 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⁡2(x)=−xsin⁡2(x)x \sin^{2}{\left(x \right)} = - x \sin^{2}{\left(x \right)}
- No
xsin⁡2(x)=xsin⁡2(x)x \sin^{2}{\left(x \right)} = x \sin^{2}{\left(x \right)}
- Yes
so, the function
is
odd
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