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Plotting a function graph/ sin(2*x)/((5*x))

Graphing y = sin(2*x)/((5*x))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       sin(2*x)
f(x) = --------
         5*x
f(x)=sin⁡(2x)5xf{\left(x \right)} = \frac{\sin{\left(2 x \right)}}{5 x} 
f = sin(2*x)/((5*x))
The graph of the function
02468-8-6-4-2-10100.5-0.5
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)5x=0\frac{\sin{\left(2 x \right)}}{5 x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π2x_{1} = \frac{\pi}{2}
Numerical solution
x1=65.9734457253857x_{1} = 65.9734457253857
x2=−64.4026493985908x_{2} = -64.4026493985908
x3=−23.5619449019235x_{3} = -23.5619449019235
x4=−29.845130209103x_{4} = -29.845130209103
x5=−21.9911485751286x_{5} = -21.9911485751286
x6=21.9911485751286x_{6} = 21.9911485751286
x7=−271.747764535517x_{7} = -271.747764535517
x8=1.5707963267949x_{8} = 1.5707963267949
x9=−15.707963267949x_{9} = -15.707963267949
x10=42.4115008234622x_{10} = 42.4115008234622
x11=4.71238898038469x_{11} = 4.71238898038469
x12=36.1283155162826x_{12} = 36.1283155162826
x13=23.5619449019235x_{13} = 23.5619449019235
x14=6.28318530717959x_{14} = 6.28318530717959
x15=−17.2787595947439x_{15} = -17.2787595947439
x16=26.7035375555132x_{16} = 26.7035375555132
x17=−80.1106126665397x_{17} = -80.1106126665397
x18=86.3937979737193x_{18} = 86.3937979737193
x19=64.4026493985908x_{19} = 64.4026493985908
x20=−83.2522053201295x_{20} = -83.2522053201295
x21=−95.8185759344887x_{21} = -95.8185759344887
x22=28.2743338823081x_{22} = 28.2743338823081
x23=−94.2477796076938x_{23} = -94.2477796076938
x24=−1.5707963267949x_{24} = -1.5707963267949
x25=−86.3937979737193x_{25} = -86.3937979737193
x26=73.8274273593601x_{26} = 73.8274273593601
x27=−53.4070751110265x_{27} = -53.4070751110265
x28=−39.2699081698724x_{28} = -39.2699081698724
x29=153.9380400259x_{29} = 153.9380400259
x30=67.5442420521806x_{30} = 67.5442420521806
x31=70.6858347057703x_{31} = 70.6858347057703
x32=59.6902604182061x_{32} = 59.6902604182061
x33=56.5486677646163x_{33} = 56.5486677646163
x34=−42.4115008234622x_{34} = -42.4115008234622
x35=72.2566310325652x_{35} = 72.2566310325652
x36=−50.2654824574367x_{36} = -50.2654824574367
x37=−51.8362787842316x_{37} = -51.8362787842316
x38=58.1194640914112x_{38} = 58.1194640914112
x39=−73.8274273593601x_{39} = -73.8274273593601
x40=51.8362787842316x_{40} = 51.8362787842316
x41=78.5398163397448x_{41} = 78.5398163397448
x42=−87.9645943005142x_{42} = -87.9645943005142
x43=37.6991118430775x_{43} = 37.6991118430775
x44=−6.28318530717959x_{44} = -6.28318530717959
x45=−37.6991118430775x_{45} = -37.6991118430775
x46=−43.9822971502571x_{46} = -43.9822971502571
x47=29.845130209103x_{47} = 29.845130209103
x48=−45.553093477052x_{48} = -45.553093477052
x49=80.1106126665397x_{49} = 80.1106126665397
x50=−58.1194640914112x_{50} = -58.1194640914112
x51=−370.707933123596x_{51} = -370.707933123596
x52=−36.1283155162826x_{52} = -36.1283155162826
x53=−72.2566310325652x_{53} = -72.2566310325652
x54=−81.6814089933346x_{54} = -81.6814089933346
x55=−65.9734457253857x_{55} = -65.9734457253857
x56=−28.2743338823081x_{56} = -28.2743338823081
x57=−67.5442420521806x_{57} = -67.5442420521806
x58=43.9822971502571x_{58} = 43.9822971502571
x59=100.530964914873x_{59} = 100.530964914873
x60=−7.85398163397448x_{60} = -7.85398163397448
x61=48.6946861306418x_{61} = 48.6946861306418
x62=−97.3893722612836x_{62} = -97.3893722612836
x63=89.5353906273091x_{63} = 89.5353906273091
x64=81.6814089933346x_{64} = 81.6814089933346
x65=−75.398223686155x_{65} = -75.398223686155
x66=7.85398163397448x_{66} = 7.85398163397448
x67=−14.1371669411541x_{67} = -14.1371669411541
x68=−1669.75649538298x_{68} = -1669.75649538298
x69=50.2654824574367x_{69} = 50.2654824574367
x70=94.2477796076938x_{70} = 94.2477796076938
x71=−59.6902604182061x_{71} = -59.6902604182061
x72=12.5663706143592x_{72} = 12.5663706143592
x73=14.1371669411541x_{73} = 14.1371669411541
x74=34.5575191894877x_{74} = 34.5575191894877
x75=−317.300858012569x_{75} = -317.300858012569
x76=20.4203522483337x_{76} = 20.4203522483337
x77=45.553093477052x_{77} = 45.553093477052
x78=95.8185759344887x_{78} = 95.8185759344887
x79=15.707963267949x_{79} = 15.707963267949
x80=−89.5353906273091x_{80} = -89.5353906273091
x81=87.9645943005142x_{81} = 87.9645943005142
x82=92.6769832808989x_{82} = 92.6769832808989
x83=−9.42477796076938x_{83} = -9.42477796076938
x84=−237.190245346029x_{84} = -237.190245346029
x85=−20.4203522483337x_{85} = -20.4203522483337
x86=−31.4159265358979x_{86} = -31.4159265358979
x87=−61.261056745001x_{87} = -61.261056745001
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)/((5*x)).
sin⁡(0⋅2)0⋅5\frac{\sin{\left(0 \cdot 2 \right)}}{0 \cdot 5}
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
215xcos⁡(2x)−sin⁡(2x)5x2=02 \frac{1}{5 x} \cos{\left(2 x \right)} - \frac{\sin{\left(2 x \right)}}{5 x^{2}} = 0
Solve this equation
The roots of this equation
x1=−3.86262591846885x_{1} = -3.86262591846885
x2=22.7655670069956x_{2} = 22.7655670069956
x3=60.4715244985757x_{3} = 60.4715244985757
x4=98.172223901556x_{4} = 98.172223901556
x5=74.6094747920599x_{5} = 74.6094747920599
x6=2.24670472895453x_{6} = 2.24670472895453
x7=−41.6200962353617x_{7} = -41.6200962353617
x8=−60.4715244985757x_{8} = -60.4715244985757
x9=13.3330271294063x_{9} = 13.3330271294063
x10=−93.4597065202651x_{10} = -93.4597065202651
x11=76.1803402100956x_{11} = 76.1803402100956
x12=18.0503111221878x_{12} = 18.0503111221878
x13=54.1878598258373x_{13} = 54.1878598258373
x14=82.4637755597094x_{14} = 82.4637755597094
x15=77.7512028363303x_{15} = 77.7512028363303
x16=−19.6222161805821x_{16} = -19.6222161805821
x17=−32.1935597952787x_{17} = -32.1935597952787
x18=−57.3297052975115x_{18} = -57.3297052975115
x19=−2.24670472895453x_{19} = -2.24670472895453
x20=−24.3370721159772x_{20} = -24.3370721159772
x21=−49.4750314121659x_{21} = -49.4750314121659
x22=−98.172223901556x_{22} = -98.172223901556
x23=−99.7430603324317x_{23} = -99.7430603324317
x24=33.7647173885721x_{24} = 33.7647173885721
x25=−46.3330961388114x_{25} = -46.3330961388114
x26=−85.6054794697228x_{26} = -85.6054794697228
x27=−69.8968599047927x_{27} = -69.8968599047927
x28=204.987701063789x_{28} = 204.987701063789
x29=68.3259813506395x_{29} = 68.3259813506395
x30=90.3180208221014x_{30} = 90.3180208221014
x31=−27.4798391439445x_{31} = -27.4798391439445
x32=−652.665490738742x_{32} = -652.665490738742
x33=30.6223651301872x_{33} = 30.6223651301872
x34=−84.0346285545694x_{34} = -84.0346285545694
x35=32.1935597952787x_{35} = 32.1935597952787
x36=−40.0490643144726x_{36} = -40.0490643144726
x37=−10.1856514796438x_{37} = -10.1856514796438
x38=−33.7647173885721x_{38} = -33.7647173885721
x39=41.6200962353617x_{39} = 41.6200962353617
x40=−5.45206082971445x_{40} = -5.45206082971445
x41=40.0490643144726x_{41} = 40.0490643144726
x42=3.86262591846885x_{42} = 3.86262591846885
x43=11.7597262493445x_{43} = 11.7597262493445
x44=85.6054794697228x_{44} = 85.6054794697228
x45=52.6169257678188x_{45} = 52.6169257678188
x46=−38.4780131551656x_{46} = -38.4780131551656
x47=−25.9084912436398x_{47} = -25.9084912436398
x48=−90.3180208221014x_{48} = -90.3180208221014
x49=−71.4677348441946x_{49} = -71.4677348441946
x50=46.3330961388114x_{50} = 46.3330961388114
x51=−13.3330271294063x_{51} = -13.3330271294063
x52=−16.4781945199112x_{52} = -16.4781945199112
x53=19.6222161805821x_{53} = 19.6222161805821
x54=91.8888644664832x_{54} = 91.8888644664832
x55=−35.3358428558098x_{55} = -35.3358428558098
x56=25.9084912436398x_{56} = 25.9084912436398
x57=−58.9006179191122x_{57} = -58.9006179191122
x58=−68.3259813506395x_{58} = -68.3259813506395
x59=−76.1803402100956x_{59} = -76.1803402100956
x60=−55.7587861230655x_{60} = -55.7587861230655
x61=10.1856514796438x_{61} = 10.1856514796438
x62=8.61037763596538x_{62} = 8.61037763596538
x63=66.7550989265392x_{63} = 66.7550989265392
x64=−77.7512028363303x_{64} = -77.7512028363303
x65=63.6133213216672x_{65} = 63.6133213216672
x66=62.0424254948814x_{66} = 62.0424254948814
x67=55.7587861230655x_{67} = 55.7587861230655
x68=−79.3220628366317x_{68} = -79.3220628366317
x69=−63.6133213216672x_{69} = -63.6133213216672
x70=−18.0503111221878x_{70} = -18.0503111221878
x71=16.4781945199112x_{71} = 16.4781945199112
x72=24.3370721159772x_{72} = 24.3370721159772
x73=−62.0424254948814x_{73} = -62.0424254948814
x74=96.6013861664138x_{74} = 96.6013861664138
x75=99.7430603324317x_{75} = 99.7430603324317
x76=88.7471755026564x_{76} = 88.7471755026564
x77=51.0459832324538x_{77} = 51.0459832324538
x78=−11.7597262493445x_{78} = -11.7597262493445
x79=38.4780131551656x_{79} = 38.4780131551656
x80=−54.1878598258373x_{80} = -54.1878598258373
x81=−82.4637755597094x_{81} = -82.4637755597094
x82=69.8968599047927x_{82} = 69.8968599047927
x83=84.0346285545694x_{83} = 84.0346285545694
x84=44.7621104652086x_{84} = 44.7621104652086
x85=−91.8888644664832x_{85} = -91.8888644664832
x86=47.9040693934309x_{86} = 47.9040693934309
x87=−47.9040693934309x_{87} = -47.9040693934309
The values of the extrema at the points:
(-3.8626259184688534, 0.0513498214103597)

(22.76556700699564, 0.00878307929139297)

(60.47152449857575, 0.00330722874014303)

(98.172223901556, 0.0020372096927757)

(74.60947479205991, -0.00268056449419756)

(2.246704728954532, -0.0868934512844887)

(-41.6200962353617, 0.00480502419282109)

(-60.47152449857575, 0.00330722874014303)

(13.333027129406338, 0.0149898079975725)

(-93.45970652026512, -0.00213992901717524)

(76.18034021009562, 0.00262529271726657)

(18.050311122187804, -0.0110758929204597)

(54.18785982583734, 0.00369070650031279)

(82.46377555970939, 0.0024252627583641)

(77.75120283633034, -0.00257225428487172)

(-19.622216180582097, 0.0101892212371523)

(-32.19355979527871, 0.00621167352298453)

(-57.32970529751154, 0.00348846017908172)

(-2.246704728954532, -0.0868934512844887)

(-24.337072115977193, -0.00821618161670149)

(-49.47503141216594, -0.00404223669460161)

(-98.172223901556, 0.0020372096927757)

(-99.74306033243167, -0.00200512683773813)

(33.76471738857206, -0.00592269357861969)

(-46.33309613881142, -0.0043163175398137)

(-85.60547946972281, 0.00233625919617385)

(-69.8968599047927, 0.00286128566232707)

(204.98770106378876, 0.000975665388749515)

(68.3259813506395, -0.00292706582698748)

(90.31802082210145, -0.00221436357359601)

(-27.479839143944467, -0.00727685852872126)

(-652.6654907387419, -0.000306435600087234)

(30.6223651301872, -0.00653030372839913)

(-84.0346285545694, -0.00237992912409497)

(32.19355979527871, 0.00621167352298453)

(-40.04906431447256, -0.0049934853287116)

(-10.18565147964378, 0.0196118496056297)

(-33.76471738857206, -0.00592269357861969)

(41.6200962353617, 0.00480502419282109)

(-5.4520608297144495, -0.0365300811292231)

(40.04906431447256, -0.0049934853287116)

(3.8626259184688534, 0.0513498214103597)

(11.759726249344503, -0.0169918467910451)

(85.60547946972281, 0.00233625919617385)

(52.6169257678188, -0.00380088664751342)

(-38.47801315516559, 0.00519733479481709)

(-25.908491243639833, 0.00771803979503518)

(-90.31802082210145, -0.00221436357359601)

(-71.46773484419464, -0.00279839715060906)

(46.33309613881142, -0.0043163175398137)

(-13.333027129406338, 0.0149898079975725)

(-16.478194519911238, 0.0121316684745241)

(19.622216180582097, 0.0101892212371523)

(91.88886446648316, 0.00217651007433517)

(-35.33584285580975, 0.00565940882594655)

(25.908491243639833, 0.00771803979503518)

(-58.90061791911219, -0.00339542777910609)

(-68.3259813506395, -0.00292706582698748)

(-76.18034021009562, 0.00262529271726657)

(-55.758786123065505, -0.00358673445619732)

(10.18565147964378, 0.0196118496056297)

(8.610377635965385, -0.0231887209384616)

(66.75509892653919, 0.00299594178340484)

(-77.75120283633034, -0.00257225428487172)

(63.613321321667165, 0.0031438984506466)

(62.04242549488138, -0.00322349592187255)

(55.758786123065505, -0.00358673445619732)

(-79.32206283663172, 0.00252131651220847)

(-63.613321321667165, 0.0031438984506466)

(-18.050311122187804, -0.0110758929204597)

(16.478194519911238, 0.0121316684745241)

(24.337072115977193, -0.00821618161670149)

(-62.04242549488138, -0.00322349592187255)

(96.60138616641379, -0.00207033593395571)

(99.74306033243167, -0.00200512683773813)

(88.7471755026564, 0.00225355708243497)

(51.04598323245382, 0.00391784805869645)

(-11.759726249344503, -0.0169918467910451)

(38.47801315516559, 0.00519733479481709)

(-54.18785982583734, 0.00369070650031279)

(-82.46377555970939, 0.0024252627583641)

(69.8968599047927, 0.00286128566232707)

(84.0346285545694, -0.00237992912409497)

(44.76211046520859, 0.00446778585366942)

(-91.88886446648316, 0.00217651007433517)

(47.90406939343085, 0.00417478325382633)

(-47.90406939343085, 0.00417478325382633)


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=74.6094747920599x_{1} = 74.6094747920599
x2=2.24670472895453x_{2} = 2.24670472895453
x3=−93.4597065202651x_{3} = -93.4597065202651
x4=18.0503111221878x_{4} = 18.0503111221878
x5=77.7512028363303x_{5} = 77.7512028363303
x6=−2.24670472895453x_{6} = -2.24670472895453
x7=−24.3370721159772x_{7} = -24.3370721159772
x8=−49.4750314121659x_{8} = -49.4750314121659
x9=−99.7430603324317x_{9} = -99.7430603324317
x10=33.7647173885721x_{10} = 33.7647173885721
x11=−46.3330961388114x_{11} = -46.3330961388114
x12=68.3259813506395x_{12} = 68.3259813506395
x13=90.3180208221014x_{13} = 90.3180208221014
x14=−27.4798391439445x_{14} = -27.4798391439445
x15=−652.665490738742x_{15} = -652.665490738742
x16=30.6223651301872x_{16} = 30.6223651301872
x17=−84.0346285545694x_{17} = -84.0346285545694
x18=−40.0490643144726x_{18} = -40.0490643144726
x19=−33.7647173885721x_{19} = -33.7647173885721
x20=−5.45206082971445x_{20} = -5.45206082971445
x21=40.0490643144726x_{21} = 40.0490643144726
x22=11.7597262493445x_{22} = 11.7597262493445
x23=52.6169257678188x_{23} = 52.6169257678188
x24=−90.3180208221014x_{24} = -90.3180208221014
x25=−71.4677348441946x_{25} = -71.4677348441946
x26=46.3330961388114x_{26} = 46.3330961388114
x27=−58.9006179191122x_{27} = -58.9006179191122
x28=−68.3259813506395x_{28} = -68.3259813506395
x29=−55.7587861230655x_{29} = -55.7587861230655
x30=8.61037763596538x_{30} = 8.61037763596538
x31=−77.7512028363303x_{31} = -77.7512028363303
x32=62.0424254948814x_{32} = 62.0424254948814
x33=55.7587861230655x_{33} = 55.7587861230655
x34=−18.0503111221878x_{34} = -18.0503111221878
x35=24.3370721159772x_{35} = 24.3370721159772
x36=−62.0424254948814x_{36} = -62.0424254948814
x37=96.6013861664138x_{37} = 96.6013861664138
x38=99.7430603324317x_{38} = 99.7430603324317
x39=−11.7597262493445x_{39} = -11.7597262493445
x40=84.0346285545694x_{40} = 84.0346285545694
Maxima of the function at points:
x40=−3.86262591846885x_{40} = -3.86262591846885
x40=22.7655670069956x_{40} = 22.7655670069956
x40=60.4715244985757x_{40} = 60.4715244985757
x40=98.172223901556x_{40} = 98.172223901556
x40=−41.6200962353617x_{40} = -41.6200962353617
x40=−60.4715244985757x_{40} = -60.4715244985757
x40=13.3330271294063x_{40} = 13.3330271294063
x40=76.1803402100956x_{40} = 76.1803402100956
x40=54.1878598258373x_{40} = 54.1878598258373
x40=82.4637755597094x_{40} = 82.4637755597094
x40=−19.6222161805821x_{40} = -19.6222161805821
x40=−32.1935597952787x_{40} = -32.1935597952787
x40=−57.3297052975115x_{40} = -57.3297052975115
x40=−98.172223901556x_{40} = -98.172223901556
x40=−85.6054794697228x_{40} = -85.6054794697228
x40=−69.8968599047927x_{40} = -69.8968599047927
x40=204.987701063789x_{40} = 204.987701063789
x40=32.1935597952787x_{40} = 32.1935597952787
x40=−10.1856514796438x_{40} = -10.1856514796438
x40=41.6200962353617x_{40} = 41.6200962353617
x40=3.86262591846885x_{40} = 3.86262591846885
x40=85.6054794697228x_{40} = 85.6054794697228
x40=−38.4780131551656x_{40} = -38.4780131551656
x40=−25.9084912436398x_{40} = -25.9084912436398
x40=−13.3330271294063x_{40} = -13.3330271294063
x40=−16.4781945199112x_{40} = -16.4781945199112
x40=19.6222161805821x_{40} = 19.6222161805821
x40=91.8888644664832x_{40} = 91.8888644664832
x40=−35.3358428558098x_{40} = -35.3358428558098
x40=25.9084912436398x_{40} = 25.9084912436398
x40=−76.1803402100956x_{40} = -76.1803402100956
x40=10.1856514796438x_{40} = 10.1856514796438
x40=66.7550989265392x_{40} = 66.7550989265392
x40=63.6133213216672x_{40} = 63.6133213216672
x40=−79.3220628366317x_{40} = -79.3220628366317
x40=−63.6133213216672x_{40} = -63.6133213216672
x40=16.4781945199112x_{40} = 16.4781945199112
x40=88.7471755026564x_{40} = 88.7471755026564
x40=51.0459832324538x_{40} = 51.0459832324538
x40=38.4780131551656x_{40} = 38.4780131551656
x40=−54.1878598258373x_{40} = -54.1878598258373
x40=−82.4637755597094x_{40} = -82.4637755597094
x40=69.8968599047927x_{40} = 69.8968599047927
x40=44.7621104652086x_{40} = 44.7621104652086
x40=−91.8888644664832x_{40} = -91.8888644664832
x40=47.9040693934309x_{40} = 47.9040693934309
x40=−47.9040693934309x_{40} = -47.9040693934309
Decreasing at intervals
[99.7430603324317,∞)\left[99.7430603324317, \infty\right)
Increasing at intervals
(−∞,−652.665490738742]\left(-\infty, -652.665490738742\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)5x=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)}{5 x} = 0
Solve this equation
The roots of this equation
x1=36.1144688810077x_{1} = 36.1144688810077
x2=2.97018499528636x_{2} = 2.97018499528636
x3=−53.3977108664721x_{3} = -53.3977108664721
x4=−37.6858427046437x_{4} = -37.6858427046437
x5=94.2424740447043x_{5} = 94.2424740447043
x6=50.2555326476356x_{6} = 50.2555326476356
x7=64.3948844946117x_{7} = 64.3948844946117
x8=−72.2497103686524x_{8} = -72.2497103686524
x9=−23.5406987060771x_{9} = -23.5406987060771
x10=−28.2566352310993x_{10} = -28.2566352310993
x11=45.5421137457344x_{11} = 45.5421137457344
x12=21.9683807357099x_{12} = 21.9683807357099
x13=23.5406987060771x_{13} = 23.5406987060771
x14=−64.3948844946117x_{14} = -64.3948844946117
x15=78.5334494538573x_{15} = 78.5334494538573
x16=−83.2461988954369x_{16} = -83.2461988954369
x17=51.8266306358671x_{17} = 51.8266306358671
x18=37.6858427046437x_{18} = 37.6858427046437
x19=43.9709250198299x_{19} = 43.9709250198299
x20=31.4000002782599x_{20} = 31.4000002782599
x21=70.6787602087186x_{21} = 70.6787602087186
x22=73.8206539800394x_{22} = 73.8206539800394
x23=−7.78961820519359x_{23} = -7.78961820519359
x24=67.536838414692x_{24} = 67.536838414692
x25=−58.1108594230163x_{25} = -58.1108594230163
x26=−87.9589097056013x_{26} = -87.9589097056013
x27=−86.3880100042327x_{27} = -86.3880100042327
x28=−249.754613990023x_{28} = -249.754613990023
x29=42.3997071961013x_{29} = 42.3997071961013
x30=−12.5264126404965x_{30} = -12.5264126404965
x31=65.9658657574213x_{31} = 65.9658657574213
x32=−97.3842378699522x_{32} = -97.3842378699522
x33=−15.6760458632822x_{33} = -15.6760458632822
x34=6.20222251095099x_{34} = 6.20222251095099
x35=−89.5298057788704x_{35} = -89.5298057788704
x36=20.3958276156359x_{36} = 20.3958276156359
x37=−9.37132279238738x_{37} = -9.37132279238738
x38=4.60292007146833x_{38} = 4.60292007146833
x39=86.3880100042327x_{39} = 86.3880100042327
x40=−59.6818822743783x_{40} = -59.6818822743783
x41=−36.1144688810077x_{41} = -36.1144688810077
x42=14.1016805019762x_{42} = 14.1016805019762
x43=−81.6752870376032x_{43} = -81.6752870376032
x44=15.6760458632822x_{44} = 15.6760458632822
x45=34.5430424733226x_{45} = 34.5430424733226
x46=−67.536838414692x_{46} = -67.536838414692
x47=−50.2555326476356x_{47} = -50.2555326476356
x48=−31.4000002782599x_{48} = -31.4000002782599
x49=−42.3997071961013x_{49} = -42.3997071961013
x50=7.78961820519359x_{50} = 7.78961820519359
x51=−14.1016805019762x_{51} = -14.1016805019762
x52=72.2497103686524x_{52} = 72.2497103686524
x53=−20.3958276156359x_{53} = -20.3958276156359
x54=−39.2571702659654x_{54} = -39.2571702659654
x55=−2.97018499528636x_{55} = -2.97018499528636
x56=−21.9683807357099x_{56} = -21.9683807357099
x57=100.525990994784x_{57} = 100.525990994784
x58=−10.9498482397464x_{58} = -10.9498482397464
x59=29.828364501764x_{59} = 29.828364501764
x60=−4.60292007146833x_{60} = -4.60292007146833
x61=58.1108594230163x_{61} = 58.1108594230163
x62=92.671587779267x_{62} = 92.671587779267
x63=81.6752870376032x_{63} = 81.6752870376032
x64=−45.5421137457344x_{64} = -45.5421137457344
x65=−6.20222251095099x_{65} = -6.20222251095099
x66=95.8133573606804x_{66} = 95.8133573606804
x67=−65.9658657574213x_{67} = -65.9658657574213
x68=−100.525990994784x_{68} = -100.525990994784
x69=−80.1043706477551x_{69} = -80.1043706477551
x70=28.2566352310993x_{70} = 28.2566352310993
x71=89.5298057788704x_{71} = 89.5298057788704
x72=87.9589097056013x_{72} = 87.9589097056013
x73=48.6844151814505x_{73} = 48.6844151814505
x74=−73.8206539800394x_{74} = -73.8206539800394
x75=12.5264126404965x_{75} = 12.5264126404965
x76=−17.2497574606835x_{76} = -17.2497574606835
x77=80.1043706477551x_{77} = 80.1043706477551
x78=−61.252893502736x_{78} = -61.252893502736
x79=26.6847959102454x_{79} = 26.6847959102454
x80=−29.828364501764x_{80} = -29.828364501764
x81=−51.8266306358671x_{81} = -51.8266306358671
x82=56.5398239792896x_{82} = 56.5398239792896
x83=−43.9709250198299x_{83} = -43.9709250198299
x84=−94.2424740447043x_{84} = -94.2424740447043
x85=−75.391591452362x_{85} = -75.391591452362
x86=59.6818822743783x_{86} = 59.6818822743783
x87=−95.8133573606804x_{87} = -95.8133573606804
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)5x)=−815\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)}{5 x}\right) = - \frac{8}{15}
lim⁡x→0+(2(−2sin⁡(2x)−2cos⁡(2x)x+sin⁡(2x)x2)5x)=−815\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)}{5 x}\right) = - \frac{8}{15}
- 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)5x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)}}{5 x}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(sin⁡(2x)5x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)}}{5 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)/((5*x)), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(15xsin⁡(2x)x)=0\lim_{x \to -\infty}\left(\frac{\frac{1}{5 x} \sin{\left(2 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(15xsin⁡(2x)x)=0\lim_{x \to \infty}\left(\frac{\frac{1}{5 x} \sin{\left(2 x \right)}}{x}\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)5x=sin⁡(2x)5x\frac{\sin{\left(2 x \right)}}{5 x} = \frac{\sin{\left(2 x \right)}}{5 x}
- No
sin⁡(2x)5x=−sin⁡(2x)5x\frac{\sin{\left(2 x \right)}}{5 x} = - \frac{\sin{\left(2 x \right)}}{5 x}
- No
so, the function
not is
neither even, nor odd
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