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Plotting a function graph/ sinx^2/x^2

Graphing y = sinx^2/x^2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          2   
       sin (x)
f(x) = -------
           2  
          x
f(x)=sin⁡2(x)x2f{\left(x \right)} = \frac{\sin^{2}{\left(x \right)}}{x^{2}} 
f = sin(x)^2/x^2
The graph of the function
02468-8-6-4-2-101002
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
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:
sin⁡2(x)x2=0\frac{\sin^{2}{\left(x \right)}}{x^{2}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=πx_{1} = \pi
Numerical solution
x1=81.6814091740375x_{1} = 81.6814091740375
x2=−100.530964668746x_{2} = -100.530964668746
x3=−81.6814090378975x_{3} = -81.6814090378975
x4=62.8318528264673x_{4} = 62.8318528264673
x5=−28.274336411954x_{5} = -28.274336411954
x6=15.7079634314657x_{6} = 15.7079634314657
x7=−97.3893724386893x_{7} = -97.3893724386893
x8=−18.8495563021403x_{8} = -18.8495563021403
x9=94.2477796093523x_{9} = 94.2477796093523
x10=91.1061867208972x_{10} = 91.1061867208972
x11=−75.398223859952x_{11} = -75.398223859952
x12=56.5486676070174x_{12} = 56.5486676070174
x13=−91.1061871357964x_{13} = -91.1061871357964
x14=3.14159202258703x_{14} = 3.14159202258703
x15=−72.2566351619525x_{15} = -72.2566351619525
x16=25.1327414061616x_{16} = 25.1327414061616
x17=84.823001404869x_{17} = 84.823001404869
x18=3.14159230647624x_{18} = 3.14159230647624
x19=−91.1061871962484x_{19} = -91.1061871962484
x20=−6.28318511325755x_{20} = -6.28318511325755
x21=69.1150381457816x_{21} = 69.1150381457816
x22=−182.212330270911x_{22} = -182.212330270911
x23=−15.7079632962971x_{23} = -15.7079632962971
x24=−40.8407046587706x_{24} = -40.8407046587706
x25=−1709.02635315652x_{25} = -1709.02635315652
x26=100.530964765533x_{26} = 100.530964765533
x27=−12.5663703305055x_{27} = -12.5663703305055
x28=59.690260594979x_{28} = 59.690260594979
x29=−25.1327414564238x_{29} = -25.1327414564238
x30=−18.8495560547772x_{30} = -18.8495560547772
x31=97.3893725100508x_{31} = 97.3893725100508
x32=40.8407042462337x_{32} = 40.8407042462337
x33=−78.5398160908277x_{33} = -78.5398160908277
x34=69.1150385723079x_{34} = 69.1150385723079
x35=−18.8495556424861x_{35} = -18.8495556424861
x36=18.8495556571753x_{36} = 18.8495556571753
x37=−84.8230013997108x_{37} = -84.8230013997108
x38=−81.6814095337443x_{38} = -81.6814095337443
x39=9.42477789820696x_{39} = 9.42477789820696
x40=40.8407039198138x_{40} = 40.8407039198138
x41=−34.5575189305341x_{41} = -34.5575189305341
x42=−37.6991119106453x_{42} = -37.6991119106453
x43=50.2654824463366x_{43} = 50.2654824463366
x44=65.9734457528465x_{44} = 65.9734457528465
x45=31.4159268994904x_{45} = 31.4159268994904
x46=78.5398161863985x_{46} = 78.5398161863985
x47=−47.1238900096216x_{47} = -47.1238900096216
x48=−50.2654822927432x_{48} = -50.2654822927432
x49=9.42477816834986x_{49} = 9.42477816834986
x50=−9.42477811279807x_{50} = -9.42477811279807
x51=−75.3982253232481x_{51} = -75.3982253232481
x52=−65.9734457649134x_{52} = -65.9734457649134
x53=43.98229716939x_{53} = 43.98229716939
x54=87.9645943356948x_{54} = 87.9645943356948
x55=−69.1150385823665x_{55} = -69.1150385823665
x56=75.3982239327941x_{56} = 75.3982239327941
x57=333.008670443942x_{57} = 333.008670443942
x58=−94.2477794517506x_{58} = -94.2477794517506
x59=37.6991120151215x_{59} = 37.6991120151215
x60=91.1061871454997x_{60} = 91.1061871454997
x61=−62.8318532373291x_{61} = -62.8318532373291
x62=−40.8407042430283x_{62} = -40.8407042430283
x63=−3.1415918086506x_{63} = -3.1415918086506
x64=−62.831852823464x_{64} = -62.831852823464
x65=28.2743338651556x_{65} = 28.2743338651556
x66=72.2566310277176x_{66} = 72.2566310277176
x67=75.3982240865665x_{67} = 75.3982240865665
x68=31.4159267729052x_{68} = 31.4159267729052
x69=−31.4159267006674x_{69} = -31.4159267006674
x70=−47.1238900400185x_{70} = -47.1238900400185
x71=18.849559744074x_{71} = 18.849559744074
x72=−84.82300181x_{72} = -84.82300181
x73=53.4070753544334x_{73} = 53.4070753544334
x74=−37.6991118770152x_{74} = -37.6991118770152
x75=84.8230011943097x_{75} = 84.8230011943097
x76=−56.5486675120423x_{76} = -56.5486675120423
x77=12.5663704410235x_{77} = 12.5663704410235
x78=−21.991148586426x_{78} = -21.991148586426
x79=47.1238895673742x_{79} = 47.1238895673742
x80=21.9911485851759x_{80} = 21.9911485851759
x81=−69.1150386188422x_{81} = -69.1150386188422
x82=47.1238899954189x_{82} = 47.1238899954189
x83=53.4070755245567x_{83} = 53.4070755245567
x84=97.3893726288047x_{84} = 97.3893726288047
x85=−3.14159278291973x_{85} = -3.14159278291973
x86=34.5575190268142x_{86} = 34.5575190268142
x87=−87.9645943586046x_{87} = -87.9645943586046
x88=6.28318528388037x_{88} = 6.28318528388037
x89=−25.1327413659795x_{89} = -25.1327413659795
x90=−59.6902604575246x_{90} = -59.6902604575246
x91=−53.4070752808355x_{91} = -53.4070752808355
x92=−25.1327415660596x_{92} = -25.1327415660596
x93=62.8318524971054x_{93} = 62.8318524971054
x94=−87.964591359568x_{94} = -87.964591359568
x95=−43.98229717452x_{95} = -43.98229717452
x96=−72.2566308724232x_{96} = -72.2566308724232
x97=15.7079650823801x_{97} = 15.7079650823801
x98=−28.2743337117191x_{98} = -28.2743337117191
x99=56.5486679526703x_{99} = 56.5486679526703
x100=25.1327409761656x_{100} = 25.1327409761656
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^2.
sin⁡2(0)02\frac{\sin^{2}{\left(0 \right)}}{0^{2}}
The result:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- the solutions of the equation d'not exist
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
2sin⁡(x)cos⁡(x)x2−2sin⁡2(x)x3=0\frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{x^{2}} - \frac{2 \sin^{2}{\left(x \right)}}{x^{3}} = 0
Solve this equation
The roots of this equation
x1=−15.707963267949x_{1} = -15.707963267949
x2=14.0661939128315x_{2} = 14.0661939128315
x3=−61.2447302603744x_{3} = -61.2447302603744
x4=−56.5486677646163x_{4} = -56.5486677646163
x5=−31.4159265358979x_{5} = -31.4159265358979
x6=21.9911485751286x_{6} = 21.9911485751286
x7=−42.3879135681319x_{7} = -42.3879135681319
x8=−21.9911485751286x_{8} = -21.9911485751286
x9=−86.3822220347287x_{9} = -86.3822220347287
x10=28.2743338823081x_{10} = 28.2743338823081
x11=72.2566310325652x_{11} = 72.2566310325652
x12=23.519452498689x_{12} = 23.519452498689
x13=−51.8169824872797x_{13} = -51.8169824872797
x14=−94.2477796076938x_{14} = -94.2477796076938
x15=4.49340945790906x_{15} = 4.49340945790906
x16=87.9645943005142x_{16} = 87.9645943005142
x17=−50.2654824574367x_{17} = -50.2654824574367
x18=−45.5311340139913x_{18} = -45.5311340139913
x19=58.1022547544956x_{19} = 58.1022547544956
x20=−43.9822971502571x_{20} = -43.9822971502571
x21=−97.3893722612836x_{21} = -97.3893722612836
x22=50.2654824574367x_{22} = 50.2654824574367
x23=59.6902604182061x_{23} = 59.6902604182061
x24=−4.49340945790906x_{24} = -4.49340945790906
x25=7.72525183693771x_{25} = 7.72525183693771
x26=−80.0981286289451x_{26} = -80.0981286289451
x27=29.811598790893x_{27} = 29.811598790893
x28=−73.8138806006806x_{28} = -73.8138806006806
x29=70.6716857116195x_{29} = 70.6716857116195
x30=73.8138806006806x_{30} = 73.8138806006806
x31=−14.0661939128315x_{31} = -14.0661939128315
x32=−53.4070751110265x_{32} = -53.4070751110265
x33=45.5311340139913x_{33} = 45.5311340139913
x34=89.5242209304172x_{34} = 89.5242209304172
x35=12.5663706143592x_{35} = 12.5663706143592
x36=−81.6814089933346x_{36} = -81.6814089933346
x37=−64.3871195905574x_{37} = -64.3871195905574
x38=94.2477796076938x_{38} = 94.2477796076938
x39=95.8081387868617x_{39} = 95.8081387868617
x40=81.6814089933346x_{40} = 81.6814089933346
x41=−83.2401924707234x_{41} = -83.2401924707234
x42=−28.2743338823081x_{42} = -28.2743338823081
x43=69.1150383789755x_{43} = 69.1150383789755
x44=−72.2566310325652x_{44} = -72.2566310325652
x45=185.353966561798x_{45} = 185.353966561798
x46=37.6991118430775x_{46} = 37.6991118430775
x47=65.9734457253857x_{47} = 65.9734457253857
x48=−939.336203423348x_{48} = -939.336203423348
x49=−11295.596297452x_{49} = -11295.596297452
x50=3.14159265358979x_{50} = 3.14159265358979
x51=−87.9645943005142x_{51} = -87.9645943005142
x52=47.1238898038469x_{52} = 47.1238898038469
x53=78.5398163397448x_{53} = 78.5398163397448
x54=−6.28318530717959x_{54} = -6.28318530717959
x55=−89.5242209304172x_{55} = -89.5242209304172
x56=15.707963267949x_{56} = 15.707963267949
x57=42.3879135681319x_{57} = 42.3879135681319
x58=−23.519452498689x_{58} = -23.519452498689
x59=−3.14159265358979x_{59} = -3.14159265358979
x60=36.1006222443756x_{60} = 36.1006222443756
x61=56.5486677646163x_{61} = 56.5486677646163
x62=−100.530964914873x_{62} = -100.530964914873
x63=−95.8081387868617x_{63} = -95.8081387868617
x64=64.3871195905574x_{64} = 64.3871195905574
x65=−65.9734457253857x_{65} = -65.9734457253857
x66=43.9822971502571x_{66} = 43.9822971502571
x67=51.8169824872797x_{67} = 51.8169824872797
x68=−58.1022547544956x_{68} = -58.1022547544956
x69=92.6661922776228x_{69} = 92.6661922776228
x70=−54.9596782878889x_{70} = -54.9596782878889
x71=−67.5294347771441x_{71} = -67.5294347771441
x72=−37.6991118430775x_{72} = -37.6991118430775
x73=−59.6902604182061x_{73} = -59.6902604182061
x74=86.3822220347287x_{74} = 86.3822220347287
x75=26.6660542588127x_{75} = 26.6660542588127
x76=−36.1006222443756x_{76} = -36.1006222443756
x77=−75.398223686155x_{77} = -75.398223686155
x78=100.530964914873x_{78} = 100.530964914873
x79=80.0981286289451x_{79} = 80.0981286289451
x80=−29.811598790893x_{80} = -29.811598790893
x81=34.5575191894877x_{81} = 34.5575191894877
x82=6.28318530717959x_{82} = 6.28318530717959
x83=48.6741442319544x_{83} = 48.6741442319544
x84=−39.2444323611642x_{84} = -39.2444323611642
x85=−207.345115136926x_{85} = -207.345115136926
x86=67.5294347771441x_{86} = 67.5294347771441
x87=−7.72525183693771x_{87} = -7.72525183693771
x88=20.3713029592876x_{88} = 20.3713029592876
x89=−9.42477796076938x_{89} = -9.42477796076938
x90=−20.3713029592876x_{90} = -20.3713029592876
x91=−12.5663706143592x_{91} = -12.5663706143592
The values of the extrema at the points:
(-15.707963267948966, 1.51957436358475e-33)

(14.066193912831473, 0.00502871873123234)

(-61.2447302603744, 0.000266530417407147)

(-56.548667764616276, 1.51957436358475e-33)

(-31.41592653589793, 1.51957436358475e-33)

(21.991148575128552, 1.51957436358475e-33)

(-42.38791356813192, 0.000556255443367358)

(-21.991148575128552, 1.51957436358475e-33)

(-86.38222203472871, 0.000133996378076552)

(28.274333882308138, 1.51957436358475e-33)

(72.25663103256524, 7.77037197267108e-33)

(23.519452498689006, 0.00180451785856468)

(-51.81698248727967, 0.000372300864235917)

(-94.2477796076938, 1.32563038769384e-33)

(4.493409457909064, 0.0471904492258113)

(87.96459430051421, 1.51957436358475e-33)

(-50.26548245743669, 1.51957436358475e-33)

(-45.53113401399128, 0.0004821405114931)

(58.10225475449559, 0.000296132041061176)

(-43.982297150257104, 1.51957436358475e-33)

(-97.3893722612836, 4.96414972110828e-33)

(50.26548245743669, 1.51957436358475e-33)

(59.69026041820607, 4.21786179228739e-34)

(-4.493409457909064, 0.0471904492258113)

(7.725251836937707, 0.0164800259929739)

(-80.09812862894512, 0.000155843098300362)

(29.81159879089296, 0.00112393467820302)

(-73.81388060068065, 0.000183503445117105)

(70.6716857116195, 0.000200180676620011)

(73.81388060068065, 0.000183503445117105)

(-14.066193912831473, 0.00502871873123234)

(-53.40707511102649, 7.58434347792408e-34)

(45.53113401399128, 0.0004821405114931)

(89.52422093041719, 0.000124756940214054)

(12.566370614359172, 1.51957436358475e-33)

(-81.68140899333463, 2.30474995774498e-33)

(-64.38711959055742, 0.000241155549725919)

(94.2477796076938, 1.32563038769384e-33)

(95.8081387868617, 0.00010893009510268)

(81.68140899333463, 2.30474995774498e-33)

(-83.2401924707234, 0.000144301609334975)

(-28.274333882308138, 1.51957436358475e-33)

(69.11503837897546, 4.07351440822617e-33)

(-72.25663103256524, 7.77037197267108e-33)

(185.3539665617978, 3.38129490222314e-33)

(37.69911184307752, 1.51957436358475e-33)

(65.97344572538566, 2.21085464398688e-34)

(-939.3362034233481, 1.82874873069053e-33)

(-11295.596297452026, 7.83757431584905e-9)

(3.141592653589793, 1.51957436358475e-33)

(-87.96459430051421, 1.51957436358475e-33)

(47.1238898038469, 1.32563038769384e-33)

(78.53981633974483, 3.90979723557589e-35)

(-6.283185307179586, 1.51957436358475e-33)

(-89.52422093041719, 0.000124756940214054)

(15.707963267948966, 1.51957436358475e-33)

(42.38791356813192, 0.000556255443367358)

(-23.519452498689006, 0.00180451785856468)

(-3.141592653589793, 1.51957436358475e-33)

(36.10062224437561, 0.000766721274909305)

(56.548667764616276, 1.51957436358475e-33)

(-100.53096491487338, 1.51957436358475e-33)

(-95.8081387868617, 0.00010893009510268)

(64.38711959055742, 0.000241155549725919)

(-65.97344572538566, 2.21085464398688e-34)

(43.982297150257104, 1.51957436358475e-33)

(51.81698248727967, 0.000372300864235917)

(-58.10225475449559, 0.000296132041061176)

(92.66619227762284, 0.000116441231903146)

(-54.959678287888934, 0.000330954187793896)

(-67.52943477714412, 0.000219239370163893)

(-37.69911184307752, 1.51957436358475e-33)

(-59.69026041820607, 4.21786179228739e-34)

(86.38222203472871, 0.000133996378076552)

(26.666054258812675, 0.00140433964877555)

(-36.10062224437561, 0.000766721274909305)

(-75.39822368615503, 1.51957436358475e-33)

(100.53096491487338, 1.51957436358475e-33)

(80.09812862894512, 0.000155843098300362)

(-29.81159879089296, 0.00112393467820302)

(34.55751918948773, 4.07351440822617e-33)

(6.283185307179586, 1.51957436358475e-33)

(48.674144231954386, 0.000421910252241397)

(-39.24443236116419, 0.000648876433872722)

(-207.34511513692635, 2.22134595698259e-35)

(67.52943477714412, 0.000219239370163893)

(-7.725251836937707, 0.0164800259929739)

(20.37130295928756, 0.00240390403096148)

(-9.42477796076938, 1.51957436358475e-33)

(-20.37130295928756, 0.00240390403096148)

(-12.566370614359172, 1.51957436358475e-33)


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=−15.707963267949x_{1} = -15.707963267949
x2=−56.5486677646163x_{2} = -56.5486677646163
x3=−31.4159265358979x_{3} = -31.4159265358979
x4=21.9911485751286x_{4} = 21.9911485751286
x5=−21.9911485751286x_{5} = -21.9911485751286
x6=28.2743338823081x_{6} = 28.2743338823081
x7=72.2566310325652x_{7} = 72.2566310325652
x8=−94.2477796076938x_{8} = -94.2477796076938
x9=87.9645943005142x_{9} = 87.9645943005142
x10=−50.2654824574367x_{10} = -50.2654824574367
x11=−43.9822971502571x_{11} = -43.9822971502571
x12=−97.3893722612836x_{12} = -97.3893722612836
x13=50.2654824574367x_{13} = 50.2654824574367
x14=59.6902604182061x_{14} = 59.6902604182061
x15=−53.4070751110265x_{15} = -53.4070751110265
x16=12.5663706143592x_{16} = 12.5663706143592
x17=−81.6814089933346x_{17} = -81.6814089933346
x18=94.2477796076938x_{18} = 94.2477796076938
x19=81.6814089933346x_{19} = 81.6814089933346
x20=−28.2743338823081x_{20} = -28.2743338823081
x21=69.1150383789755x_{21} = 69.1150383789755
x22=−72.2566310325652x_{22} = -72.2566310325652
x23=185.353966561798x_{23} = 185.353966561798
x24=37.6991118430775x_{24} = 37.6991118430775
x25=65.9734457253857x_{25} = 65.9734457253857
x26=−939.336203423348x_{26} = -939.336203423348
x27=3.14159265358979x_{27} = 3.14159265358979
x28=−87.9645943005142x_{28} = -87.9645943005142
x29=47.1238898038469x_{29} = 47.1238898038469
x30=78.5398163397448x_{30} = 78.5398163397448
x31=−6.28318530717959x_{31} = -6.28318530717959
x32=15.707963267949x_{32} = 15.707963267949
x33=−3.14159265358979x_{33} = -3.14159265358979
x34=56.5486677646163x_{34} = 56.5486677646163
x35=−100.530964914873x_{35} = -100.530964914873
x36=−65.9734457253857x_{36} = -65.9734457253857
x37=43.9822971502571x_{37} = 43.9822971502571
x38=−37.6991118430775x_{38} = -37.6991118430775
x39=−59.6902604182061x_{39} = -59.6902604182061
x40=−75.398223686155x_{40} = -75.398223686155
x41=100.530964914873x_{41} = 100.530964914873
x42=34.5575191894877x_{42} = 34.5575191894877
x43=6.28318530717959x_{43} = 6.28318530717959
x44=−207.345115136926x_{44} = -207.345115136926
x45=−9.42477796076938x_{45} = -9.42477796076938
x46=−12.5663706143592x_{46} = -12.5663706143592
Maxima of the function at points:
x46=14.0661939128315x_{46} = 14.0661939128315
x46=−61.2447302603744x_{46} = -61.2447302603744
x46=−42.3879135681319x_{46} = -42.3879135681319
x46=−86.3822220347287x_{46} = -86.3822220347287
x46=23.519452498689x_{46} = 23.519452498689
x46=−51.8169824872797x_{46} = -51.8169824872797
x46=4.49340945790906x_{46} = 4.49340945790906
x46=−45.5311340139913x_{46} = -45.5311340139913
x46=58.1022547544956x_{46} = 58.1022547544956
x46=−4.49340945790906x_{46} = -4.49340945790906
x46=7.72525183693771x_{46} = 7.72525183693771
x46=−80.0981286289451x_{46} = -80.0981286289451
x46=29.811598790893x_{46} = 29.811598790893
x46=−73.8138806006806x_{46} = -73.8138806006806
x46=70.6716857116195x_{46} = 70.6716857116195
x46=73.8138806006806x_{46} = 73.8138806006806
x46=−14.0661939128315x_{46} = -14.0661939128315
x46=45.5311340139913x_{46} = 45.5311340139913
x46=89.5242209304172x_{46} = 89.5242209304172
x46=−64.3871195905574x_{46} = -64.3871195905574
x46=95.8081387868617x_{46} = 95.8081387868617
x46=−83.2401924707234x_{46} = -83.2401924707234
x46=−11295.596297452x_{46} = -11295.596297452
x46=−89.5242209304172x_{46} = -89.5242209304172
x46=42.3879135681319x_{46} = 42.3879135681319
x46=−23.519452498689x_{46} = -23.519452498689
x46=36.1006222443756x_{46} = 36.1006222443756
x46=−95.8081387868617x_{46} = -95.8081387868617
x46=64.3871195905574x_{46} = 64.3871195905574
x46=51.8169824872797x_{46} = 51.8169824872797
x46=−58.1022547544956x_{46} = -58.1022547544956
x46=92.6661922776228x_{46} = 92.6661922776228
x46=−54.9596782878889x_{46} = -54.9596782878889
x46=−67.5294347771441x_{46} = -67.5294347771441
x46=86.3822220347287x_{46} = 86.3822220347287
x46=26.6660542588127x_{46} = 26.6660542588127
x46=−36.1006222443756x_{46} = -36.1006222443756
x46=80.0981286289451x_{46} = 80.0981286289451
x46=−29.811598790893x_{46} = -29.811598790893
x46=48.6741442319544x_{46} = 48.6741442319544
x46=−39.2444323611642x_{46} = -39.2444323611642
x46=67.5294347771441x_{46} = 67.5294347771441
x46=−7.72525183693771x_{46} = -7.72525183693771
x46=20.3713029592876x_{46} = 20.3713029592876
x46=−20.3713029592876x_{46} = -20.3713029592876
Decreasing at intervals
[185.353966561798,∞)\left[185.353966561798, \infty\right)
Increasing at intervals
(−∞,−939.336203423348]\left(-\infty, -939.336203423348\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(−sin⁡2(x)+cos⁡2(x)−4sin⁡(x)cos⁡(x)x+3sin⁡2(x)x2)x2=0\frac{2 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} - \frac{4 \sin{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{3 \sin^{2}{\left(x \right)}}{x^{2}}\right)}{x^{2}} = 0
Solve this equation
The roots of this equation
x1=−82.4547893068791x_{1} = -82.4547893068791
x2=−38.4590122703912x_{2} = -38.4590122703912
x3=24.3049190457346x_{3} = 24.3049190457346
x4=25.8806097494708x_{4} = 25.8806097494708
x5=60.4593229336188x_{5} = 60.4593229336188
x6=−110.732171281162x_{6} = -110.732171281162
x7=−68.3148408970371x_{7} = -68.3148408970371
x8=−16.4352509360813x_{8} = -16.4352509360813
x9=−10.118480698715x_{9} = -10.118480698715
x10=74.5992854099041x_{10} = 74.5992854099041
x11=742.199918425809x_{11} = 742.199918425809
x12=−18.0062837080869x_{12} = -18.0062837080869
x13=54.1742687855514x_{13} = 54.1742687855514
x14=−47.888731676718x_{14} = -47.888731676718
x15=84.025595846992x_{15} = 84.025595846992
x16=66.7440290551475x_{16} = 66.7440290551475
x17=−99.7354646652444x_{17} = -99.7354646652444
x18=99.7354646652444x_{18} = 99.7354646652444
x19=30.5970389725585x_{19} = 30.5970389725585
x20=98.1646611098724x_{20} = 98.1646611098724
x21=−77.7414305824544x_{21} = -77.7414305824544
x22=−3.70722846405825x_{22} = -3.70722846405825
x23=41.6024965392658x_{23} = 41.6024965392658
x24=−1.30308171092781x_{24} = -1.30308171092781
x25=29.0261632399594x_{25} = 29.0261632399594
x26=−98.1646611098724x_{26} = -98.1646611098724
x27=77.7414305824544x_{27} = 77.7414305824544
x28=−85.5968192243094x_{28} = -85.5968192243094
x29=38.4590122703912x_{29} = 38.4590122703912
x30=−19.5858273496712x_{30} = -19.5858273496712
x31=−60.4593229336188x_{31} = -60.4593229336188
x32=18.0062837080869x_{32} = 18.0062837080869
x33=−8.51135078767434x_{33} = -8.51135078767434
x34=76.1706223070459x_{34} = 76.1706223070459
x35=40.0298544804573x_{35} = 40.0298544804573
x36=−76.1706223070459x_{36} = -76.1706223070459
x37=−62.0301381205536x_{37} = -62.0301381205536
x38=32.1709600835167x_{38} = 32.1709600835167
x39=−41.6024965392658x_{39} = -41.6024965392658
x40=91.880790070283x_{40} = 91.880790070283
x41=96.5935408750673x_{41} = 96.5935408750673
x42=−57.3168464266514x_{42} = -57.3168464266514
x43=−93.4515946389277x_{43} = -93.4515946389277
x44=47.888731676718x_{44} = 47.888731676718
x45=16.4352509360813x_{45} = 16.4352509360813
x46=−374.632259996929x_{46} = -374.632259996929
x47=11.6898582204548x_{47} = 11.6898582204548
x48=19.5858273496712x_{48} = 19.5858273496712
x49=55.745088528446x_{49} = 55.745088528446
x50=−102.877368081182x_{50} = -102.877368081182
x51=−11.6898582204548x_{51} = -11.6898582204548
x52=−33.7418214227106x_{52} = -33.7418214227106
x53=−49.4595578500579x_{53} = -49.4595578500579
x54=52.6023942608824x_{54} = 52.6023942608824
x55=−5.28103240630265x_{55} = -5.28103240630265
x56=22.7339953909907x_{56} = 22.7339953909907
x57=−13.2806415733888x_{57} = -13.2806415733888
x58=46.3165498784734x_{58} = 46.3165498784734
x59=−25.8806097494708x_{59} = -25.8806097494708
x60=3.70722846405825x_{60} = 3.70722846405825
x61=1.30308171092781x_{61} = 1.30308171092781
x62=82.4547893068791x_{62} = 82.4547893068791
x63=−91.880790070283x_{63} = -91.880790070283
x64=−27.4515052410928x_{64} = -27.4515052410928
x65=1335.96152783466x_{65} = 1335.96152783466
x66=8.51135078767434x_{66} = 8.51135078767434
x67=−79.3127250697764x_{67} = -79.3127250697764
x68=62.0301381205536x_{68} = 62.0301381205536
x69=88.7388184372412x_{69} = 88.7388184372412
x70=−69.8862806383665x_{70} = -69.8862806383665
x71=69.8862806383665x_{71} = 69.8862806383665
x72=−54.1742687855514x_{72} = -54.1742687855514
x73=−24.3049190457346x_{73} = -24.3049190457346
x74=33.7418214227106x_{74} = 33.7418214227106
x75=68.3148408970371x_{75} = 68.3148408970371
x76=−55.745088528446x_{76} = -55.745088528446
x77=10.118480698715x_{77} = 10.118480698715
x78=85.5968192243094x_{78} = 85.5968192243094
x79=−32.1709600835167x_{79} = -32.1709600835167
x80=−63.6017131240042x_{80} = -63.6017131240042
x81=−90.3096235943467x_{81} = -90.3096235943467
x82=−84.025595846992x_{82} = -84.025595846992
x83=−71.4570911320099x_{83} = -71.4570911320099
x84=44.7457194481397x_{84} = 44.7457194481397
x85=63.6017131240042x_{85} = 63.6017131240042
x86=−35.3151982819233x_{86} = -35.3151982819233
x87=−46.3165498784734x_{87} = -46.3165498784734
x88=−40.0298544804573x_{88} = -40.0298544804573
x89=90.3096235943467x_{89} = 90.3096235943467
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=0x_{1} = 0

lim⁡x→0−(2(−sin⁡2(x)+cos⁡2(x)−4sin⁡(x)cos⁡(x)x+3sin⁡2(x)x2)x2)=−23\lim_{x \to 0^-}\left(\frac{2 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} - \frac{4 \sin{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{3 \sin^{2}{\left(x \right)}}{x^{2}}\right)}{x^{2}}\right) = - \frac{2}{3}
lim⁡x→0+(2(−sin⁡2(x)+cos⁡2(x)−4sin⁡(x)cos⁡(x)x+3sin⁡2(x)x2)x2)=−23\lim_{x \to 0^+}\left(\frac{2 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} - \frac{4 \sin{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{3 \sin^{2}{\left(x \right)}}{x^{2}}\right)}{x^{2}}\right) = - \frac{2}{3}
- limits are equal, then skip the corresponding point

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
[99.7354646652444,∞)\left[99.7354646652444, \infty\right)
Convex at the intervals
(−∞,−374.632259996929]\left(-\infty, -374.632259996929\right]
Vertical asymptotes
Have:
x1=0x_{1} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(sin⁡2(x)x2)=0\lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(x \right)}}{x^{2}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(sin⁡2(x)x2)=0\lim_{x \to \infty}\left(\frac{\sin^{2}{\left(x \right)}}{x^{2}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x)^2/x^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡2(x)xx2)=0\lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(x \right)}}{x x^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡2(x)xx2)=0\lim_{x \to \infty}\left(\frac{\sin^{2}{\left(x \right)}}{x x^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
sin⁡2(x)x2=sin⁡2(x)x2\frac{\sin^{2}{\left(x \right)}}{x^{2}} = \frac{\sin^{2}{\left(x \right)}}{x^{2}}
- Yes
sin⁡2(x)x2=−sin⁡2(x)x2\frac{\sin^{2}{\left(x \right)}}{x^{2}} = - \frac{\sin^{2}{\left(x \right)}}{x^{2}}
- No
so, the function
is
even
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