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Plotting a function graph/ (sin^2)x+1

Graphing y = (sin^2)x+1

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          2         
f(x) = sin (x)*x + 1
f(x)=xsin⁡2(x)+1f{\left(x \right)} = x \sin^{2}{\left(x \right)} + 1 
f = x*sin(x)^2 + 1
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)+1=0x \sin^{2}{\left(x \right)} + 1 = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=−31.5947853377724x_{1} = -31.5947853377724
x2=−5.85722663432788x_{2} = -5.85722663432788
x3=−6.68045113641011x_{3} = -6.68045113641011
x4=−28.084498001866x_{4} = -28.084498001866
x5=−72.1386194578734x_{5} = -72.1386194578734
x6=−5.85722663432788x_{6} = -5.85722663432788
x7=−69.2355105075789x_{7} = -69.2355105075789
x8=−53.5441648991804x_{8} = -53.5441648991804
x9=−21.7751749653377x_{9} = -21.7751749653377
x10=−59.8199170060094x_{10} = -59.8199170060094
x11=−9.75076473628389x_{11} = -9.75076473628389
x12=−87.8577042224353x_{12} = -87.8577042224353
x13=−28.4628884118944x_{13} = -28.4628884118944
x14=−50.1237618757427x_{14} = -50.1237618757427
x15=−50.4068019125433x_{15} = -50.4068019125433
x16=−37.8623518931411x_{16} = -37.8623518931411
x17=−65.8498999738295x_{17} = -65.8498999738295
x18=−97.4908249098309x_{18} = -97.4908249098309
x19=−18.6156571152535x_{19} = -18.6156571152535
x20=−78.6528134837387x_{20} = -78.6528134837387
x21=−2.44826060461718x_{21} = -2.44826060461718
x22=−25.3327542823831x_{22} = -25.3327542823831
x23=−68.9943548648633x_{23} = -68.9943548648633
x24=−78.4266556909368x_{24} = -78.4266556909368
x25=−81.7922072647269x_{25} = -81.7922072647269
x26=−34.3861487021489x_{26} = -34.3861487021489
x27=−15.9609591233296x_{27} = -15.9609591233296
x28=−43.8306702189324x_{28} = -43.8306702189324
x29=−84.7141386705402x_{29} = -84.7141386705402
x30=−40.683274504348x_{30} = -40.683274504348
x31=−1.17479617129145x_{31} = -1.17479617129145
x32=−47.2698556132478x_{32} = -47.2698556132478
x33=−62.9582190685772x_{33} = -62.9582190685772
x34=−94.1445333540311x_{34} = -94.1445333540311
x35=−56.4151332155179x_{35} = -56.4151332155179
x36=−62.7052310476363x_{36} = -62.7052310476363
x37=−19.0805353611918x_{37} = -19.0805353611918
x38=−100.431013386096x_{38} = -100.431013386096
x39=−3.68918886547985x_{39} = -3.68918886547985
x40=−40.9975249546568x_{40} = -40.9975249546568
x41=−12.2769462681894x_{41} = -12.2769462681894
x42=−75.5135559357283x_{42} = -75.5135559357283
x43=−91.2110863448323x_{43} = -91.2110863448323
x44=−1.1747961712916x_{44} = -1.1747961712916
x45=−84.9317245335367x_{45} = -84.9317245335367
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x)^2*x + 1.
0sin⁡2(0)+10 \sin^{2}{\left(0 \right)} + 1
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
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=65.9734457253857x_{1} = 65.9734457253857
x2=−67.5516436614121x_{2} = -67.5516436614121
x3=−21.9911485751286x_{3} = -21.9911485751286
x4=21.9911485751286x_{4} = 21.9911485751286
x5=−15.707963267949x_{5} = -15.707963267949
x6=−61.2692172687226x_{6} = -61.2692172687226
x7=80.1168534696549x_{7} = 80.1168534696549
x8=−73.8341991854591x_{8} = -73.8341991854591
x9=1.83659720315213x_{9} = 1.83659720315213
x10=64.410411962776x_{10} = 64.410411962776
x11=23.5831433102848x_{11} = 23.5831433102848
x12=6.28318530717959x_{12} = 6.28318530717959
x13=29.861872403816x_{13} = 29.861872403816
x14=−7.91705268466621x_{14} = -7.91705268466621
x15=−83.2582106616487x_{15} = -83.2582106616487
x16=28.2743338823081x_{16} = 28.2743338823081
x17=58.1280655761511x_{17} = 58.1280655761511
x18=−94.2477796076938x_{18} = -94.2477796076938
x19=−306.306916073247x_{19} = -306.306916073247
x20=−64.410411962776x_{20} = -64.410411962776
x21=−95.8237937978449x_{21} = -95.8237937978449
x22=−36.1421488970061x_{22} = -36.1421488970061
x23=−80.1168534696549x_{23} = -80.1168534696549
x24=−20.4448034666183x_{24} = -20.4448034666183
x25=70.692907433161x_{25} = 70.692907433161
x26=−1.83659720315213x_{26} = -1.83659720315213
x27=−53.4070751110265x_{27} = -53.4070751110265
x28=45.5640665961997x_{28} = 45.5640665961997
x29=36.1421488970061x_{29} = 36.1421488970061
x30=73.8341991854591x_{30} = 73.8341991854591
x31=−39.2826357527234x_{31} = -39.2826357527234
x32=59.6902604182061x_{32} = 59.6902604182061
x33=56.5486677646163x_{33} = 56.5486677646163
x34=25.1327412287183x_{34} = 25.1327412287183
x35=26.7222463741877x_{35} = 26.7222463741877
x36=72.2566310325652x_{36} = 72.2566310325652
x37=−50.2654824574367x_{37} = -50.2654824574367
x38=86.3995849739529x_{38} = 86.3995849739529
x39=78.5398163397448x_{39} = 78.5398163397448
x40=51.8459224452234x_{40} = 51.8459224452234
x41=−87.9645943005142x_{41} = -87.9645943005142
x42=37.6991118430775x_{42} = 37.6991118430775
x43=−6.28318530717959x_{43} = -6.28318530717959
x44=−37.6991118430775x_{44} = -37.6991118430775
x45=−43.9822971502571x_{45} = -43.9822971502571
x46=3.14159265358979x_{46} = 3.14159265358979
x47=−58.1280655761511x_{47} = -58.1280655761511
x48=−72.2566310325652x_{48} = -72.2566310325652
x49=−81.6814089933346x_{49} = -81.6814089933346
x50=−42.4232862577008x_{50} = -42.4232862577008
x51=−65.9734457253857x_{51} = -65.9734457253857
x52=0x_{52} = 0
x53=−28.2743338823081x_{53} = -28.2743338823081
x54=43.9822971502571x_{54} = 43.9822971502571
x55=92.682377997352x_{55} = 92.682377997352
x56=100.530964914873x_{56} = 100.530964914873
x57=−97.3893722612836x_{57} = -97.3893722612836
x58=81.6814089933346x_{58} = 81.6814089933346
x59=67.5516436614121x_{59} = 67.5516436614121
x60=−75.398223686155x_{60} = -75.398223686155
x61=−45.5640665961997x_{61} = -45.5640665961997
x62=−51.8459224452234x_{62} = -51.8459224452234
x63=95.8237937978449x_{63} = 95.8237937978449
x64=−4.81584231784594x_{64} = -4.81584231784594
x65=−86.3995849739529x_{65} = -86.3995849739529
x66=94.2477796076938x_{66} = 94.2477796076938
x67=50.2654824574367x_{67} = 50.2654824574367
x68=48.7049516666752x_{68} = 48.7049516666752
x69=−84.8230016469244x_{69} = -84.8230016469244
x70=−59.6902604182061x_{70} = -59.6902604182061
x71=14.1724320747999x_{71} = 14.1724320747999
x72=12.5663706143592x_{72} = 12.5663706143592
x73=−17.3076405374146x_{73} = -17.3076405374146
x74=−89.5409746049841x_{74} = -89.5409746049841
x75=−23.5831433102848x_{75} = -23.5831433102848
x76=278.032748190065x_{76} = 278.032748190065
x77=89.5409746049841x_{77} = 89.5409746049841
x78=−14.1724320747999x_{78} = -14.1724320747999
x79=34.5575191894877x_{79} = 34.5575191894877
x80=15.707963267949x_{80} = 15.707963267949
x81=87.9645943005142x_{81} = 87.9645943005142
x82=−29.861872403816x_{82} = -29.861872403816
x83=7.91705268466621x_{83} = 7.91705268466621
x84=42.4232862577008x_{84} = 42.4232862577008
x85=−9.42477796076938x_{85} = -9.42477796076938
x86=−105.248104538899x_{86} = -105.248104538899
x87=20.4448034666183x_{87} = 20.4448034666183
x88=−31.4159265358979x_{88} = -31.4159265358979
The values of the extrema at the points:
(65.97344572538566, 1)

(-67.5516436614121, -66.5479429919577)

(-21.991148575128552, 1)

(21.991148575128552, 1)

(-15.707963267948966, 1)

(-61.269217268722585, -60.2651371880071)

(80.11685346965491, 81.1137331491182)

(-73.83419918545908, -72.8308133759219)

(1.8365972031521258, 2.70986852923209)

(64.41041196277601, 65.4065308365988)

(23.583143310284843, 24.5725472811462)

(6.283185307179586, 1)

(29.861872403816044, 30.853502870657)

(-7.917052684666207, -6.88560072412753)

(-83.25821066164869, -82.255208063081)

(28.274333882308138, 1)

(58.12806557615112, 59.1237650459065)

(-94.2477796076938, 1)

(-306.30691607324667, -305.306099900576)

(-64.41041196277601, -63.4065308365988)

(-95.82379379784489, -94.8211849135206)

(-36.142148897006074, -35.135233089007)

(-80.11685346965491, -79.1137331491182)

(-20.4448034666183, -19.4325827297121)

(70.692907433161, 71.6893711873986)

(-1.8365972031521258, -0.709868529232089)

(-53.40707511102649, 1)

(45.56406659619972, 46.5585804770373)

(36.142148897006074, 37.135233089007)

(73.83419918545908, 74.8308133759219)

(-39.282635752723394, -38.2762726485285)

(59.69026041820607, 1)

(56.548667764616276, 1)

(25.132741228718345, 1)

(26.72224637418772, 27.7128941475173)

(72.25663103256524, 1)

(-50.26548245743669, 1)

(86.3995849739529, 87.3966915384367)

(78.53981633974483, 1)

(51.84592244522343, 52.8411009136761)

(-87.96459430051421, 1)

(37.69911184307752, 1)

(-6.283185307179586, 1)

(-37.69911184307752, 1)

(-43.982297150257104, 1)

(3.141592653589793, 1)

(-58.12806557615112, -57.1237650459065)

(-72.25663103256524, 1)

(-81.68140899333463, 1)

(-42.423286257700816, -41.4173940862181)

(-65.97344572538566, 1)

(0, 1)

(-28.274333882308138, 1)

(43.982297150257104, 1)

(92.68237799735202, 93.6796806914592)

(100.53096491487338, 1)

(-97.3893722612836, 1)

(81.68140899333463, 1)

(67.5516436614121, 68.5479429919577)

(-75.39822368615503, 1)

(-45.56406659619972, -44.5585804770373)

(-51.84592244522343, -50.8411009136761)

(95.82379379784489, 96.8211849135206)

(-4.815842317845935, -3.76448393290203)

(-86.3995849739529, -85.3966915384367)

(94.2477796076938, 1)

(50.26548245743669, 1)

(48.70495166667517, 49.6998192592491)

(-84.82300164692441, 1)

(-59.69026041820607, 1)

(14.172432074799941, 15.1548141232633)

(12.566370614359172, 1)

(-17.307640537414635, -16.2932080946897)

(-89.54097460498406, -88.5381826741839)

(-23.583143310284843, -22.5725472811462)

(278.0327481900649, 279.031849018319)

(89.54097460498406, 90.5381826741839)

(-14.172432074799941, -13.1548141232633)

(34.55751918948773, 1)

(15.707963267948966, 1)

(87.96459430051421, 1)

(-29.861872403816044, -28.853502870657)

(7.917052684666207, 8.88560072412753)

(42.423286257700816, 43.4173940862181)

(-9.42477796076938, 1)

(-105.24810453889911, -104.245729252817)

(20.4448034666183, 21.4325827297121)

(-31.41592653589793, 1)


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

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)+1)=⟨−∞,1⟩\lim_{x \to -\infty}\left(x \sin^{2}{\left(x \right)} + 1\right) = \left\langle -\infty, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−∞,1⟩y = \left\langle -\infty, 1\right\rangle
lim⁡x→∞(xsin⁡2(x)+1)=⟨0,∞⟩\lim_{x \to \infty}\left(x \sin^{2}{\left(x \right)} + 1\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 sin(x)^2*x + 1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(xsin⁡2(x)+1x)=⟨0,∞⟩\lim_{x \to -\infty}\left(\frac{x \sin^{2}{\left(x \right)} + 1}{x}\right) = \left\langle 0, \infty\right\rangle
Let's take the limit
so,
inclined asymptote equation on the left:
y=⟨0,∞⟩xy = \left\langle 0, \infty\right\rangle x
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(xsin⁡2(x)+1x)y = x \lim_{x \to \infty}\left(\frac{x \sin^{2}{\left(x \right)} + 1}{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:
xsin⁡2(x)+1=−xsin⁡2(x)+1x \sin^{2}{\left(x \right)} + 1 = - x \sin^{2}{\left(x \right)} + 1
- No
xsin⁡2(x)+1=xsin⁡2(x)−1x \sin^{2}{\left(x \right)} + 1 = x \sin^{2}{\left(x \right)} - 1
- No
so, the function
not is
neither even, nor odd
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