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

You entered:

sin(x)^2/x

What you mean?

sin(x)^2/x
 
sin(x)^(2/x)
 
sin(x)^(2/x)

Graphing y = sin(x)^2/x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          2   
       sin (x)
f(x) = -------
          x
f(x)=sin⁡2(x)xf{\left(x \right)} = \frac{\sin^{2}{\left(x \right)}}{x} 
f = sin(x)^2/x
The graph of the function
05-60-55-50-45-40-35-30-25-20-15-10-5102-2
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)x=0\frac{\sin^{2}{\left(x \right)}}{x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=πx_{1} = \pi
Numerical solution
x1=−62.8318528306552x_{1} = -62.8318528306552
x2=−12.566370348569x_{2} = -12.566370348569
x3=−3.14159223666992x_{3} = -3.14159223666992
x4=−69.11503862209x_{4} = -69.11503862209
x5=87.9645943357262x_{5} = 87.9645943357262
x6=−47.1238900446296x_{6} = -47.1238900446296
x7=−37.6991118770833x_{7} = -37.6991118770833
x8=−6.28318512582806x_{8} = -6.28318512582806
x9=21.9911485851862x_{9} = 21.9911485851862
x10=−75.3982238609893x_{10} = -75.3982238609893
x11=−84.8230014048764x_{11} = -84.8230014048764
x12=−62.8318532480223x_{12} = -62.8318532480223
x13=56.5486676080768x_{13} = 56.5486676080768
x14=97.3893726684677x_{14} = 97.3893726684677
x15=−9.42477811967166x_{15} = -9.42477811967166
x16=−47.1238900763012x_{16} = -47.1238900763012
x17=−91.1061871670528x_{17} = -91.1061871670528
x18=−53.407075282232x_{18} = -53.407075282232
x19=−43.9822970046757x_{19} = -43.9822970046757
x20=6.2831852840651x_{20} = 6.2831852840651
x21=59.6902605963318x_{21} = 59.6902605963318
x22=15.7079634360324x_{22} = 15.7079634360324
x23=9.4247781890235x_{23} = 9.4247781890235
x24=34.5575190286501x_{24} = 34.5575190286501
x25=−53.4070752240643x_{25} = -53.4070752240643
x26=−78.5398160933203x_{26} = -78.5398160933203
x27=84.823001407093x_{27} = 84.823001407093
x28=−40.8407042544435x_{28} = -40.8407042544435
x29=50.2654780610567x_{29} = 50.2654780610567
x30=−40.8407046746634x_{30} = -40.8407046746634
x31=251.327413254646x_{31} = 251.327413254646
x32=−18.8495560894836x_{32} = -18.8495560894836
x33=−56.5486675156139x_{33} = -56.5486675156139
x34=91.1061871519824x_{34} = 91.1061871519824
x35=47.1238895782348x_{35} = 47.1238895782348
x36=−72.2566308732795x_{36} = -72.2566308732795
x37=50.2654824463419x_{37} = 50.2654824463419
x38=−3.14159284098784x_{38} = -3.14159284098784
x39=−69.115038626205x_{39} = -69.115038626205
x40=75.3982239358177x_{40} = 75.3982239358177
x41=91.1061867261437x_{41} = 91.1061867261437
x42=−56.548667490356x_{42} = -56.548667490356
x43=28.2743338651783x_{43} = 28.2743338651783
x44=84.8230010502372x_{44} = 84.8230010502372
x45=94.2477796093524x_{45} = 94.2477796093524
x46=−97.3893724395293x_{46} = -97.3893724395293
x47=−28.2743337141732x_{47} = -28.2743337141732
x48=−103.672558052222x_{48} = -103.672558052222
x49=40.8407042511736x_{49} = 40.8407042511736
x50=−25.1327414647216x_{50} = -25.1327414647216
x51=−15.7079632964119x_{51} = -15.7079632964119
x52=65.973445752872x_{52} = 65.973445752872
x53=−50.2654822940439x_{53} = -50.2654822940439
x54=78.5398161871203x_{54} = 78.5398161871203
x55=40.8407038529316x_{55} = 40.8407038529316
x56=−40.8407066842267x_{56} = -40.8407066842267
x57=31.4159267796903x_{57} = 31.4159267796903
x58=−81.6814090379518x_{58} = -81.6814090379518
x59=75.398224139694x_{59} = 75.398224139694
x60=31.4159270278136x_{60} = 31.4159270278136
x61=12.5663704464902x_{61} = 12.5663704464902
x62=−94.2477794523719x_{62} = -94.2477794523719
x63=−21.9911485864387x_{63} = -21.9911485864387
x64=37.6991120171585x_{64} = 37.6991120171585
x65=−87.9645943586888x_{65} = -87.9645943586888
x66=9.4247781842281x_{66} = 9.4247781842281
x67=97.3893725124567x_{67} = 97.3893725124567
x68=−100.530964670637x_{68} = -100.530964670637
x69=53.4070753585761x_{69} = 53.4070753585761
x70=18.8495556684588x_{70} = 18.8495556684588
x71=3.14159223734373x_{71} = 3.14159223734373
x72=−43.9822971745494x_{72} = -43.9822971745494
x73=81.6814091750724x_{73} = 81.6814091750724
x74=−31.415926702919x_{74} = -31.415926702919
x75=69.1150381529445x_{75} = 69.1150381529445
x76=53.4070756006464x_{76} = 53.4070756006464
x77=−59.6902604575823x_{77} = -59.6902604575823
x78=43.9822971694085x_{78} = 43.9822971694085
x79=25.132740997054x_{79} = 25.132740997054
x80=−65.9734457649654x_{80} = -65.9734457649654
x81=18.8495552650209x_{81} = 18.8495552650209
x82=−34.5575189366009x_{82} = -34.5575189366009
x83=47.1238900070552x_{83} = 47.1238900070552
x84=−84.8230018182772x_{84} = -84.8230018182772
x85=109.955741076952x_{85} = 109.955741076952
x86=−69.1150380617605x_{86} = -69.1150380617605
x87=72.2566310277183x_{87} = 72.2566310277183
x88=62.8318528295667x_{88} = 62.8318528295667
x89=−18.8495556680998x_{89} = -18.8495556680998
x90=100.530964766066x_{90} = 100.530964766066
x91=62.8318524328373x_{91} = 62.8318524328373
x92=−91.1061871987871x_{92} = -91.1061871987871
x93=25.13274142749x_{93} = 25.13274142749
x94=69.1150385805253x_{94} = 69.1150385805253
x95=−25.1327414848872x_{95} = -25.1327414848872
x96=3.14159272740328x_{96} = 3.14159272740328
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.
sin⁡2(0)0\frac{\sin^{2}{\left(0 \right)}}{0}
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)x−sin⁡2(x)x2=0\frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{\sin^{2}{\left(x \right)}}{x^{2}} = 0
Solve this equation
The roots of this equation
x1=−61.2528940466862x_{1} = -61.2528940466862
x2=−86.3880101981266x_{2} = -86.3880101981266
x3=53.4070751110265x_{3} = 53.4070751110265
x4=72.2566310325652x_{4} = 72.2566310325652
x5=−9.42477796076938x_{5} = -9.42477796076938
x6=−59.6902604182061x_{6} = -59.6902604182061
x7=42.3997088362447x_{7} = 42.3997088362447
x8=20.3958423573092x_{8} = 20.3958423573092
x9=14.1017251335659x_{9} = 14.1017251335659
x10=−95.8133575027966x_{10} = -95.8133575027966
x11=−114.663771308444x_{11} = -114.663771308444
x12=80.1043708909521x_{12} = 80.1043708909521
x13=12.5663706143592x_{13} = 12.5663706143592
x14=−14.1017251335659x_{14} = -14.1017251335659
x15=95.8133575027966x_{15} = 95.8133575027966
x16=−64.3948849627586x_{16} = -64.3948849627586
x17=37.6991118430775x_{17} = 37.6991118430775
x18=−23.5407082923052x_{18} = -23.5407082923052
x19=64.3948849627586x_{19} = 64.3948849627586
x20=−42.3997088362447x_{20} = -42.3997088362447
x21=−20.3958423573092x_{21} = -20.3958423573092
x22=58.1108600600615x_{22} = 58.1108600600615
x23=7.78988375114457x_{23} = 7.78988375114457
x24=67.5368388204916x_{24} = 67.5368388204916
x25=15.707963267949x_{25} = 15.707963267949
x26=92.6715879363332x_{26} = 92.6715879363332
x27=−97.3893722612836x_{27} = -97.3893722612836
x28=−21.9911485751286x_{28} = -21.9911485751286
x29=6.28318530717959x_{29} = 6.28318530717959
x30=−83.2461991121237x_{30} = -83.2461991121237
x31=−6.28318530717959x_{31} = -6.28318530717959
x32=34.5575191894877x_{32} = 34.5575191894877
x33=−51.8266315338985x_{33} = -51.8266315338985
x34=−2678.20755049327x_{34} = -2678.20755049327
x35=−125.663706143592x_{35} = -125.663706143592
x36=−94.2477796076938x_{36} = -94.2477796076938
x37=−67.5368388204916x_{37} = -67.5368388204916
x38=48.6844162648433x_{38} = 48.6844162648433
x39=28.2743338823081x_{39} = 28.2743338823081
x40=−31.4159265358979x_{40} = -31.4159265358979
x41=59.6902604182061x_{41} = 59.6902604182061
x42=−75.398223686155x_{42} = -75.398223686155
x43=−1.16556118520721x_{43} = -1.16556118520721
x44=−81.6814089933346x_{44} = -81.6814089933346
x45=−37.6991118430775x_{45} = -37.6991118430775
x46=−1740.44233008875x_{46} = -1740.44233008875
x47=197.920337176157x_{47} = 197.920337176157
x48=−80.1043708909521x_{48} = -80.1043708909521
x49=−17.2497818346079x_{49} = -17.2497818346079
x50=−7.78988375114457x_{50} = -7.78988375114457
x51=−50.2654824574367x_{51} = -50.2654824574367
x52=94.2477796076938x_{52} = 94.2477796076938
x53=−29.8283692130955x_{53} = -29.8283692130955
x54=36.1144715353049x_{54} = 36.1144715353049
x55=56.5486677646163x_{55} = 56.5486677646163
x56=−45.5421150692309x_{56} = -45.5421150692309
x57=87.9645943005142x_{57} = 87.9645943005142
x58=65.9734457253857x_{58} = 65.9734457253857
x59=70.6787605627689x_{59} = 70.6787605627689
x60=73.8206542907788x_{60} = 73.8206542907788
x61=−53.4070751110265x_{61} = -53.4070751110265
x62=43.9822971502571x_{62} = 43.9822971502571
x63=86.3880101981266x_{63} = 86.3880101981266
x64=−15.707963267949x_{64} = -15.707963267949
x65=−36.1144715353049x_{65} = -36.1144715353049
x66=−62.8318530717959x_{66} = -62.8318530717959
x67=−18.8495559215388x_{67} = -18.8495559215388
x68=23.5407082923052x_{68} = 23.5407082923052
x69=−89.5298059530594x_{69} = -89.5298059530594
x70=45.5421150692309x_{70} = 45.5421150692309
x71=−72.2566310325652x_{71} = -72.2566310325652
x72=4.60421677720058x_{72} = 4.60421677720058
x73=89.5298059530594x_{73} = 89.5298059530594
x74=78.5398163397448x_{74} = 78.5398163397448
x75=−73.8206542907788x_{75} = -73.8206542907788
x76=29.8283692130955x_{76} = 29.8283692130955
x77=−87.9645943005142x_{77} = -87.9645943005142
x78=−43.9822971502571x_{78} = -43.9822971502571
x79=26.6848024909251x_{79} = 26.6848024909251
x80=21.9911485751286x_{80} = 21.9911485751286
x81=−58.1108600600615x_{81} = -58.1108600600615
x82=−10.9499436485412x_{82} = -10.9499436485412
x83=51.8266315338985x_{83} = 51.8266315338985
x84=81.6814089933346x_{84} = 81.6814089933346
x85=−28.2743338823081x_{85} = -28.2743338823081
x86=−65.9734457253857x_{86} = -65.9734457253857
x87=50.2654824574367x_{87} = 50.2654824574367
x88=100.530964914873x_{88} = 100.530964914873
x89=−39.2571723324086x_{89} = -39.2571723324086
The values of the extrema at the points:
(-61.2528940466862, -0.0163246714689743)

(-86.3880101981266, -0.0115752926793239)

(53.4070751110265, 4.05057601793315e-32)

(72.2566310325652, 5.6146090061508e-31)

(-9.42477796076938, -1.43216509716637e-32)

(-59.6902604182061, -2.51765268789636e-32)

(42.3997088362447, 0.0235817882463307)

(20.3958423573092, 0.0490001524829528)

(14.1017251335659, 0.0708242711210408)

(-95.8133575027966, -0.0104366739072752)

(-114.663771308444, -0.00872098461732392)

(80.1043708909521, 0.0124832269403218)

(12.5663706143592, 1.90955346288849e-32)

(-14.1017251335659, -0.0708242711210408)

(95.8133575027966, 0.0104366739072752)

(-64.3948849627586, -0.0155282475514317)

(37.6991118430775, 5.72866038866547e-32)

(-23.5407082923052, -0.0424604502887016)

(64.3948849627586, 0.0155282475514317)

(-42.3997088362447, -0.0235817882463307)

(-20.3958423573092, -0.0490001524829528)

(58.1108600600615, 0.0172072134440586)

(7.78988375114457, 0.127844922574794)

(67.5368388204916, 0.0148059223769658)

(15.707963267949, 2.38694182861061e-32)

(92.6715879363332, 0.0107904797231539)

(-97.3893722612836, -4.83455425149761e-31)

(-21.9911485751286, -3.34171856005486e-32)

(6.28318530717959, 9.54776731444245e-33)

(-83.2461991121237, -0.0120121271188891)

(-6.28318530717959, -9.54776731444245e-33)

(34.5575191894877, 1.40770552330931e-31)

(-51.8266315338985, -0.0192933035363155)

(-2678.20755049327, -0.000373384043728018)

(-125.663706143592, -1.90955346288849e-31)

(-94.2477796076938, -1.24937720620631e-31)

(-67.5368388204916, -0.0148059223769658)

(48.6844162648433, 0.0205382874085413)

(28.2743338823081, 4.2964952914991e-32)

(-31.4159265358979, -4.77388365722123e-32)

(59.6902604182061, 2.51765268789636e-32)

(-75.398223686155, -1.14573207773309e-31)

(-1.16556118520721, -0.724611353776708)

(-81.6814089933346, -1.88255223925938e-31)

(-37.6991118430775, -5.72866038866547e-32)

(-1740.44233008875, -2.11977620970517e-30)

(197.920337176157, 4.37573096585357e-32)

(-80.1043708909521, -0.0124832269403218)

(-17.2497818346079, -0.0579230818110724)

(-7.78988375114457, -0.127844922574794)

(-50.2654824574367, -7.63821385155396e-32)

(94.2477796076938, 1.24937720620631e-31)

(-29.8283692130955, -0.0335157141235985)

(36.1144715353049, 0.0276844243853039)

(56.5486677646163, 8.59299058299821e-32)

(-45.5421150692309, -0.021955051448177)

(87.9645943005142, 1.33668742402194e-31)

(65.9734457253857, 1.45857698861786e-32)

(70.6787605627689, 0.0141478139878745)

(73.8206542907788, 0.0135457228854227)

(-53.4070751110265, -4.05057601793315e-32)

(43.9822971502571, 6.68343712010972e-32)

(86.3880101981266, 0.0115752926793239)

(-15.707963267949, -2.38694182861061e-32)

(-36.1144715353049, -0.0276844243853039)

(-62.8318530717959, -9.54776731444245e-32)

(-18.8495559215388, -2.86433019433273e-32)

(23.5407082923052, 0.0424604502887016)

(-89.5298059530594, -0.0111691162634939)

(45.5421150692309, 0.021955051448177)

(-72.2566310325652, -5.6146090061508e-31)

(4.60421677720058, 0.214660688386019)

(89.5298059530594, 0.0111691162634939)

(78.5398163397448, 3.07074756807772e-33)

(-73.8206542907788, -0.0135457228854227)

(29.8283692130955, 0.0335157141235985)

(-87.9645943005142, -1.33668742402194e-31)

(-43.9822971502571, -6.68343712010972e-32)

(26.6848024909251, 0.0374613617155508)

(21.9911485751286, 3.34171856005486e-32)

(-58.1108600600615, -0.0172072134440586)

(-10.9499436485412, -0.0911346506917966)

(51.8266315338985, 0.0192933035363155)

(81.6814089933346, 1.88255223925938e-31)

(-28.2743338823081, -4.2964952914991e-32)

(-65.9734457253857, -1.45857698861786e-32)

(50.2654824574367, 7.63821385155396e-32)

(100.530964914873, 1.52764277031079e-31)

(-39.2571723324086, -0.0254689206534694)


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=−61.2528940466862x_{1} = -61.2528940466862
x2=−86.3880101981266x_{2} = -86.3880101981266
x3=53.4070751110265x_{3} = 53.4070751110265
x4=72.2566310325652x_{4} = 72.2566310325652
x5=−95.8133575027966x_{5} = -95.8133575027966
x6=−114.663771308444x_{6} = -114.663771308444
x7=12.5663706143592x_{7} = 12.5663706143592
x8=−14.1017251335659x_{8} = -14.1017251335659
x9=−64.3948849627586x_{9} = -64.3948849627586
x10=37.6991118430775x_{10} = 37.6991118430775
x11=−23.5407082923052x_{11} = -23.5407082923052
x12=−42.3997088362447x_{12} = -42.3997088362447
x13=−20.3958423573092x_{13} = -20.3958423573092
x14=15.707963267949x_{14} = 15.707963267949
x15=6.28318530717959x_{15} = 6.28318530717959
x16=−83.2461991121237x_{16} = -83.2461991121237
x17=34.5575191894877x_{17} = 34.5575191894877
x18=−51.8266315338985x_{18} = -51.8266315338985
x19=−2678.20755049327x_{19} = -2678.20755049327
x20=−67.5368388204916x_{20} = -67.5368388204916
x21=28.2743338823081x_{21} = 28.2743338823081
x22=59.6902604182061x_{22} = 59.6902604182061
x23=−1.16556118520721x_{23} = -1.16556118520721
x24=197.920337176157x_{24} = 197.920337176157
x25=−80.1043708909521x_{25} = -80.1043708909521
x26=−17.2497818346079x_{26} = -17.2497818346079
x27=−7.78988375114457x_{27} = -7.78988375114457
x28=94.2477796076938x_{28} = 94.2477796076938
x29=−29.8283692130955x_{29} = -29.8283692130955
x30=56.5486677646163x_{30} = 56.5486677646163
x31=−45.5421150692309x_{31} = -45.5421150692309
x32=87.9645943005142x_{32} = 87.9645943005142
x33=65.9734457253857x_{33} = 65.9734457253857
x34=43.9822971502571x_{34} = 43.9822971502571
x35=−36.1144715353049x_{35} = -36.1144715353049
x36=−89.5298059530594x_{36} = -89.5298059530594
x37=78.5398163397448x_{37} = 78.5398163397448
x38=−73.8206542907788x_{38} = -73.8206542907788
x39=21.9911485751286x_{39} = 21.9911485751286
x40=−58.1108600600615x_{40} = -58.1108600600615
x41=−10.9499436485412x_{41} = -10.9499436485412
x42=81.6814089933346x_{42} = 81.6814089933346
x43=50.2654824574367x_{43} = 50.2654824574367
x44=100.530964914873x_{44} = 100.530964914873
x45=−39.2571723324086x_{45} = -39.2571723324086
Maxima of the function at points:
x45=−9.42477796076938x_{45} = -9.42477796076938
x45=−59.6902604182061x_{45} = -59.6902604182061
x45=42.3997088362447x_{45} = 42.3997088362447
x45=20.3958423573092x_{45} = 20.3958423573092
x45=14.1017251335659x_{45} = 14.1017251335659
x45=80.1043708909521x_{45} = 80.1043708909521
x45=95.8133575027966x_{45} = 95.8133575027966
x45=64.3948849627586x_{45} = 64.3948849627586
x45=58.1108600600615x_{45} = 58.1108600600615
x45=7.78988375114457x_{45} = 7.78988375114457
x45=67.5368388204916x_{45} = 67.5368388204916
x45=92.6715879363332x_{45} = 92.6715879363332
x45=−97.3893722612836x_{45} = -97.3893722612836
x45=−21.9911485751286x_{45} = -21.9911485751286
x45=−6.28318530717959x_{45} = -6.28318530717959
x45=−125.663706143592x_{45} = -125.663706143592
x45=−94.2477796076938x_{45} = -94.2477796076938
x45=48.6844162648433x_{45} = 48.6844162648433
x45=−31.4159265358979x_{45} = -31.4159265358979
x45=−75.398223686155x_{45} = -75.398223686155
x45=−81.6814089933346x_{45} = -81.6814089933346
x45=−37.6991118430775x_{45} = -37.6991118430775
x45=−1740.44233008875x_{45} = -1740.44233008875
x45=−50.2654824574367x_{45} = -50.2654824574367
x45=36.1144715353049x_{45} = 36.1144715353049
x45=70.6787605627689x_{45} = 70.6787605627689
x45=73.8206542907788x_{45} = 73.8206542907788
x45=−53.4070751110265x_{45} = -53.4070751110265
x45=86.3880101981266x_{45} = 86.3880101981266
x45=−15.707963267949x_{45} = -15.707963267949
x45=−62.8318530717959x_{45} = -62.8318530717959
x45=−18.8495559215388x_{45} = -18.8495559215388
x45=23.5407082923052x_{45} = 23.5407082923052
x45=45.5421150692309x_{45} = 45.5421150692309
x45=−72.2566310325652x_{45} = -72.2566310325652
x45=4.60421677720058x_{45} = 4.60421677720058
x45=89.5298059530594x_{45} = 89.5298059530594
x45=29.8283692130955x_{45} = 29.8283692130955
x45=−87.9645943005142x_{45} = -87.9645943005142
x45=−43.9822971502571x_{45} = -43.9822971502571
x45=26.6848024909251x_{45} = 26.6848024909251
x45=51.8266315338985x_{45} = 51.8266315338985
x45=−28.2743338823081x_{45} = -28.2743338823081
x45=−65.9734457253857x_{45} = -65.9734457253857
Decreasing at intervals
[197.920337176157,∞)\left[197.920337176157, \infty\right)
Increasing at intervals
(−∞,−2678.20755049327]\left(-\infty, -2678.20755049327\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)−2sin⁡(x)cos⁡(x)x+sin⁡2(x)x2)x=0\frac{2 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} - \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right)}{x} = 0
Solve this equation
The roots of this equation
x1=−65.1803173890255x_{1} = -65.1803173890255
x2=−90.3152218631357x_{2} = -90.3152218631357
x3=−77.7479456026924x_{3} = -77.7479456026924
x4=38.4716808813945x_{4} = 38.4716808813945
x5=−5.3960161178562x_{5} = -5.3960161178562
x6=24.3263609848619x_{6} = 24.3263609848619
x7=−16.4638956000575x_{7} = -16.4638956000575
x8=−84.0316177976063x_{8} = -84.0316177976063
x9=−62.0383300698268x_{9} = -62.0383300698268
x10=−46.3275816132535x_{10} = -46.3275816132535
x11=−49.4698742897642x_{11} = -49.4698742897642
x12=−54.1833299748694x_{12} = -54.1833299748694
x13=−115.449180298571x_{13} = -115.449180298571
x14=60.4674576690199x_{14} = 60.4674576690199
x15=−10.1633207350938x_{15} = -10.1633207350938
x16=32.1860288367118x_{16} = 32.1860288367118
x17=62.0383300698268x_{17} = 62.0383300698268
x18=−35.328963068332x_{18} = -35.328963068332
x19=−32.1860288367118x_{19} = -32.1860288367118
x20=−99.7405285299269x_{20} = -99.7405285299269
x21=−533.284414481252x_{21} = -533.284414481252
x22=22.7550493438209x_{22} = 22.7550493438209
x23=40.0426624361667x_{23} = 40.0426624361667
x24=16.4638956000575x_{24} = 16.4638956000575
x25=25.8992016896926x_{25} = 25.8992016896926
x26=−33.7570877695276x_{26} = -33.7570877695276
x27=−63.609452231378x_{27} = -63.609452231378
x28=55.7542207657436x_{28} = 55.7542207657436
x29=−55.7542207657436x_{29} = -55.7542207657436
x30=118.590888671409x_{30} = 118.590888671409
x31=91.8861731050591x_{31} = 91.8861731050591
x32=−68.3222681391823x_{32} = -68.3222681391823
x33=−71.4641871782887x_{33} = -71.4641871782887
x34=−13.3155935768387x_{34} = -13.3155935768387
x35=193.990762262228x_{35} = 193.990762262228
x36=30.6139262842768x_{36} = 30.6139262842768
x37=−19.610096072963x_{37} = -19.610096072963
x38=−25.8992016896926x_{38} = -25.8992016896926
x39=−60.4674576690199x_{39} = -60.4674576690199
x40=96.598771164673x_{40} = 96.598771164673
x41=−11.7365039593478x_{41} = -11.7365039593478
x42=66.7514092373268x_{42} = 66.7514092373268
x43=77.7479456026924x_{43} = 77.7479456026924
x44=52.612082538167x_{44} = 52.612082538167
x45=−3.81153864777937x_{45} = -3.81153864777937
x46=90.3152218631357x_{46} = 90.3152218631357
x47=−24.3263609848619x_{47} = -24.3263609848619
x48=76.1771010903408x_{48} = 76.1771010903408
x49=−91.8861731050591x_{49} = -91.8861731050591
x50=19.610096072963x_{50} = 19.610096072963
x51=54.1833299748694x_{51} = 54.1833299748694
x52=−41.6142307464496x_{52} = -41.6142307464496
x53=10.1633207350938x_{53} = 10.1633207350938
x54=−47.8989575337176x_{54} = -47.8989575337176
x55=−40.0426624361667x_{55} = -40.0426624361667
x56=47.8989575337176x_{56} = 47.8989575337176
x57=84.0316177976063x_{57} = 84.0316177976063
x58=−93.4570026652504x_{58} = -93.4570026652504
x59=−82.4607802854273x_{59} = -82.4607802854273
x60=68.3222681391823x_{60} = 68.3222681391823
x61=85.6025928495405x_{61} = 85.6025928495405
x62=−27.4703989990655x_{62} = -27.4703989990655
x63=74.6060785402851x_{63} = 74.6060785402851
x64=−69.8933337165055x_{64} = -69.8933337165055
x65=−38.4716808813945x_{65} = -38.4716808813945
x66=11.7365039593478x_{66} = 11.7365039593478
x67=−18.0356521422536x_{67} = -18.0356521422536
x68=63.609452231378x_{68} = 63.609452231378
x69=99.7405285299269x_{69} = 99.7405285299269
x70=44.7566476624499x_{70} = 44.7566476624499
x71=8.57755878460975x_{71} = 8.57755878460975
x72=33.7570877695276x_{72} = 33.7570877695276
x73=−57.3254194271399x_{73} = -57.3254194271399
x74=−3128.24072495224x_{74} = -3128.24072495224
x75=18.0356521422536x_{75} = 18.0356521422536
x76=3.81153864777937x_{76} = 3.81153864777937
x77=−98.1697030564332x_{77} = -98.1697030564332
x78=82.4607802854273x_{78} = 82.4607802854273
x79=46.3275816132535x_{79} = 46.3275816132535
x80=−410.762020724136x_{80} = -410.762020724136
x81=69.8933337165055x_{81} = 69.8933337165055
x82=−85.6025928495405x_{82} = -85.6025928495405
x83=−79.318950408493x_{83} = -79.318950408493
x84=−2.04278694273841x_{84} = -2.04278694273841
x85=71.4641871782887x_{85} = 71.4641871782887
x86=−76.1771010903408x_{86} = -76.1771010903408
x87=88.7443899294668x_{87} = 88.7443899294668
x88=98.1697030564332x_{88} = 98.1697030564332
x89=41.6142307464496x_{89} = 41.6142307464496
x90=2.04278694273841x_{90} = 2.04278694273841
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)−2sin⁡(x)cos⁡(x)x+sin⁡2(x)x2)x)=0\lim_{x \to 0^-}\left(\frac{2 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} - \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right)}{x}\right) = 0
Let's take the limit
lim⁡x→0+(2(−sin⁡2(x)+cos⁡2(x)−2sin⁡(x)cos⁡(x)x+sin⁡2(x)x2)x)=0\lim_{x \to 0^+}\left(\frac{2 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} - \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right)}{x}\right) = 0
Let's take the limit
- 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
[193.990762262228,∞)\left[193.990762262228, \infty\right)
Convex at the intervals
(−∞,−3128.24072495224]\left(-\infty, -3128.24072495224\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)x)=0\lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(sin⁡2(x)x)=0\lim_{x \to \infty}\left(\frac{\sin^{2}{\left(x \right)}}{x}\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, divided by x 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,
inclined coincides with the horizontal asymptote on the right
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,
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)x=−sin⁡2(x)x\frac{\sin^{2}{\left(x \right)}}{x} = - \frac{\sin^{2}{\left(x \right)}}{x}
- No
sin⁡2(x)x=sin⁡2(x)x\frac{\sin^{2}{\left(x \right)}}{x} = \frac{\sin^{2}{\left(x \right)}}{x}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = sin(x)^2/x
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