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

Graphing y = sin(5*x)/x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       sin(5*x)
f(x) = --------
          x
f(x)=sin⁡(5x)xf{\left(x \right)} = \frac{\sin{\left(5 x \right)}}{x} 
f = sin(5*x)/x
The graph of the function
02468-8-6-4-2-1010-510
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⁡(5x)x=0\frac{\sin{\left(5 x \right)}}{x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π5x_{1} = \frac{\pi}{5}
Numerical solution
x1=74.1415866247191x_{1} = 74.1415866247191
x2=11.9380520836412x_{2} = 11.9380520836412
x3=−89.8495498926681x_{3} = -89.8495498926681
x4=62.2035345410779x_{4} = 62.2035345410779
x5=−86.0796387083603x_{5} = -86.0796387083603
x6=96.1327351998477x_{6} = 96.1327351998477
x7=−67.8584013175395x_{7} = -67.8584013175395
x8=23.8761041672824x_{8} = 23.8761041672824
x9=−43.9822971502571x_{9} = -43.9822971502571
x10=28.2743338823081x_{10} = 28.2743338823081
x11=−79.7964534011807x_{11} = -79.7964534011807
x12=−49.6371639267187x_{12} = -49.6371639267187
x13=84.1946831162065x_{13} = 84.1946831162065
x14=33.9292006587698x_{14} = 33.9292006587698
x15=−52.7787565803085x_{15} = -52.7787565803085
x16=−3.76991118430775x_{16} = -3.76991118430775
x17=10.0530964914873x_{17} = 10.0530964914873
x18=−13.1946891450771x_{18} = -13.1946891450771
x19=−45.867252742411x_{19} = -45.867252742411
x20=−15.707963267949x_{20} = -15.707963267949
x21=−37.6991118430775x_{21} = -37.6991118430775
x22=1.88495559215388x_{22} = 1.88495559215388
x23=77.9114978090269x_{23} = 77.9114978090269
x24=45.867252742411x_{24} = 45.867252742411
x25=82.3097275240526x_{25} = 82.3097275240526
x26=−54.0353936417444x_{26} = -54.0353936417444
x27=−42.0973415581032x_{27} = -42.0973415581032
x28=65.9734457253857x_{28} = 65.9734457253857
x29=70.3716754404114x_{29} = 70.3716754404114
x30=21.9911485751286x_{30} = 21.9911485751286
x31=−35.8141562509236x_{31} = -35.8141562509236
x32=−57.8053048260522x_{32} = -57.8053048260522
x33=−65.9734457253857x_{33} = -65.9734457253857
x34=−55.9203492338983x_{34} = -55.9203492338983
x35=−64.0884901332318x_{35} = -64.0884901332318
x36=3.76991118430775x_{36} = 3.76991118430775
x37=−69.1150383789755x_{37} = -69.1150383789755
x38=16.3362817986669x_{38} = 16.3362817986669
x39=76.026542216873x_{39} = 76.026542216873
x40=−38.3274303737955x_{40} = -38.3274303737955
x41=−20.1061929829747x_{41} = -20.1061929829747
x42=−25.7610597594363x_{42} = -25.7610597594363
x43=−71.6283125018473x_{43} = -71.6283125018473
x44=30.159289474462x_{44} = 30.159289474462
x45=−21.9911485751286x_{45} = -21.9911485751286
x46=−69.7433569096934x_{46} = -69.7433569096934
x47=−74.1415866247191x_{47} = -74.1415866247191
x48=55.9203492338983x_{48} = 55.9203492338983
x49=33.3008821280518x_{49} = 33.3008821280518
x50=−47.7522083345649x_{50} = -47.7522083345649
x51=54.0353936417444x_{51} = 54.0353936417444
x52=18.2212373908208x_{52} = 18.2212373908208
x53=−27.6460153515902x_{53} = -27.6460153515902
x54=−59.6902604182061x_{54} = -59.6902604182061
x55=98.0176907920015x_{55} = 98.0176907920015
x56=−5.65486677646163x_{56} = -5.65486677646163
x57=86.0796387083603x_{57} = 86.0796387083603
x58=26.3893782901543x_{58} = 26.3893782901543
x59=−1.88495559215388x_{59} = -1.88495559215388
x60=40.2123859659494x_{60} = 40.2123859659494
x61=60.318578948924x_{61} = 60.318578948924
x62=−76.026542216873x_{62} = -76.026542216873
x63=−11.9380520836412x_{63} = -11.9380520836412
x64=−87.9645943005142x_{64} = -87.9645943005142
x65=92.3628240155399x_{65} = 92.3628240155399
x66=−98.0176907920015x_{66} = -98.0176907920015
x67=6.28318530717959x_{67} = 6.28318530717959
x68=64.0884901332318x_{68} = 64.0884901332318
x69=20.1061929829747x_{69} = 20.1061929829747
x70=−96.1327351998477x_{70} = -96.1327351998477
x71=−10.0530964914873x_{71} = -10.0530964914873
x72=99.9026463841554x_{72} = 99.9026463841554
x73=87.9645943005142x_{73} = 87.9645943005142
x74=43.9822971502571x_{74} = 43.9822971502571
x75=−23.8761041672824x_{75} = -23.8761041672824
x76=−75.398223686155x_{76} = -75.398223686155
x77=−13.8230076757951x_{77} = -13.8230076757951
x78=−91.734505484822x_{78} = -91.734505484822
x79=38.3274303737955x_{79} = 38.3274303737955
x80=42.0973415581032x_{80} = 42.0973415581032
x81=72.2566310325652x_{81} = 72.2566310325652
x82=94.2477796076938x_{82} = 94.2477796076938
x83=89.2212313619501x_{83} = 89.2212313619501
x84=−33.9292006587698x_{84} = -33.9292006587698
x85=32.0442450666159x_{85} = 32.0442450666159
x86=67.8584013175395x_{86} = 67.8584013175395
x87=48.3805268652828x_{87} = 48.3805268652828
x88=−32.0442450666159x_{88} = -32.0442450666159
x89=8.16814089933346x_{89} = 8.16814089933346
x90=−81.6814089933346x_{90} = -81.6814089933346
x91=−93.6194610769758x_{91} = -93.6194610769758
x92=−99.9026463841554x_{92} = -99.9026463841554
x93=−77.9114978090269x_{93} = -77.9114978090269
x94=52.1504380495906x_{94} = 52.1504380495906
x95=50.2654824574367x_{95} = 50.2654824574367
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(5*x)/x.
sin⁡(0⋅5)0\frac{\sin{\left(0 \cdot 5 \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
5cos⁡(5x)x−sin⁡(5x)x2=0\frac{5 \cos{\left(5 x \right)}}{x} - \frac{\sin{\left(5 x \right)}}{x^{2}} = 0
Solve this equation
The roots of this equation
x1=−54.3488169238437x_{1} = -54.3488169238437
x2=−71.9419157645404x_{2} = -71.9419157645404
x3=14.1343371423239x_{3} = 14.1343371423239
x4=58.118775848386x_{4} = 58.118775848386
x5=38.0122188249304x_{5} = 38.0122188249304
x6=71.3135923350694x_{6} = 71.3135923350694
x7=−100.844727531359x_{7} = -100.844727531359
x8=146.083784576627x_{8} = 146.083784576627
x9=71.9419157645404x_{9} = 71.9419157645404
x10=−36.1272083288406x_{10} = -36.1272083288406
x11=26.0736849410777x_{11} = 26.0736849410777
x12=−51.8355071162431x_{12} = -51.8355071162431
x13=12.2489460520749x_{13} = 12.2489460520749
x14=39.8972241329727x_{14} = 39.8972241329727
x15=4.07426059185751x_{15} = 4.07426059185751
x16=−29.843789916824x_{16} = -29.843789916824
x17=17.9048441860834x_{17} = 17.9048441860834
x18=90.1632655190998x_{18} = 90.1632655190998
x19=76.3401775129436x_{19} = 76.3401775129436
x20=5.33321085176253x_{20} = 5.33321085176253
x21=−61.8887289567295x_{21} = -61.8887289567295
x22=49.9505224039313x_{22} = 49.9505224039313
x23=2.18082433188578x_{23} = 2.18082433188578
x24=16.019625725789x_{24} = 16.019625725789
x25=−5.96231975817859x_{25} = -5.96231975817859
x26=65.0303528338626x_{26} = 65.0303528338626
x27=81.9950804255155x_{27} = 81.9950804255155
x28=−9.73482884639088x_{28} = -9.73482884639088
x29=70.0569452125131x_{29} = 70.0569452125131
x30=−4.07426059185751x_{30} = -4.07426059185751
x31=92.0482301960676x_{31} = 92.0482301960676
x32=21.6751439303349x_{32} = 21.6751439303349
x33=−73.8268855526477x_{33} = -73.8268855526477
x34=54.3488169238437x_{34} = 54.3488169238437
x35=−87.6499786753114x_{35} = -87.6499786753114
x36=−76.3401775129436x_{36} = -76.3401775129436
x37=−93.9331945084242x_{37} = -93.9331945084242
x38=−14.1343371423239x_{38} = -14.1343371423239
x39=46.1805458475896x_{39} = 46.1805458475896
x40=−48.0655354076095x_{40} = -48.0655354076095
x41=5.96231975817859x_{41} = 5.96231975817859
x42=27.9587439619171x_{42} = 27.9587439619171
x43=−7.84888647223284x_{43} = -7.84888647223284
x44=−41.7822249551553x_{44} = -41.7822249551553
x45=44.2955533965743x_{45} = 44.2955533965743
x46=1062.80075707302x_{46} = 1062.80075707302
x47=−53.7204897849762x_{47} = -53.7204897849762
x48=48.0655354076095x_{48} = 48.0655354076095
x49=−68.8002977224212x_{49} = -68.8002977224212
x50=−26.0736849410777x_{50} = -26.0736849410777
x51=88.2783004541645x_{51} = 88.2783004541645
x52=−16.019625725789x_{52} = -16.019625725789
x53=93.9331945084242x_{53} = 93.9331945084242
x54=10.3633964974559x_{54} = 10.3633964974559
x55=34.2421917878891x_{55} = 34.2421917878891
x56=−75.7118546338925x_{56} = -75.7118546338925
x57=−95.8181584776878x_{57} = -95.8181584776878
x58=−49.9505224039313x_{58} = -49.9505224039313
x59=80.1101133548396x_{59} = 80.1101133548396
x60=100.216406513801x_{60} = 100.216406513801
x61=32.3571681455931x_{61} = 32.3571681455931
x62=−17.2764444069457x_{62} = -17.2764444069457
x63=−85.7650130531989x_{63} = -85.7650130531989
x64=98.3314432704416x_{64} = 98.3314432704416
x65=61.8887289567295x_{65} = 61.8887289567295
x66=−97.703122123716x_{66} = -97.703122123716
x67=−63.773703652162x_{67} = -63.773703652162
x68=−11.6204509508991x_{68} = -11.6204509508991
x69=22.3035144492262x_{69} = 22.3035144492262
x70=−31.7288251346527x_{70} = -31.7288251346527
x71=−17.9048441860834x_{71} = -17.9048441860834
x72=−43.6672218724157x_{72} = -43.6672218724157
x73=−92.0482301960676x_{73} = -92.0482301960676
x74=−27.9587439619171x_{74} = -27.9587439619171
x75=−81.9950804255155x_{75} = -81.9950804255155
x76=−33.6138514218278x_{76} = -33.6138514218278
x77=−39.8972241329727x_{77} = -39.8972241329727
x78=78.2251457309749x_{78} = 78.2251457309749
x79=−80.1101133548396x_{79} = -80.1101133548396
x80=36.1272083288406x_{80} = 36.1272083288406
x81=56.2337971862508x_{81} = 56.2337971862508
x82=60.0037530610651x_{82} = 60.0037530610651
x83=−60.0037530610651x_{83} = -60.0037530610651
x84=−21.6751439303349x_{84} = -21.6751439303349
x85=83.2517248505282x_{85} = 83.2517248505282
x86=83.8800469802968x_{86} = 83.8800469802968
x87=66.2870015560186x_{87} = 66.2870015560186
x88=24.1886097994303x_{88} = 24.1886097994303
x89=23.5602471676449x_{89} = 23.5602471676449
x90=−236.561757726314x_{90} = -236.561757726314
x91=68.1719738331978x_{91} = 68.1719738331978
x92=−58.118775848386x_{92} = -58.118775848386
x93=−38.0122188249304x_{93} = -38.0122188249304
x94=−19.7900125648664x_{94} = -19.7900125648664
x95=−83.8800469802968x_{95} = -83.8800469802968
x96=−70.0569452125131x_{96} = -70.0569452125131
x97=−65.6586772507346x_{97} = -65.6586772507346
The values of the extrema at the points:
(-54.34881692384368, 0.0183995399663786)

(-71.94191576454038, 0.0139000487426319)

(14.1343371423239, 0.0707426103243319)

(58.118775848386036, 0.0172060416694555)

(38.012218824930414, 0.0263069662778767)

(71.31359233506937, -0.0140225170914689)

(-100.84472753135903, 0.00991621533273093)

(146.08378457662738, 0.00684538031180583)

(71.94191576454038, 0.0139000487426319)

(-36.12720832884058, -0.0276795446699501)

(26.073684941077683, -0.0383517168648523)

(-51.835507116243086, 0.0192916518473169)

(12.24894605207488, -0.0816287966049891)

(39.89722413297267, -0.0250640854717266)

(4.074260591857513, 0.245148120070371)

(-29.843789916823965, -0.0335070561774695)

(17.904844186083437, 0.0558473231708678)

(90.16326551909978, -0.0110909640866485)

(76.34017751294361, -0.0130992172245699)

(5.3332108517625345, 0.187372599969656)

(-61.88872895672949, 0.0161579466125529)

(49.950522403931345, -0.0200196501680894)

(2.18082433188578, -0.456626014115288)

(16.019625725789023, -0.062418566608895)

(-5.962319758178592, -0.167625675106994)

(65.03035283386262, -0.0153773619109585)

(81.9950804255155, 0.0121958173593689)

(-9.734828846390878, -0.102702270208769)

(70.0569452125131, -0.014274044093581)

(-4.074260591857513, 0.245148120070371)

(92.04823019606764, 0.0108638442847289)

(21.675143930334936, 0.0461338312539098)

(-73.82688555264774, -0.0135451512424658)

(54.34881692384368, 0.0183995399663786)

(-87.64997867531142, -0.0114089861949842)

(-76.34017751294361, -0.0130992172245699)

(-93.9331945084242, -0.010645839722077)

(-14.1343371423239, 0.0707426103243319)

(46.18054584758957, -0.0216539368199916)

(-48.06553540760948, 0.0208047478246456)

(5.962319758178592, -0.167625675106994)

(27.95874396191708, 0.0357660707790884)

(-7.848886472232839, 0.127365265464404)

(-41.782224955155264, 0.0239333483294491)

(44.29555339657426, 0.0225753993413768)

(1062.8007570730222, -0.000940910114749811)

(-53.720489784976216, -0.0186147422297532)

(48.06553540760948, 0.0208047478246456)

(-68.80029772242119, -0.0145347594110856)

(-26.073684941077683, -0.0383517168648523)

(88.27830045416445, 0.011327783027977)

(-16.019625725789023, -0.062418566608895)

(93.9331945084242, -0.010645839722077)

(10.363396497455934, 0.0964754974379398)

(34.242191787889126, 0.0292032399521732)

(-75.71185463389251, 0.0132079251768918)

(-95.81815847768776, 0.0104364124453132)

(-49.950522403931345, -0.0200196501680894)

(80.11011335483961, -0.0124827795358379)

(100.21640651380112, -0.00997838620861423)

(32.357168145593135, -0.0309044627639139)

(-17.276444406945743, -0.0578784022923388)

(-85.76501305319893, 0.0116597344932234)

(98.3314432704416, 0.0101696659613288)

(61.88872895672949, 0.0161579466125529)

(-97.70312212371603, -0.0102350660155981)

(-63.773703652162, -0.0156803670673403)

(-11.62045095089912, 0.0860424373581397)

(22.303514449226203, -0.0448341807024664)

(-31.728825134652688, 0.0315164564026059)

(-17.904844186083437, 0.0558473231708678)

(-43.66722187241573, -0.022900231996764)

(-92.04823019606764, 0.0108638442847289)

(-27.95874396191708, 0.0357660707790884)

(-81.9950804255155, 0.0121958173593689)

(-33.61385142182776, -0.0297491140511871)

(-39.89722413297267, -0.0250640854717266)

(78.22514573097487, 0.0127835713472473)

(-80.11011335483961, -0.0124827795358379)

(36.12720832884058, -0.0276795446699501)

(56.23379718625078, -0.0177827876733487)

(60.00375306106511, -0.0166655316404624)

(-60.00375306106511, -0.0166655316404624)

(-21.675143930334936, 0.0461338312539098)

(83.25172485052818, 0.012011728479606)

(83.88004698029675, -0.0119217524719224)

(66.28700155601862, -0.0150858452616839)

(24.1886097994303, 0.0413403592517879)

(23.560247167644878, -0.0424428472388261)

(-236.56175772631389, 0.00422722443484859)

(68.17197383319782, 0.0146687214750408)

(-58.118775848386036, 0.0172060416694555)

(-38.012218824930414, 0.0263069662778767)

(-19.790012564866377, -0.0505279586825202)

(-83.88004698029675, -0.0119217524719224)

(-70.0569452125131, -0.014274044093581)

(-65.65867725073458, 0.0152302087504037)


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=71.3135923350694x_{1} = 71.3135923350694
x2=−36.1272083288406x_{2} = -36.1272083288406
x3=26.0736849410777x_{3} = 26.0736849410777
x4=12.2489460520749x_{4} = 12.2489460520749
x5=39.8972241329727x_{5} = 39.8972241329727
x6=−29.843789916824x_{6} = -29.843789916824
x7=90.1632655190998x_{7} = 90.1632655190998
x8=76.3401775129436x_{8} = 76.3401775129436
x9=49.9505224039313x_{9} = 49.9505224039313
x10=2.18082433188578x_{10} = 2.18082433188578
x11=16.019625725789x_{11} = 16.019625725789
x12=−5.96231975817859x_{12} = -5.96231975817859
x13=65.0303528338626x_{13} = 65.0303528338626
x14=−9.73482884639088x_{14} = -9.73482884639088
x15=70.0569452125131x_{15} = 70.0569452125131
x16=−73.8268855526477x_{16} = -73.8268855526477
x17=−87.6499786753114x_{17} = -87.6499786753114
x18=−76.3401775129436x_{18} = -76.3401775129436
x19=−93.9331945084242x_{19} = -93.9331945084242
x20=46.1805458475896x_{20} = 46.1805458475896
x21=5.96231975817859x_{21} = 5.96231975817859
x22=1062.80075707302x_{22} = 1062.80075707302
x23=−53.7204897849762x_{23} = -53.7204897849762
x24=−68.8002977224212x_{24} = -68.8002977224212
x25=−26.0736849410777x_{25} = -26.0736849410777
x26=−16.019625725789x_{26} = -16.019625725789
x27=93.9331945084242x_{27} = 93.9331945084242
x28=−49.9505224039313x_{28} = -49.9505224039313
x29=80.1101133548396x_{29} = 80.1101133548396
x30=100.216406513801x_{30} = 100.216406513801
x31=32.3571681455931x_{31} = 32.3571681455931
x32=−17.2764444069457x_{32} = -17.2764444069457
x33=−97.703122123716x_{33} = -97.703122123716
x34=−63.773703652162x_{34} = -63.773703652162
x35=22.3035144492262x_{35} = 22.3035144492262
x36=−43.6672218724157x_{36} = -43.6672218724157
x37=−33.6138514218278x_{37} = -33.6138514218278
x38=−39.8972241329727x_{38} = -39.8972241329727
x39=−80.1101133548396x_{39} = -80.1101133548396
x40=36.1272083288406x_{40} = 36.1272083288406
x41=56.2337971862508x_{41} = 56.2337971862508
x42=60.0037530610651x_{42} = 60.0037530610651
x43=−60.0037530610651x_{43} = -60.0037530610651
x44=83.8800469802968x_{44} = 83.8800469802968
x45=66.2870015560186x_{45} = 66.2870015560186
x46=23.5602471676449x_{46} = 23.5602471676449
x47=−19.7900125648664x_{47} = -19.7900125648664
x48=−83.8800469802968x_{48} = -83.8800469802968
x49=−70.0569452125131x_{49} = -70.0569452125131
Maxima of the function at points:
x49=−54.3488169238437x_{49} = -54.3488169238437
x49=−71.9419157645404x_{49} = -71.9419157645404
x49=14.1343371423239x_{49} = 14.1343371423239
x49=58.118775848386x_{49} = 58.118775848386
x49=38.0122188249304x_{49} = 38.0122188249304
x49=−100.844727531359x_{49} = -100.844727531359
x49=146.083784576627x_{49} = 146.083784576627
x49=71.9419157645404x_{49} = 71.9419157645404
x49=−51.8355071162431x_{49} = -51.8355071162431
x49=4.07426059185751x_{49} = 4.07426059185751
x49=17.9048441860834x_{49} = 17.9048441860834
x49=5.33321085176253x_{49} = 5.33321085176253
x49=−61.8887289567295x_{49} = -61.8887289567295
x49=81.9950804255155x_{49} = 81.9950804255155
x49=−4.07426059185751x_{49} = -4.07426059185751
x49=92.0482301960676x_{49} = 92.0482301960676
x49=21.6751439303349x_{49} = 21.6751439303349
x49=54.3488169238437x_{49} = 54.3488169238437
x49=−14.1343371423239x_{49} = -14.1343371423239
x49=−48.0655354076095x_{49} = -48.0655354076095
x49=27.9587439619171x_{49} = 27.9587439619171
x49=−7.84888647223284x_{49} = -7.84888647223284
x49=−41.7822249551553x_{49} = -41.7822249551553
x49=44.2955533965743x_{49} = 44.2955533965743
x49=48.0655354076095x_{49} = 48.0655354076095
x49=88.2783004541645x_{49} = 88.2783004541645
x49=10.3633964974559x_{49} = 10.3633964974559
x49=34.2421917878891x_{49} = 34.2421917878891
x49=−75.7118546338925x_{49} = -75.7118546338925
x49=−95.8181584776878x_{49} = -95.8181584776878
x49=−85.7650130531989x_{49} = -85.7650130531989
x49=98.3314432704416x_{49} = 98.3314432704416
x49=61.8887289567295x_{49} = 61.8887289567295
x49=−11.6204509508991x_{49} = -11.6204509508991
x49=−31.7288251346527x_{49} = -31.7288251346527
x49=−17.9048441860834x_{49} = -17.9048441860834
x49=−92.0482301960676x_{49} = -92.0482301960676
x49=−27.9587439619171x_{49} = -27.9587439619171
x49=−81.9950804255155x_{49} = -81.9950804255155
x49=78.2251457309749x_{49} = 78.2251457309749
x49=−21.6751439303349x_{49} = -21.6751439303349
x49=83.2517248505282x_{49} = 83.2517248505282
x49=24.1886097994303x_{49} = 24.1886097994303
x49=−236.561757726314x_{49} = -236.561757726314
x49=68.1719738331978x_{49} = 68.1719738331978
x49=−58.118775848386x_{49} = -58.118775848386
x49=−38.0122188249304x_{49} = -38.0122188249304
x49=−65.6586772507346x_{49} = -65.6586772507346
Decreasing at intervals
[1062.80075707302,∞)\left[1062.80075707302, \infty\right)
Increasing at intervals
(−∞,−97.703122123716]\left(-\infty, -97.703122123716\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
−25sin⁡(5x)−10cos⁡(5x)x+2sin⁡(5x)x2x=0\frac{- 25 \sin{\left(5 x \right)} - \frac{10 \cos{\left(5 x \right)}}{x} + \frac{2 \sin{\left(5 x \right)}}{x^{2}}}{x} = 0
Solve this equation
The roots of this equation
x1=−20.1022130590542x_{1} = -20.1022130590542
x2=−3.74852911695495x_{2} = -3.74852911695495
x3=18.2168454982938x_{3} = 18.2168454982938
x4=−42.0954410861804x_{4} = -42.0954410861804
x5=−15.7028681063862x_{5} = -15.7028681063862
x6=−71.6271956018327x_{6} = -71.6271956018327
x7=96.1319030075333x_{7} = 96.1319030075333
x8=38.3253429442721x_{8} = 38.3253429442721
x9=99.9018455960091x_{9} = 99.9018455960091
x10=74.1405075872172x_{10} = 74.1405075872172
x11=−13.817216989329x_{11} = -13.817216989329
x12=64.0872418267494x_{12} = 64.0872418267494
x13=−79.7954508335631x_{13} = -79.7954508335631
x14=−93.6186065433915x_{14} = -93.6186065433915
x15=−37.6969896178817x_{15} = -37.6969896178817
x16=−30.7850092646624x_{16} = -30.7850092646624
x17=−52.7772407609x_{17} = -52.7772407609
x18=32.041748259102x_{18} = 32.041748259102
x19=52.1489039658734x_{19} = 52.1489039658734
x20=−81.6804295626413x_{20} = -81.6804295626413
x21=−10.0451303298366x_{21} = -10.0451303298366
x22=−38.9536951479809x_{22} = -38.9536951479809
x23=76.025489933367x_{23} = 76.025489933367
x24=54.0339130765566x_{24} = 54.0339130765566
x25=26.3863463029685x_{25} = 26.3863463029685
x26=98.0168746037444x_{26} = 98.0168746037444
x27=−47.750532941x_{27} = -47.750532941
x28=−43.9804781363157x_{28} = -43.9804781363157
x29=−87.9636848311592x_{29} = -87.9636848311592
x30=−59.6889201259272x_{30} = -59.6889201259272
x31=−96.1319030075333x_{31} = -96.1319030075333
x32=89.2203347023684x_{32} = 89.2203347023684
x33=30.1566365809448x_{33} = 30.1566365809448
x34=8.15833104625438x_{34} = 8.15833104625438
x35=87.9636848311592x_{35} = 87.9636848311592
x36=40.2103963979136x_{36} = 40.2103963979136
x37=55.9189185788666x_{37} = 55.9189185788666
x38=23.8727529097513x_{38} = 23.8727529097513
x39=20.1022130590542x_{39} = 20.1022130590542
x40=−25.7579537978447x_{40} = -25.7579537978447
x41=45.8655084902109x_{41} = 45.8655084902109
x42=65.9722330865767x_{42} = 65.9722330865767
x43=−99.9018455960091x_{43} = -99.9018455960091
x44=−86.0787093230507x_{44} = -86.0787093230507
x45=−55.9189185788666x_{45} = -55.9189185788666
x46=28.2715040834874x_{46} = 28.2715040834874
x47=50.2638908408101x_{47} = 50.2638908408101
x48=−32.041748259102x_{48} = -32.041748259102
x49=92.3619578553596x_{49} = 92.3619578553596
x50=70.3705385949019x_{50} = 70.3705385949019
x51=21.9875099451753x_{51} = 21.9875099451753
x52=−91.7336333918877x_{52} = -91.7336333918877
x53=84.1937329231437x_{53} = 84.1937329231437
x54=−89.8486595035994x_{54} = -89.8486595035994
x55=−11.9313458007056x_{55} = -11.9313458007056
x56=6.2704183453129x_{56} = 6.2704183453129
x57=−47.1221920695691x_{57} = -47.1221920695691
x58=−77.9104709848986x_{58} = -77.9104709848986
x59=−21.9875099451753x_{59} = -21.9875099451753
x60=68.485551712018x_{60} = 68.485551712018
x61=77.9104709848986x_{61} = 77.9104709848986
x62=−35.8119223115482x_{62} = -35.8119223115482
x63=33.9268425892523x_{63} = 33.9268425892523
x64=86.0787093230507x_{64} = 86.0787093230507
x65=−98.0168746037444x_{65} = -98.0168746037444
x66=−61.5739167497818x_{66} = -61.5739167497818
x67=94.2469307711373x_{67} = 94.2469307711373
x68=−69.7422098218853x_{68} = -69.7422098218853
x69=−8.78735229428396x_{69} = -8.78735229428396
x70=−74.1405075872172x_{70} = -74.1405075872172
x71=−34.5552040016931x_{71} = -34.5552040016931
x72=11.9313458007056x_{72} = 11.9313458007056
x73=43.9804781363157x_{73} = 43.9804781363157
x74=48.3788732320724x_{74} = 48.3788732320724
x75=−1.84116802858733x_{75} = -1.84116802858733
x76=82.3087555701658x_{76} = 82.3087555701658
x77=−64.0872418267494x_{77} = -64.0872418267494
x78=10.0451303298366x_{78} = 10.0451303298366
x79=60.3172526188461x_{79} = 60.3172526188461
x80=72.2555238451233x_{80} = 72.2555238451233
x81=−5.64067220079047x_{81} = -5.64067220079047
x82=−33.9268425892523x_{82} = -33.9268425892523
x83=−27.6431212214019x_{83} = -27.6431212214019
x84=−54.0339130765566x_{84} = -54.0339130765566
x85=−45.8655084902109x_{85} = -45.8655084902109
x86=−23.8727529097513x_{86} = -23.8727529097513
x87=−57.8039208258439x_{87} = -57.8039208258439
x88=42.0954410861804x_{88} = 42.0954410861804
x89=−76.025489933367x_{89} = -76.025489933367
x90=−65.9722330865767x_{90} = -65.9722330865767
x91=−67.2288928152387x_{91} = -67.2288928152387
x92=16.3313827648073x_{92} = 16.3313827648073
x93=−67.8572223647249x_{93} = -67.8572223647249
x94=62.2022484050713x_{94} = 62.2022484050713
x95=4.37993929589856x_{95} = 4.37993929589856
x96=−49.6355521613125x_{96} = -49.6355521613125
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−(−25sin⁡(5x)−10cos⁡(5x)x+2sin⁡(5x)x2x)=−1253\lim_{x \to 0^-}\left(\frac{- 25 \sin{\left(5 x \right)} - \frac{10 \cos{\left(5 x \right)}}{x} + \frac{2 \sin{\left(5 x \right)}}{x^{2}}}{x}\right) = - \frac{125}{3}
lim⁡x→0+(−25sin⁡(5x)−10cos⁡(5x)x+2sin⁡(5x)x2x)=−1253\lim_{x \to 0^+}\left(\frac{- 25 \sin{\left(5 x \right)} - \frac{10 \cos{\left(5 x \right)}}{x} + \frac{2 \sin{\left(5 x \right)}}{x^{2}}}{x}\right) = - \frac{125}{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
[99.9018455960091,∞)\left[99.9018455960091, \infty\right)
Convex at the intervals
(−∞,−98.0168746037444]\left(-\infty, -98.0168746037444\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⁡(5x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(5 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⁡(5x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(5 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(5*x)/x, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(5x)x2)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(5 x \right)}}{x^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(5x)x2)=0\lim_{x \to \infty}\left(\frac{\sin{\left(5 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⁡(5x)x=sin⁡(5x)x\frac{\sin{\left(5 x \right)}}{x} = \frac{\sin{\left(5 x \right)}}{x}
- No
sin⁡(5x)x=−sin⁡(5x)x\frac{\sin{\left(5 x \right)}}{x} = - \frac{\sin{\left(5 x \right)}}{x}
- No
so, the function
not is
neither even, nor odd
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