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

Graphing y = sin(2*x)/x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       sin(2*x)
f(x) = --------
          x
f(x)=sin⁡(2x)xf{\left(x \right)} = \frac{\sin{\left(2 x \right)}}{x} 
f = sin(2*x)/x
The graph of the function
02468-8-6-4-2-10102.5-2.5
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⁡(2x)x=0\frac{\sin{\left(2 x \right)}}{x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π2x_{1} = \frac{\pi}{2}
Numerical solution
x1=12.5663706143592x_{1} = 12.5663706143592
x2=78.5398163397448x_{2} = 78.5398163397448
x3=−65.9734457253857x_{3} = -65.9734457253857
x4=−15.707963267949x_{4} = -15.707963267949
x5=86.3937979737193x_{5} = 86.3937979737193
x6=153.9380400259x_{6} = 153.9380400259
x7=50.2654824574367x_{7} = 50.2654824574367
x8=81.6814089933346x_{8} = 81.6814089933346
x9=−64.4026493985908x_{9} = -64.4026493985908
x10=42.4115008234622x_{10} = 42.4115008234622
x11=73.8274273593601x_{11} = 73.8274273593601
x12=45.553093477052x_{12} = 45.553093477052
x13=89.5353906273091x_{13} = 89.5353906273091
x14=−75.398223686155x_{14} = -75.398223686155
x15=−1.5707963267949x_{15} = -1.5707963267949
x16=−58.1194640914112x_{16} = -58.1194640914112
x17=56.5486677646163x_{17} = 56.5486677646163
x18=−61.261056745001x_{18} = -61.261056745001
x19=−51.8362787842316x_{19} = -51.8362787842316
x20=15.707963267949x_{20} = 15.707963267949
x21=7.85398163397448x_{21} = 7.85398163397448
x22=−86.3937979737193x_{22} = -86.3937979737193
x23=58.1194640914112x_{23} = 58.1194640914112
x24=23.5619449019235x_{24} = 23.5619449019235
x25=−67.5442420521806x_{25} = -67.5442420521806
x26=−59.6902604182061x_{26} = -59.6902604182061
x27=21.9911485751286x_{27} = 21.9911485751286
x28=6.28318530717959x_{28} = 6.28318530717959
x29=−271.747764535517x_{29} = -271.747764535517
x30=−87.9645943005142x_{30} = -87.9645943005142
x31=−28.2743338823081x_{31} = -28.2743338823081
x32=−9.42477796076938x_{32} = -9.42477796076938
x33=−95.8185759344887x_{33} = -95.8185759344887
x34=59.6902604182061x_{34} = 59.6902604182061
x35=29.845130209103x_{35} = 29.845130209103
x36=28.2743338823081x_{36} = 28.2743338823081
x37=80.1106126665397x_{37} = 80.1106126665397
x38=94.2477796076938x_{38} = 94.2477796076938
x39=−72.2566310325652x_{39} = -72.2566310325652
x40=−80.1106126665397x_{40} = -80.1106126665397
x41=51.8362787842316x_{41} = 51.8362787842316
x42=−29.845130209103x_{42} = -29.845130209103
x43=−97.3893722612836x_{43} = -97.3893722612836
x44=37.6991118430775x_{44} = 37.6991118430775
x45=−370.707933123596x_{45} = -370.707933123596
x46=−20.4203522483337x_{46} = -20.4203522483337
x47=−50.2654824574367x_{47} = -50.2654824574367
x48=−94.2477796076938x_{48} = -94.2477796076938
x49=−1668.18569905618x_{49} = -1668.18569905618
x50=−17.2787595947439x_{50} = -17.2787595947439
x51=20.4203522483337x_{51} = 20.4203522483337
x52=67.5442420521806x_{52} = 67.5442420521806
x53=14.1371669411541x_{53} = 14.1371669411541
x54=4.71238898038469x_{54} = 4.71238898038469
x55=−237.190245346029x_{55} = -237.190245346029
x56=−37.6991118430775x_{56} = -37.6991118430775
x57=70.6858347057703x_{57} = 70.6858347057703
x58=−23.5619449019235x_{58} = -23.5619449019235
x59=−83.2522053201295x_{59} = -83.2522053201295
x60=−36.1283155162826x_{60} = -36.1283155162826
x61=1.5707963267949x_{61} = 1.5707963267949
x62=36.1283155162826x_{62} = 36.1283155162826
x63=−81.6814089933346x_{63} = -81.6814089933346
x64=43.9822971502571x_{64} = 43.9822971502571
x65=95.8185759344887x_{65} = 95.8185759344887
x66=−14.1371669411541x_{66} = -14.1371669411541
x67=−31.4159265358979x_{67} = -31.4159265358979
x68=−21.9911485751286x_{68} = -21.9911485751286
x69=−39.2699081698724x_{69} = -39.2699081698724
x70=26.7035375555132x_{70} = 26.7035375555132
x71=−89.5353906273091x_{71} = -89.5353906273091
x72=−317.300858012569x_{72} = -317.300858012569
x73=100.530964914873x_{73} = 100.530964914873
x74=34.5575191894877x_{74} = 34.5575191894877
x75=48.6946861306418x_{75} = 48.6946861306418
x76=65.9734457253857x_{76} = 65.9734457253857
x77=−45.553093477052x_{77} = -45.553093477052
x78=−53.4070751110265x_{78} = -53.4070751110265
x79=−7.85398163397448x_{79} = -7.85398163397448
x80=64.4026493985908x_{80} = 64.4026493985908
x81=−6.28318530717959x_{81} = -6.28318530717959
x82=−73.8274273593601x_{82} = -73.8274273593601
x83=−43.9822971502571x_{83} = -43.9822971502571
x84=92.6769832808989x_{84} = 92.6769832808989
x85=72.2566310325652x_{85} = 72.2566310325652
x86=−42.4115008234622x_{86} = -42.4115008234622
x87=87.9645943005142x_{87} = 87.9645943005142
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(2*x)/x.
sin⁡(0⋅2)0\frac{\sin{\left(0 \cdot 2 \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
2cos⁡(2x)x−sin⁡(2x)x2=0\frac{2 \cos{\left(2 x \right)}}{x} - \frac{\sin{\left(2 x \right)}}{x^{2}} = 0
Solve this equation
The roots of this equation
x1=84.0346285545694x_{1} = 84.0346285545694
x2=24.3370721159772x_{2} = 24.3370721159772
x3=−63.6133213216672x_{3} = -63.6133213216672
x4=11.7597262493445x_{4} = 11.7597262493445
x5=−62.0424254948814x_{5} = -62.0424254948814
x6=−79.3220628366317x_{6} = -79.3220628366317
x7=−46.3330961388114x_{7} = -46.3330961388114
x8=−54.1878598258373x_{8} = -54.1878598258373
x9=30.6223651301872x_{9} = 30.6223651301872
x10=96.6013861664138x_{10} = 96.6013861664138
x11=−85.6054794697228x_{11} = -85.6054794697228
x12=−93.4597065202651x_{12} = -93.4597065202651
x13=−76.1803402100956x_{13} = -76.1803402100956
x14=−68.3259813506395x_{14} = -68.3259813506395
x15=−3.86262591846885x_{15} = -3.86262591846885
x16=−91.8888644664832x_{16} = -91.8888644664832
x17=−27.4798391439445x_{17} = -27.4798391439445
x18=3.86262591846885x_{18} = 3.86262591846885
x19=−33.7647173885721x_{19} = -33.7647173885721
x20=−13.3330271294063x_{20} = -13.3330271294063
x21=−47.9040693934309x_{21} = -47.9040693934309
x22=66.7550989265392x_{22} = 66.7550989265392
x23=25.9084912436398x_{23} = 25.9084912436398
x24=22.7655670069956x_{24} = 22.7655670069956
x25=32.1935597952787x_{25} = 32.1935597952787
x26=8.61037763596538x_{26} = 8.61037763596538
x27=−60.4715244985757x_{27} = -60.4715244985757
x28=10.1856514796438x_{28} = 10.1856514796438
x29=99.7430603324317x_{29} = 99.7430603324317
x30=−90.3180208221014x_{30} = -90.3180208221014
x31=44.7621104652086x_{31} = 44.7621104652086
x32=88.7471755026564x_{32} = 88.7471755026564
x33=16.4781945199112x_{33} = 16.4781945199112
x34=−10.1856514796438x_{34} = -10.1856514796438
x35=−40.0490643144726x_{35} = -40.0490643144726
x36=−38.4780131551656x_{36} = -38.4780131551656
x37=−69.8968599047927x_{37} = -69.8968599047927
x38=−19.6222161805821x_{38} = -19.6222161805821
x39=54.1878598258373x_{39} = 54.1878598258373
x40=−41.6200962353617x_{40} = -41.6200962353617
x41=62.0424254948814x_{41} = 62.0424254948814
x42=−35.3358428558098x_{42} = -35.3358428558098
x43=51.0459832324538x_{43} = 51.0459832324538
x44=63.6133213216672x_{44} = 63.6133213216672
x45=204.987701063789x_{45} = 204.987701063789
x46=2.24670472895453x_{46} = 2.24670472895453
x47=74.6094747920599x_{47} = 74.6094747920599
x48=47.9040693934309x_{48} = 47.9040693934309
x49=33.7647173885721x_{49} = 33.7647173885721
x50=46.3330961388114x_{50} = 46.3330961388114
x51=40.0490643144726x_{51} = 40.0490643144726
x52=−98.172223901556x_{52} = -98.172223901556
x53=98.172223901556x_{53} = 98.172223901556
x54=41.6200962353617x_{54} = 41.6200962353617
x55=77.7512028363303x_{55} = 77.7512028363303
x56=−58.9006179191122x_{56} = -58.9006179191122
x57=91.8888644664832x_{57} = 91.8888644664832
x58=−49.4750314121659x_{58} = -49.4750314121659
x59=−24.3370721159772x_{59} = -24.3370721159772
x60=−57.3297052975115x_{60} = -57.3297052975115
x61=69.8968599047927x_{61} = 69.8968599047927
x62=−99.7430603324317x_{62} = -99.7430603324317
x63=−82.4637755597094x_{63} = -82.4637755597094
x64=90.3180208221014x_{64} = 90.3180208221014
x65=60.4715244985757x_{65} = 60.4715244985757
x66=−11.7597262493445x_{66} = -11.7597262493445
x67=−71.4677348441946x_{67} = -71.4677348441946
x68=−25.9084912436398x_{68} = -25.9084912436398
x69=68.3259813506395x_{69} = 68.3259813506395
x70=38.4780131551656x_{70} = 38.4780131551656
x71=−55.7587861230655x_{71} = -55.7587861230655
x72=13.3330271294063x_{72} = 13.3330271294063
x73=18.0503111221878x_{73} = 18.0503111221878
x74=−16.4781945199112x_{74} = -16.4781945199112
x75=−2.24670472895453x_{75} = -2.24670472895453
x76=55.7587861230655x_{76} = 55.7587861230655
x77=−84.0346285545694x_{77} = -84.0346285545694
x78=85.6054794697228x_{78} = 85.6054794697228
x79=76.1803402100956x_{79} = 76.1803402100956
x80=52.6169257678188x_{80} = 52.6169257678188
x81=82.4637755597094x_{81} = 82.4637755597094
x82=−5.45206082971445x_{82} = -5.45206082971445
x83=−18.0503111221878x_{83} = -18.0503111221878
x84=−77.7512028363303x_{84} = -77.7512028363303
x85=−32.1935597952787x_{85} = -32.1935597952787
x86=19.6222161805821x_{86} = 19.6222161805821
The values of the extrema at the points:
(84.0346285545694, -0.0118996456204748)

(24.337072115977193, -0.0410809080835075)

(-63.613321321667165, 0.015719492253233)

(11.759726249344503, -0.0849592339552253)

(-62.04242549488138, -0.0161174796093628)

(-79.32206283663172, 0.0126065825610424)

(-46.33309613881142, -0.0215815876990685)

(-54.18785982583734, 0.0184535325015639)

(30.6223651301872, -0.0326515186419956)

(96.60138616641379, -0.0103516796697785)

(-85.60547946972281, 0.0116812959808693)

(-93.45970652026512, -0.0106996450858762)

(-76.18034021009562, 0.0131264635863328)

(-68.3259813506395, -0.0146353291349374)

(-3.8626259184688534, 0.256749107051798)

(-91.88886446648316, 0.0108825503716759)

(-27.479839143944467, -0.0363842926436063)

(3.8626259184688534, 0.256749107051798)

(-33.76471738857206, -0.0296134678930985)

(-13.333027129406338, 0.0749490399878624)

(-47.90406939343085, 0.0208739162691316)

(66.75509892653919, 0.0149797089170242)

(25.908491243639833, 0.0385901989751759)

(22.76556700699564, 0.0439153964569649)

(32.19355979527871, 0.0310583676149227)

(8.610377635965385, -0.115943604692308)

(-60.47152449857575, 0.0165361437007152)

(10.18565147964378, 0.0980592480281483)

(99.74306033243167, -0.0100256341886906)

(-90.31802082210145, -0.01107181786798)

(44.76211046520859, 0.0223389292683471)

(88.7471755026564, 0.0112677854121748)

(16.478194519911238, 0.0606583423726206)

(-10.18565147964378, 0.0980592480281483)

(-40.04906431447256, -0.024967426643558)

(-38.47801315516559, 0.0259866739740854)

(-69.8968599047927, 0.0143064283116353)

(-19.622216180582097, 0.0509461061857615)

(54.18785982583734, 0.0184535325015639)

(-41.6200962353617, 0.0240251209641055)

(62.04242549488138, -0.0161174796093628)

(-35.33584285580975, 0.0282970441297328)

(51.04598323245382, 0.0195892402934823)

(63.613321321667165, 0.015719492253233)

(204.98770106378876, 0.00487832694374757)

(2.246704728954532, -0.434467256422443)

(74.60947479205991, -0.0134028224709878)

(47.90406939343085, 0.0208739162691316)

(33.76471738857206, -0.0296134678930985)

(46.33309613881142, -0.0215815876990685)

(40.04906431447256, -0.024967426643558)

(-98.172223901556, 0.0101860484638785)

(98.172223901556, 0.0101860484638785)

(41.6200962353617, 0.0240251209641055)

(77.75120283633034, -0.0128612714243586)

(-58.90061791911219, -0.0169771388955304)

(91.88886446648316, 0.0108825503716759)

(-49.47503141216594, -0.0202111834730081)

(-24.337072115977193, -0.0410809080835075)

(-57.32970529751154, 0.0174423008954086)

(69.8968599047927, 0.0143064283116353)

(-99.74306033243167, -0.0100256341886906)

(-82.46377555970939, 0.0121263137918205)

(90.31802082210145, -0.01107181786798)

(60.47152449857575, 0.0165361437007152)

(-11.759726249344503, -0.0849592339552253)

(-71.46773484419464, -0.0139919857530453)

(-25.908491243639833, 0.0385901989751759)

(68.3259813506395, -0.0146353291349374)

(38.47801315516559, 0.0259866739740854)

(-55.758786123065505, -0.0179336722809866)

(13.333027129406338, 0.0749490399878624)

(18.050311122187804, -0.0553794646022984)

(-16.478194519911238, 0.0606583423726206)

(-2.246704728954532, -0.434467256422443)

(55.758786123065505, -0.0179336722809866)

(-84.0346285545694, -0.0118996456204748)

(85.60547946972281, 0.0116812959808693)

(76.18034021009562, 0.0131264635863328)

(52.6169257678188, -0.0190044332375671)

(82.46377555970939, 0.0121263137918205)

(-5.4520608297144495, -0.182650405646115)

(-18.050311122187804, -0.0553794646022984)

(-77.75120283633034, -0.0128612714243586)

(-32.19355979527871, 0.0310583676149227)

(19.622216180582097, 0.0509461061857615)


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=84.0346285545694x_{1} = 84.0346285545694
x2=24.3370721159772x_{2} = 24.3370721159772
x3=11.7597262493445x_{3} = 11.7597262493445
x4=−62.0424254948814x_{4} = -62.0424254948814
x5=−46.3330961388114x_{5} = -46.3330961388114
x6=30.6223651301872x_{6} = 30.6223651301872
x7=96.6013861664138x_{7} = 96.6013861664138
x8=−93.4597065202651x_{8} = -93.4597065202651
x9=−68.3259813506395x_{9} = -68.3259813506395
x10=−27.4798391439445x_{10} = -27.4798391439445
x11=−33.7647173885721x_{11} = -33.7647173885721
x12=8.61037763596538x_{12} = 8.61037763596538
x13=99.7430603324317x_{13} = 99.7430603324317
x14=−90.3180208221014x_{14} = -90.3180208221014
x15=−40.0490643144726x_{15} = -40.0490643144726
x16=62.0424254948814x_{16} = 62.0424254948814
x17=2.24670472895453x_{17} = 2.24670472895453
x18=74.6094747920599x_{18} = 74.6094747920599
x19=33.7647173885721x_{19} = 33.7647173885721
x20=46.3330961388114x_{20} = 46.3330961388114
x21=40.0490643144726x_{21} = 40.0490643144726
x22=77.7512028363303x_{22} = 77.7512028363303
x23=−58.9006179191122x_{23} = -58.9006179191122
x24=−49.4750314121659x_{24} = -49.4750314121659
x25=−24.3370721159772x_{25} = -24.3370721159772
x26=−99.7430603324317x_{26} = -99.7430603324317
x27=90.3180208221014x_{27} = 90.3180208221014
x28=−11.7597262493445x_{28} = -11.7597262493445
x29=−71.4677348441946x_{29} = -71.4677348441946
x30=68.3259813506395x_{30} = 68.3259813506395
x31=−55.7587861230655x_{31} = -55.7587861230655
x32=18.0503111221878x_{32} = 18.0503111221878
x33=−2.24670472895453x_{33} = -2.24670472895453
x34=55.7587861230655x_{34} = 55.7587861230655
x35=−84.0346285545694x_{35} = -84.0346285545694
x36=52.6169257678188x_{36} = 52.6169257678188
x37=−5.45206082971445x_{37} = -5.45206082971445
x38=−18.0503111221878x_{38} = -18.0503111221878
x39=−77.7512028363303x_{39} = -77.7512028363303
Maxima of the function at points:
x39=−63.6133213216672x_{39} = -63.6133213216672
x39=−79.3220628366317x_{39} = -79.3220628366317
x39=−54.1878598258373x_{39} = -54.1878598258373
x39=−85.6054794697228x_{39} = -85.6054794697228
x39=−76.1803402100956x_{39} = -76.1803402100956
x39=−3.86262591846885x_{39} = -3.86262591846885
x39=−91.8888644664832x_{39} = -91.8888644664832
x39=3.86262591846885x_{39} = 3.86262591846885
x39=−13.3330271294063x_{39} = -13.3330271294063
x39=−47.9040693934309x_{39} = -47.9040693934309
x39=66.7550989265392x_{39} = 66.7550989265392
x39=25.9084912436398x_{39} = 25.9084912436398
x39=22.7655670069956x_{39} = 22.7655670069956
x39=32.1935597952787x_{39} = 32.1935597952787
x39=−60.4715244985757x_{39} = -60.4715244985757
x39=10.1856514796438x_{39} = 10.1856514796438
x39=44.7621104652086x_{39} = 44.7621104652086
x39=88.7471755026564x_{39} = 88.7471755026564
x39=16.4781945199112x_{39} = 16.4781945199112
x39=−10.1856514796438x_{39} = -10.1856514796438
x39=−38.4780131551656x_{39} = -38.4780131551656
x39=−69.8968599047927x_{39} = -69.8968599047927
x39=−19.6222161805821x_{39} = -19.6222161805821
x39=54.1878598258373x_{39} = 54.1878598258373
x39=−41.6200962353617x_{39} = -41.6200962353617
x39=−35.3358428558098x_{39} = -35.3358428558098
x39=51.0459832324538x_{39} = 51.0459832324538
x39=63.6133213216672x_{39} = 63.6133213216672
x39=204.987701063789x_{39} = 204.987701063789
x39=47.9040693934309x_{39} = 47.9040693934309
x39=−98.172223901556x_{39} = -98.172223901556
x39=98.172223901556x_{39} = 98.172223901556
x39=41.6200962353617x_{39} = 41.6200962353617
x39=91.8888644664832x_{39} = 91.8888644664832
x39=−57.3297052975115x_{39} = -57.3297052975115
x39=69.8968599047927x_{39} = 69.8968599047927
x39=−82.4637755597094x_{39} = -82.4637755597094
x39=60.4715244985757x_{39} = 60.4715244985757
x39=−25.9084912436398x_{39} = -25.9084912436398
x39=38.4780131551656x_{39} = 38.4780131551656
x39=13.3330271294063x_{39} = 13.3330271294063
x39=−16.4781945199112x_{39} = -16.4781945199112
x39=85.6054794697228x_{39} = 85.6054794697228
x39=76.1803402100956x_{39} = 76.1803402100956
x39=82.4637755597094x_{39} = 82.4637755597094
x39=−32.1935597952787x_{39} = -32.1935597952787
x39=19.6222161805821x_{39} = 19.6222161805821
Decreasing at intervals
[99.7430603324317,∞)\left[99.7430603324317, \infty\right)
Increasing at intervals
(−∞,−99.7430603324317]\left(-\infty, -99.7430603324317\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(−2sin⁡(2x)−2cos⁡(2x)x+sin⁡(2x)x2)x=0\frac{2 \left(- 2 \sin{\left(2 x \right)} - \frac{2 \cos{\left(2 x \right)}}{x} + \frac{\sin{\left(2 x \right)}}{x^{2}}\right)}{x} = 0
Solve this equation
The roots of this equation
x1=−45.5421137457344x_{1} = -45.5421137457344
x2=−249.754613990023x_{2} = -249.754613990023
x3=31.4000002782599x_{3} = 31.4000002782599
x4=43.9709250198299x_{4} = 43.9709250198299
x5=100.525990994784x_{5} = 100.525990994784
x6=15.6760458632822x_{6} = 15.6760458632822
x7=14.1016805019762x_{7} = 14.1016805019762
x8=2.97018499528636x_{8} = 2.97018499528636
x9=−39.2571702659654x_{9} = -39.2571702659654
x10=6.20222251095099x_{10} = 6.20222251095099
x11=−72.2497103686524x_{11} = -72.2497103686524
x12=−83.2461988954369x_{12} = -83.2461988954369
x13=−81.6752870376032x_{13} = -81.6752870376032
x14=−59.6818822743783x_{14} = -59.6818822743783
x15=−23.5406987060771x_{15} = -23.5406987060771
x16=67.536838414692x_{16} = 67.536838414692
x17=−28.2566352310993x_{17} = -28.2566352310993
x18=51.8266306358671x_{18} = 51.8266306358671
x19=−2.97018499528636x_{19} = -2.97018499528636
x20=−29.828364501764x_{20} = -29.828364501764
x21=72.2497103686524x_{21} = 72.2497103686524
x22=−36.1144688810077x_{22} = -36.1144688810077
x23=20.3958276156359x_{23} = 20.3958276156359
x24=−100.525990994784x_{24} = -100.525990994784
x25=73.8206539800394x_{25} = 73.8206539800394
x26=−73.8206539800394x_{26} = -73.8206539800394
x27=−21.9683807357099x_{27} = -21.9683807357099
x28=−75.391591452362x_{28} = -75.391591452362
x29=−51.8266306358671x_{29} = -51.8266306358671
x30=−43.9709250198299x_{30} = -43.9709250198299
x31=80.1043706477551x_{31} = 80.1043706477551
x32=−7.78961820519359x_{32} = -7.78961820519359
x33=−9.37132279238738x_{33} = -9.37132279238738
x34=−15.6760458632822x_{34} = -15.6760458632822
x35=81.6752870376032x_{35} = 81.6752870376032
x36=−4.60292007146833x_{36} = -4.60292007146833
x37=86.3880100042327x_{37} = 86.3880100042327
x38=−94.2424740447043x_{38} = -94.2424740447043
x39=−42.3997071961013x_{39} = -42.3997071961013
x40=−87.9589097056013x_{40} = -87.9589097056013
x41=−80.1043706477551x_{41} = -80.1043706477551
x42=26.6847959102454x_{42} = 26.6847959102454
x43=−95.8133573606804x_{43} = -95.8133573606804
x44=−53.3977108664721x_{44} = -53.3977108664721
x45=58.1108594230163x_{45} = 58.1108594230163
x46=78.5334494538573x_{46} = 78.5334494538573
x47=−6.20222251095099x_{47} = -6.20222251095099
x48=−20.3958276156359x_{48} = -20.3958276156359
x49=−50.2555326476356x_{49} = -50.2555326476356
x50=4.60292007146833x_{50} = 4.60292007146833
x51=56.5398239792896x_{51} = 56.5398239792896
x52=−89.5298057788704x_{52} = -89.5298057788704
x53=7.78961820519359x_{53} = 7.78961820519359
x54=−12.5264126404965x_{54} = -12.5264126404965
x55=45.5421137457344x_{55} = 45.5421137457344
x56=−58.1108594230163x_{56} = -58.1108594230163
x57=59.6818822743783x_{57} = 59.6818822743783
x58=89.5298057788704x_{58} = 89.5298057788704
x59=−14.1016805019762x_{59} = -14.1016805019762
x60=42.3997071961013x_{60} = 42.3997071961013
x61=34.5430424733226x_{61} = 34.5430424733226
x62=−67.536838414692x_{62} = -67.536838414692
x63=21.9683807357099x_{63} = 21.9683807357099
x64=−31.4000002782599x_{64} = -31.4000002782599
x65=87.9589097056013x_{65} = 87.9589097056013
x66=92.671587779267x_{66} = 92.671587779267
x67=−17.2497574606835x_{67} = -17.2497574606835
x68=−97.3842378699522x_{68} = -97.3842378699522
x69=36.1144688810077x_{69} = 36.1144688810077
x70=29.828364501764x_{70} = 29.828364501764
x71=37.6858427046437x_{71} = 37.6858427046437
x72=95.8133573606804x_{72} = 95.8133573606804
x73=70.6787602087186x_{73} = 70.6787602087186
x74=64.3948844946117x_{74} = 64.3948844946117
x75=−65.9658657574213x_{75} = -65.9658657574213
x76=28.2566352310993x_{76} = 28.2566352310993
x77=−10.9498482397464x_{77} = -10.9498482397464
x78=12.5264126404965x_{78} = 12.5264126404965
x79=−61.252893502736x_{79} = -61.252893502736
x80=−86.3880100042327x_{80} = -86.3880100042327
x81=48.6844151814505x_{81} = 48.6844151814505
x82=−64.3948844946117x_{82} = -64.3948844946117
x83=50.2555326476356x_{83} = 50.2555326476356
x84=94.2424740447043x_{84} = 94.2424740447043
x85=65.9658657574213x_{85} = 65.9658657574213
x86=23.5406987060771x_{86} = 23.5406987060771
x87=−37.6858427046437x_{87} = -37.6858427046437
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(−2sin⁡(2x)−2cos⁡(2x)x+sin⁡(2x)x2)x)=−83\lim_{x \to 0^-}\left(\frac{2 \left(- 2 \sin{\left(2 x \right)} - \frac{2 \cos{\left(2 x \right)}}{x} + \frac{\sin{\left(2 x \right)}}{x^{2}}\right)}{x}\right) = - \frac{8}{3}
lim⁡x→0+(2(−2sin⁡(2x)−2cos⁡(2x)x+sin⁡(2x)x2)x)=−83\lim_{x \to 0^+}\left(\frac{2 \left(- 2 \sin{\left(2 x \right)} - \frac{2 \cos{\left(2 x \right)}}{x} + \frac{\sin{\left(2 x \right)}}{x^{2}}\right)}{x}\right) = - \frac{8}{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
[95.8133573606804,∞)\left[95.8133573606804, \infty\right)
Convex at the intervals
(−∞,−100.525990994784]\left(-\infty, -100.525990994784\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⁡(2x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(2 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⁡(2x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(2 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(2*x)/x, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(2x)x2)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)}}{x^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(2x)x2)=0\lim_{x \to \infty}\left(\frac{\sin{\left(2 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⁡(2x)x=sin⁡(2x)x\frac{\sin{\left(2 x \right)}}{x} = \frac{\sin{\left(2 x \right)}}{x}
- No
sin⁡(2x)x=−sin⁡(2x)x\frac{\sin{\left(2 x \right)}}{x} = - \frac{\sin{\left(2 x \right)}}{x}
- No
so, the function
not is
neither even, nor odd
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