Mister Exam

Other calculators

  • How to use it?

  • Graphing y =:
  • -x^2-2x+1
  • 9^(1/(x-3))
  • 5x^2-3x-1
  • 3x^2+4x-7

  • Identical expressions

  • sin(seven *x)/((six *x))
  • sinus of (7 multiply by x) divide by ((6 multiply by x))
  • sinus of (seven multiply by x) divide by ((six multiply by x))
  • sin(7x)/((6x))
  • sin7x/6x
  • sin(7*x) divide by ((6*x))
Plotting a function graph/ sin(7*x)/((6*x))

Graphing y = sin(7*x)/((6*x))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       sin(7*x)
f(x) = --------
         6*x
f(x)=sin⁡(7x)6xf{\left(x \right)} = \frac{\sin{\left(7 x \right)}}{6 x} 
f = sin(7*x)/((6*x))
The graph of the function
02468-8-6-4-2-10102-2
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⁡(7x)6x=0\frac{\sin{\left(7 x \right)}}{6 x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π7x_{1} = \frac{\pi}{7}
Numerical solution
x1=31.4159265358979x_{1} = 31.4159265358979
x2=46.2262919028212x_{2} = 46.2262919028212
x3=64.1782499233343x_{3} = 64.1782499233343
x4=17.9519580205131x_{4} = 17.9519580205131
x5=90.2085890530783x_{5} = 90.2085890530783
x6=−70.0126362800011x_{6} = -70.0126362800011
x7=−52.060678259488x_{7} = -52.060678259488
x8=−4.03919055461545x_{8} = -4.03919055461545
x9=−92.0037848551297x_{9} = -92.0037848551297
x10=2.24399475256414x_{10} = 2.24399475256414
x11=26.030339129744x_{11} = 26.030339129744
x12=83.4766047953859x_{12} = 83.4766047953859
x13=−31.8647254864108x_{13} = -31.8647254864108
x14=−5.83438635666676x_{14} = -5.83438635666676
x15=43.9822971502571x_{15} = 43.9822971502571
x16=22.8887464761542x_{16} = 22.8887464761542
x17=−74.0518268346165x_{17} = -74.0518268346165
x18=38.1479107935903x_{18} = 38.1479107935903
x19=96.0429754097451x_{19} = 96.0429754097451
x20=−19.7471538225644x_{20} = -19.7471538225644
x21=−67.768641527437x_{21} = -67.768641527437
x22=−45.7774929523084x_{22} = -45.7774929523084
x23=−15.707963267949x_{23} = -15.707963267949
x24=−83.9254037458988x_{24} = -83.9254037458988
x25=−21.9911485751286x_{25} = -21.9911485751286
x26=32.3135244369236x_{26} = 32.3135244369236
x27=92.0037848551297x_{27} = 92.0037848551297
x28=30.0695296843594x_{28} = 30.0695296843594
x29=60.1390593687189x_{29} = 60.1390593687189
x30=−26.030339129744x_{30} = -26.030339129744
x31=48.0214877048726x_{31} = 48.0214877048726
x32=86.1693984984629x_{32} = 86.1693984984629
x33=−30.0695296843594x_{33} = -30.0695296843594
x34=−11.6687727133335x_{34} = -11.6687727133335
x35=74.0518268346165x_{35} = 74.0518268346165
x36=16.1567622184618x_{36} = 16.1567622184618
x37=−71.8078320820524x_{37} = -71.8078320820524
x38=−85.7205995479501x_{38} = -85.7205995479501
x39=12.1175716638463x_{39} = 12.1175716638463
x40=131.049293549746x_{40} = 131.049293549746
x41=−63.7294509728215x_{41} = -63.7294509728215
x42=24.2351433276927x_{42} = 24.2351433276927
x43=−75.8470226366679x_{43} = -75.8470226366679
x44=−59.6902604182061x_{44} = -59.6902604182061
x45=−49.8166835069239x_{45} = -49.8166835069239
x46=−1.79519580205131x_{46} = -1.79519580205131
x47=39.9431065956417x_{47} = 39.9431065956417
x48=68.2174404779498x_{48} = 68.2174404779498
x49=−48.0214877048726x_{49} = -48.0214877048726
x50=−79.8862131912833x_{50} = -79.8862131912833
x51=−23.7863443771799x_{51} = -23.7863443771799
x52=70.0126362800011x_{52} = 70.0126362800011
x53=76.2958215871807x_{53} = 76.2958215871807
x54=−41.738302397693x_{54} = -41.738302397693
x55=34.1087202389749x_{55} = 34.1087202389749
x56=−92.9013827561553x_{56} = -92.9013827561553
x57=−93.798980657181x_{57} = -93.798980657181
x58=78.091017389232x_{58} = 78.091017389232
x59=21.9911485751286x_{59} = 21.9911485751286
x60=−96.0429754097451x_{60} = -96.0429754097451
x61=−89.7597901025655x_{61} = -89.7597901025655
x62=−57.8950646161548x_{62} = -57.8950646161548
x63=42.1871013482058x_{63} = 42.1871013482058
x64=−39.9431065956417x_{64} = -39.9431065956417
x65=104.570155469489x_{65} = 104.570155469489
x66=98.2869701623092x_{66} = 98.2869701623092
x67=−35.9039160410262x_{67} = -35.9039160410262
x68=−81.6814089933346x_{68} = -81.6814089933346
x69=50.2654824574367x_{69} = 50.2654824574367
x70=−87.9645943005142x_{70} = -87.9645943005142
x71=28.2743338823081x_{71} = 28.2743338823081
x72=−9.87357691128221x_{72} = -9.87357691128221
x73=−65.9734457253857x_{73} = -65.9734457253857
x74=4.03919055461545x_{74} = 4.03919055461545
x75=72.2566310325652x_{75} = 72.2566310325652
x76=−13.9127674658977x_{76} = -13.9127674658977
x77=82.1302079438475x_{77} = 82.1302079438475
x78=56.0998688141035x_{78} = 56.0998688141035
x79=−17.9519580205131x_{79} = -17.9519580205131
x80=6.28318530717959x_{80} = 6.28318530717959
x81=52.060678259488x_{81} = 52.060678259488
x82=94.2477796076938x_{82} = 94.2477796076938
x83=−37.6991118430775x_{83} = -37.6991118430775
x84=65.9734457253857x_{84} = 65.9734457253857
x85=−43.9822971502571x_{85} = -43.9822971502571
x86=87.9645943005142x_{86} = 87.9645943005142
x87=−8.52718005974372x_{87} = -8.52718005974372
x88=54.3046730120521x_{88} = 54.3046730120521
x89=61.9342551707702x_{89} = 61.9342551707702
x90=100.082165964361x_{90} = 100.082165964361
x91=20.1959527730772x_{91} = 20.1959527730772
x92=−27.8255349317953x_{92} = -27.8255349317953
x93=8.0783811092309x_{93} = 8.0783811092309
x94=−97.8381712117964x_{94} = -97.8381712117964
x95=−61.9342551707702x_{95} = -61.9342551707702
x96=−53.8558740615393x_{96} = -53.8558740615393
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(7*x)/((6*x)).
sin⁡(0⋅7)0⋅6\frac{\sin{\left(0 \cdot 7 \right)}}{0 \cdot 6}
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
716xcos⁡(7x)−sin⁡(7x)6x2=07 \frac{1}{6 x} \cos{\left(7 x \right)} - \frac{\sin{\left(7 x \right)}}{6 x^{2}} = 0
Solve this equation
The roots of this equation
x1=−50.0406751517951x_{1} = -50.0406751517951
x2=−43.7574312835751x_{2} = -43.7574312835751
x3=−79.6615575304586x_{3} = -79.6615575304586
x4=19.9705314013693x_{4} = 19.9705314013693
x5=100.306361981441x_{5} = 100.306361981441
x6=80.1103579161908x_{6} = 80.1103579161908
x7=−63.9535313393611x_{7} = -63.9535313393611
x8=−13.6868769695517x_{8} = -13.6868769695517
x9=−21.7658114885988x_{9} = -21.7658114885988
x10=6.05541622401885x_{10} = 6.05541622401885
x11=8.30032210778509x_{11} = 8.30032210778509
x12=173.011877464559x_{12} = 173.011877464559
x13=−11.8914560672462x_{13} = -11.8914560672462
x14=−17.7264072842518x_{14} = -17.7264072842518
x15=−95.8183629466279x_{15} = -95.8183629466279
x16=15.9310817494473x_{16} = 15.9310817494473
x17=18.1752346633335x_{17} = 18.1752346633335
x18=−6.05541622401885x_{18} = -6.05541622401885
x19=48.2454641744854x_{19} = 48.2454641744854
x20=−7.85138261255556x_{20} = -7.85138261255556
x21=−72.031948236685x_{21} = -72.031948236685
x22=28.049206829016x_{22} = 28.049206829016
x23=96.2671628900941x_{23} = 96.2671628900941
x24=88.1887623614173x_{24} = 88.1887623614173
x25=2.00945627326164x_{25} = 2.00945627326164
x26=−65.748735854334x_{26} = -65.748735854334
x27=78.3151562745643x_{27} = 78.3151562745643
x28=−25.8051488063147x_{28} = -25.8051488063147
x29=36.1277506303835x_{29} = 36.1277506303835
x30=−77.8663558220856x_{30} = -77.8663558220856
x31=26.253961276138x_{31} = 26.253961276138
x32=−15.9310817494473x_{32} = -15.9310817494473
x33=58.1191129483684x_{33} = 58.1191129483684
x34=−98.0623625730701x_{34} = -98.0623625730701
x35=−87.7399622271952x_{35} = -87.7399622271952
x36=−73.8271509280264x_{36} = -73.8271509280264
x37=30.2932554779114x_{37} = 30.2932554779114
x38=10.0959551016599x_{38} = 10.0959551016599
x39=52.2846874079873x_{39} = 52.2846874079873
x40=84.1495606988712x_{40} = 84.1495606988712
x41=4.25879982727042x_{41} = 4.25879982727042
x42=32.0884889694752x_{42} = 32.0884889694752
x43=−29.8444463965395x_{43} = -29.8444463965395
x44=85.9447615666809x_{44} = 85.9447615666809
x45=−341.760341100684x_{45} = -341.760341100684
x46=89.9839627802295x_{46} = 89.9839627802295
x47=−33.883718467298x_{47} = -33.883718467298
x48=−83.7007604479657x_{48} = -83.7007604479657
x49=63.9535313393611x_{49} = 63.9535313393611
x50=−99.8575621165067x_{50} = -99.8575621165067
x51=41.9622155286474x_{51} = 41.9622155286474
x52=74.2759515488275x_{52} = 74.2759515488275
x53=−41.9622155286474x_{53} = -41.9622155286474
x54=−69.7879443740829x_{54} = -69.7879443740829
x55=−67.9927408510451x_{55} = -67.9927408510451
x56=56.7727077692574x_{56} = 56.7727077692574
x57=67.9927408510451x_{57} = 67.9927408510451
x58=−2.00945627326164x_{58} = -2.00945627326164
x59=46.0014487872405x_{59} = 46.0014487872405
x60=−56.3239059544124x_{60} = -56.3239059544124
x61=−46.0014487872405x_{61} = -46.0014487872405
x62=−61.7095249827622x_{62} = -61.7095249827622
x63=−39.7181932985428x_{63} = -39.7181932985428
x64=62.1583263211096x_{64} = 62.1583263211096
x65=−72.4807489415142x_{65} = -72.4807489415142
x66=−81.9055593018343x_{66} = -81.9055593018343
x67=22.2146293818087x_{67} = 22.2146293818087
x68=−55.8751040935535x_{68} = -55.8751040935535
x69=−85.9447615666809x_{69} = -85.9447615666809
x70=−59.9143192716405x_{70} = -59.9143192716405
x71=−37.9229731731142x_{71} = -37.9229731731142
x72=−94.0231630779702x_{72} = -94.0231630779702
x73=76.0711538349461x_{73} = 76.0711538349461
x74=70.2367451931726x_{74} = 70.2367451931726
x75=24.0098938727341x_{75} = 24.0098938727341
x76=−28.049206829016x_{76} = -28.049206829016
x77=−19.9705314013693x_{77} = -19.9705314013693
x78=54.0798961670661x_{78} = 54.0798961670661
x79=−76.0711538349461x_{79} = -76.0711538349461
x80=98.0623625730701x_{80} = 98.0623625730701
x81=50.0406751517951x_{81} = 50.0406751517951
x82=−91.779163018434x_{82} = -91.779163018434
x83=44.2062349690925x_{83} = 44.2062349690925
x84=−47.7966612520415x_{84} = -47.7966612520415
x85=−51.835885077988x_{85} = -51.835885077988
x86=72.031948236685x_{86} = 72.031948236685
x87=−3.80943632268752x_{87} = -3.80943632268752
x88=92.2279630510011x_{88} = 92.2279630510011
x89=−89.9839627802295x_{89} = -89.9839627802295
x90=37.4741677776243x_{90} = 37.4741677776243
x91=59.0167161900147x_{91} = 59.0167161900147
x92=1728.10034714005x_{92} = 1728.10034714005
x93=94.0231630779702x_{93} = 94.0231630779702
x94=40.1669979901791x_{94} = 40.1669979901791
x95=66.1975369092382x_{95} = 66.1975369092382
x96=−24.0098938727341x_{96} = -24.0098938727341
x97=14.1357232606188x_{97} = 14.1357232606188
The values of the extrema at the points:
(-50.040675151795064, -0.00333061028850224)

(-43.75743128357506, -0.00380885654317239)

(-79.6615575304586, -0.00209218101980467)

(19.970531401369342, 0.00834541651512062)

(100.3063619814408, -0.00166157454366666)

(80.1103579161908, 0.00208046008035556)

(-63.95353133936111, 0.00260605235346723)

(-13.68687696955167, 0.0121764511569935)

(-21.76581148859875, 0.00765710375869415)

(6.055416224018845, -0.0275159129338524)

(8.300322107785085, 0.0200765687168993)

(173.01187746455872, -0.000963324670497339)

(-11.891456067246201, 0.0140146538957282)

(-17.726407284251824, -0.00940186310546161)

(-95.81836294662794, -0.00173940021834752)

(15.931081749447287, -0.0104613088305755)

(18.175234663333477, 0.00916970381438593)

(-6.055416224018845, -0.0275159129338524)

(48.245464174485434, -0.00345454104075334)

(-7.851382612555562, -0.0212241707087703)

(-72.03194823668503, 0.00231378357763722)

(28.049206829015997, 0.00594186160392913)

(96.2671628900941, 0.00173129110852122)

(88.18876236141728, 0.0018898830591425)

(2.0094562732616392, 0.0827323693588725)

(-65.74873585433403, 0.00253489699977008)

(78.31515627456432, 0.00212814986661164)

(-25.805148806314698, -0.00645856042298836)

(36.12775063038345, 0.00461322282136609)

(-77.86635582208555, -0.00214041590124719)

(26.253961276138043, 0.00634815438347758)

(-15.931081749447287, -0.0104613088305755)

(58.11911294836839, -0.00286766529513478)

(-98.06236257307013, 0.00169959692422771)

(-87.73996222719515, -0.00189955000572286)

(-73.82715092802644, 0.00225752114969546)

(30.293255477911444, -0.00550171352771523)

(10.095955101659928, 0.0165066090756774)

(52.28468740798735, 0.00318766454985892)

(84.14956069887116, -0.00198059770143643)

(4.258799827270423, -0.0391126575249652)

(32.088488969475215, -0.00519391907722712)

(-29.844446396539475, 0.00558444794353813)

(85.94476156668094, -0.00193922739894047)

(-341.7603411006841, -0.000487671131089389)

(89.98396278022946, 0.00185217956047062)

(-33.883718467298024, -0.00491873952841744)

(-83.70076044796572, 0.00199121755910715)

(63.95353133936111, 0.00260605235346723)

(-99.85756211650671, 0.00166904230967432)

(41.962215528647384, -0.00397180412744887)

(74.27595154882749, -0.00224388048790643)

(-41.962215528647384, -0.00397180412744887)

(-69.78794437408288, -0.00238818207030623)

(-67.99274085104514, -0.00245123665716453)

(56.77270776925742, 0.00293567359336773)

(67.99274085104514, -0.00245123665716453)

(-2.0094562732616392, 0.0827323693588725)

(46.0014487872405, 0.00362305682522757)

(-56.32390595441243, -0.00295906556471497)

(-46.0014487872405, 0.00362305682522757)

(-61.709524982762176, -0.00270081839254932)

(-39.71819329854276, 0.00419620266618035)

(62.158326321109584, 0.00268131779536268)

(-72.48074894151424, -0.00229945668851025)

(-81.90555930183429, 0.00203486081503426)

(22.214629381808667, -0.00750240833087585)

(-55.87510409355348, 0.00298283331435769)

(-85.94476156668094, -0.00193922739894047)

(-59.914319271640544, -0.00278174224344856)

(-37.92297317311419, 0.00439484223382984)

(-94.02316307797024, -0.00177261079965872)

(76.07115383494614, -0.00219092736703475)

(70.23674519317257, 0.00237292205764339)

(24.009893872734114, -0.00694145994527699)

(-28.049206829015997, 0.00594186160392913)

(-19.970531401369342, 0.00834541651512062)

(54.07989616706608, 0.00308184920794143)

(-76.07115383494614, -0.00219092736703475)

(98.06236257307013, 0.00169959692422771)

(50.040675151795064, -0.00333061028850224)

(-91.77916301843398, 0.00181595102076066)

(44.20623496909249, 0.00377018754292901)

(-47.79666125204154, 0.00348697833424961)

(-51.835885077988, -0.00321526358585061)

(72.03194823668503, 0.00231378357763722)

(-3.809436322687525, 0.043720273326253)

(92.22796305100107, -0.00180711425488056)

(-89.98396278022946, 0.00185217956047062)

(37.47416777762428, -0.00444747583530982)

(59.016716190014684, -0.00282405035631075)

(1728.1003471400488, 9.64450162706156e-5)

(94.02316307797024, -0.00177261079965872)

(40.16699799017913, -0.00414931712378136)

(66.19753690923824, -0.00251771117708913)

(-24.009893872734114, -0.00694145994527699)

(14.13572326061884, -0.0117898570259214)


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=−50.0406751517951x_{1} = -50.0406751517951
x2=−43.7574312835751x_{2} = -43.7574312835751
x3=−79.6615575304586x_{3} = -79.6615575304586
x4=100.306361981441x_{4} = 100.306361981441
x5=6.05541622401885x_{5} = 6.05541622401885
x6=173.011877464559x_{6} = 173.011877464559
x7=−17.7264072842518x_{7} = -17.7264072842518
x8=−95.8183629466279x_{8} = -95.8183629466279
x9=15.9310817494473x_{9} = 15.9310817494473
x10=−6.05541622401885x_{10} = -6.05541622401885
x11=48.2454641744854x_{11} = 48.2454641744854
x12=−7.85138261255556x_{12} = -7.85138261255556
x13=−25.8051488063147x_{13} = -25.8051488063147
x14=−77.8663558220856x_{14} = -77.8663558220856
x15=−15.9310817494473x_{15} = -15.9310817494473
x16=58.1191129483684x_{16} = 58.1191129483684
x17=−87.7399622271952x_{17} = -87.7399622271952
x18=30.2932554779114x_{18} = 30.2932554779114
x19=84.1495606988712x_{19} = 84.1495606988712
x20=4.25879982727042x_{20} = 4.25879982727042
x21=32.0884889694752x_{21} = 32.0884889694752
x22=85.9447615666809x_{22} = 85.9447615666809
x23=−341.760341100684x_{23} = -341.760341100684
x24=−33.883718467298x_{24} = -33.883718467298
x25=41.9622155286474x_{25} = 41.9622155286474
x26=74.2759515488275x_{26} = 74.2759515488275
x27=−41.9622155286474x_{27} = -41.9622155286474
x28=−69.7879443740829x_{28} = -69.7879443740829
x29=−67.9927408510451x_{29} = -67.9927408510451
x30=67.9927408510451x_{30} = 67.9927408510451
x31=−56.3239059544124x_{31} = -56.3239059544124
x32=−61.7095249827622x_{32} = -61.7095249827622
x33=−72.4807489415142x_{33} = -72.4807489415142
x34=22.2146293818087x_{34} = 22.2146293818087
x35=−85.9447615666809x_{35} = -85.9447615666809
x36=−59.9143192716405x_{36} = -59.9143192716405
x37=−94.0231630779702x_{37} = -94.0231630779702
x38=76.0711538349461x_{38} = 76.0711538349461
x39=24.0098938727341x_{39} = 24.0098938727341
x40=−76.0711538349461x_{40} = -76.0711538349461
x41=50.0406751517951x_{41} = 50.0406751517951
x42=−51.835885077988x_{42} = -51.835885077988
x43=92.2279630510011x_{43} = 92.2279630510011
x44=37.4741677776243x_{44} = 37.4741677776243
x45=59.0167161900147x_{45} = 59.0167161900147
x46=94.0231630779702x_{46} = 94.0231630779702
x47=40.1669979901791x_{47} = 40.1669979901791
x48=66.1975369092382x_{48} = 66.1975369092382
x49=−24.0098938727341x_{49} = -24.0098938727341
x50=14.1357232606188x_{50} = 14.1357232606188
Maxima of the function at points:
x50=19.9705314013693x_{50} = 19.9705314013693
x50=80.1103579161908x_{50} = 80.1103579161908
x50=−63.9535313393611x_{50} = -63.9535313393611
x50=−13.6868769695517x_{50} = -13.6868769695517
x50=−21.7658114885988x_{50} = -21.7658114885988
x50=8.30032210778509x_{50} = 8.30032210778509
x50=−11.8914560672462x_{50} = -11.8914560672462
x50=18.1752346633335x_{50} = 18.1752346633335
x50=−72.031948236685x_{50} = -72.031948236685
x50=28.049206829016x_{50} = 28.049206829016
x50=96.2671628900941x_{50} = 96.2671628900941
x50=88.1887623614173x_{50} = 88.1887623614173
x50=2.00945627326164x_{50} = 2.00945627326164
x50=−65.748735854334x_{50} = -65.748735854334
x50=78.3151562745643x_{50} = 78.3151562745643
x50=36.1277506303835x_{50} = 36.1277506303835
x50=26.253961276138x_{50} = 26.253961276138
x50=−98.0623625730701x_{50} = -98.0623625730701
x50=−73.8271509280264x_{50} = -73.8271509280264
x50=10.0959551016599x_{50} = 10.0959551016599
x50=52.2846874079873x_{50} = 52.2846874079873
x50=−29.8444463965395x_{50} = -29.8444463965395
x50=89.9839627802295x_{50} = 89.9839627802295
x50=−83.7007604479657x_{50} = -83.7007604479657
x50=63.9535313393611x_{50} = 63.9535313393611
x50=−99.8575621165067x_{50} = -99.8575621165067
x50=56.7727077692574x_{50} = 56.7727077692574
x50=−2.00945627326164x_{50} = -2.00945627326164
x50=46.0014487872405x_{50} = 46.0014487872405
x50=−46.0014487872405x_{50} = -46.0014487872405
x50=−39.7181932985428x_{50} = -39.7181932985428
x50=62.1583263211096x_{50} = 62.1583263211096
x50=−81.9055593018343x_{50} = -81.9055593018343
x50=−55.8751040935535x_{50} = -55.8751040935535
x50=−37.9229731731142x_{50} = -37.9229731731142
x50=70.2367451931726x_{50} = 70.2367451931726
x50=−28.049206829016x_{50} = -28.049206829016
x50=−19.9705314013693x_{50} = -19.9705314013693
x50=54.0798961670661x_{50} = 54.0798961670661
x50=98.0623625730701x_{50} = 98.0623625730701
x50=−91.779163018434x_{50} = -91.779163018434
x50=44.2062349690925x_{50} = 44.2062349690925
x50=−47.7966612520415x_{50} = -47.7966612520415
x50=72.031948236685x_{50} = 72.031948236685
x50=−3.80943632268752x_{50} = -3.80943632268752
x50=−89.9839627802295x_{50} = -89.9839627802295
x50=1728.10034714005x_{50} = 1728.10034714005
Decreasing at intervals
[173.011877464559,∞)\left[173.011877464559, \infty\right)
Increasing at intervals
(−∞,−341.760341100684]\left(-\infty, -341.760341100684\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
−49sin⁡(7x)−14cos⁡(7x)x+2sin⁡(7x)x26x=0\frac{- 49 \sin{\left(7 x \right)} - \frac{14 \cos{\left(7 x \right)}}{x} + \frac{2 \sin{\left(7 x \right)}}{x^{2}}}{6 x} = 0
Solve this equation
The roots of this equation
x1=65.9728270395425x_{1} = 65.9728270395425
x2=75.3976823376288x_{2} = 75.3976823376288
x3=−13.9098329089859x_{3} = -13.9098329089859
x4=−93.7985455076866x_{4} = -93.7985455076866
x5=550.227439147896x_{5} = 550.227439147896
x6=−39.9420846991904x_{6} = -39.9420846991904
x7=−36.3515921591033x_{7} = -36.3515921591033
x8=−61.9335961347405x_{8} = -61.9335961347405
x9=−26.0287709747892x_{9} = -26.0287709747892
x10=20.1939314882053x_{10} = 20.1939314882053
x11=43.981369106987x_{11} = 43.981369106987
x12=17.9496839944078x_{12} = 17.9496839944078
x13=−79.8857022561298x_{13} = -79.8857022561298
x14=−63.7288105016917x_{14} = -63.7288105016917
x15=78.0904947082355x_{15} = 78.0904947082355
x16=48.0206377251705x_{16} = 48.0206377251705
x17=2.22560520148388x_{17} = 2.22560520148388
x18=86.1689248195652x_{18} = 86.1689248195652
x19=38.1468408043781x_{19} = 38.1468408043781
x20=30.0681722044146x_{20} = 30.0681722044146
x21=−87.9641302886702x_{21} = -87.9641302886702
x22=−4.02905157199319x_{22} = -4.02905157199319
x23=−53.8551161666372x_{23} = -53.8551161666372
x24=−57.8943595994227x_{24} = -57.8943595994227
x25=56.0991412360678x_{25} = 56.0991412360678
x26=46.2254089125487x_{26} = 46.2254089125487
x27=33.2098932816049x_{27} = 33.2098932816049
x28=−59.6895766056425x_{28} = -59.6895766056425
x29=74.0512756431962x_{29} = 74.0512756431962
x30=83.4761158362889x_{30} = 83.4761158362889
x31=60.1383806593884x_{31} = 60.1383806593884
x32=42.1861338113666x_{32} = 42.1861338113666
x33=21.9892923319017x_{33} = 21.9892923319017
x34=28.2728902017742x_{34} = 28.2728902017742
x35=52.0598942292468x_{35} = 52.0598942292468
x36=−92.0033412148053x_{36} = -92.0033412148053
x37=−52.0598942292468x_{37} = -52.0598942292468
x38=−147.205778494091x_{38} = -147.205778494091
x39=−96.0425504274164x_{39} = -96.0425504274164
x40=−9.8694407066636x_{40} = -9.8694407066636
x41=−81.6809092877104x_{41} = -81.6809092877104
x42=−65.9728270395425x_{42} = -65.9728270395425
x43=61.9335961347405x_{43} = 61.9335961347405
x44=34.1075235292857x_{44} = 34.1075235292857
x45=54.3039213809764x_{45} = 54.3039213809764
x46=−27.8240679628435x_{46} = -27.8240679628435
x47=−35.9027791720072x_{47} = -35.9027791720072
x48=−45.776601304821x_{48} = -45.776601304821
x49=−30.0681722044146x_{49} = -30.0681722044146
x50=10.3184196802879x_{50} = 10.3184196802879
x51=−97.8377540273981x_{51} = -97.8377540273981
x52=39.9420846991904x_{52} = 39.9420846991904
x53=−67.7680392310165x_{53} = -67.7680392310165
x54=87.9641302886702x_{54} = 87.9641302886702
x55=6.27668021020283x_{55} = 6.27668021020283
x56=−23.7846282558391x_{56} = -23.7846282558391
x57=4.02905157199319x_{57} = 4.02905157199319
x58=−1.77206357455742x_{58} = -1.77206357455742
x59=50.26467042493x_{59} = 50.26467042493
x60=−10.7673836298982x_{60} = -10.7673836298982
x61=94.2473465303653x_{61} = 94.2473465303653
x62=64.1776139311424x_{62} = 64.1776139311424
x63=90.2081365839739x_{63} = 90.2081365839739
x64=−83.9249174015968x_{64} = -83.9249174015968
x65=−31.8634444926534x_{65} = -31.8634444926534
x66=26.0287709747892x_{66} = 26.0287709747892
x67=68.2168421440988x_{67} = 68.2168421440988
x68=−75.8464844914481x_{68} = -75.8464844914481
x69=−15.7053642465538x_{69} = -15.7053642465538
x70=−85.7201233890216x_{70} = -85.7201233890216
x71=82.1297109688992x_{71} = 82.1297109688992
x72=−5.82737931875313x_{72} = -5.82737931875313
x73=−70.0120532883889x_{73} = -70.0120532883889
x74=32.312261236614x_{74} = 32.312261236614
x75=−71.8072636655421x_{75} = -71.8072636655421
x76=−21.9892923319017x_{76} = -21.9892923319017
x77=72.2560661466661x_{77} = 72.2560661466661
x78=−74.0512756431962x_{78} = -74.0512756431962
x79=−89.7593353710851x_{79} = -89.7593353710851
x80=24.2334589923231x_{80} = 24.2334589923231
x81=−49.8158641584895x_{81} = -49.8158641584895
x82=70.0120532883889x_{82} = 70.0120532883889
x83=8.07332435174266x_{83} = 8.07332435174266
x84=−8.07332435174266x_{84} = -8.07332435174266
x85=−17.9496839944078x_{85} = -17.9496839944078
x86=16.1542354226542x_{86} = 16.1542354226542
x87=−48.0206377251705x_{87} = -48.0206377251705
x88=1304.65851785572x_{88} = 1304.65851785572
x89=98.286554882884x_{89} = 98.286554882884
x90=92.0033412148053x_{90} = 92.0033412148053
x91=12.1142020560289x_{91} = 12.1142020560289
x92=−41.7373244565853x_{92} = -41.7373244565853
x93=−43.981369106987x_{93} = -43.981369106987
x94=−37.6980291149286x_{94} = -37.6980291149286
x95=59.6895766056425x_{95} = 59.6895766056425
x96=76.2952866075804x_{96} = 76.2952866075804
x97=96.0425504274164x_{97} = 96.0425504274164
x98=−19.7450865867156x_{98} = -19.7450865867156
x99=34.5563380229088x_{99} = 34.5563380229088
x100=100.081758133975x_{100} = 100.081758133975
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−(−49sin⁡(7x)−14cos⁡(7x)x+2sin⁡(7x)x26x)=−34318\lim_{x \to 0^-}\left(\frac{- 49 \sin{\left(7 x \right)} - \frac{14 \cos{\left(7 x \right)}}{x} + \frac{2 \sin{\left(7 x \right)}}{x^{2}}}{6 x}\right) = - \frac{343}{18}
lim⁡x→0+(−49sin⁡(7x)−14cos⁡(7x)x+2sin⁡(7x)x26x)=−34318\lim_{x \to 0^+}\left(\frac{- 49 \sin{\left(7 x \right)} - \frac{14 \cos{\left(7 x \right)}}{x} + \frac{2 \sin{\left(7 x \right)}}{x^{2}}}{6 x}\right) = - \frac{343}{18}
- 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
[1304.65851785572,∞)\left[1304.65851785572, \infty\right)
Convex at the intervals
(−∞,−147.205778494091]\left(-\infty, -147.205778494091\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⁡(7x)6x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(sin⁡(7x)6x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(7 x \right)}}{6 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(7*x)/((6*x)), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(16xsin⁡(7x)x)=0\lim_{x \to -\infty}\left(\frac{\frac{1}{6 x} \sin{\left(7 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(16xsin⁡(7x)x)=0\lim_{x \to \infty}\left(\frac{\frac{1}{6 x} \sin{\left(7 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⁡(7x)6x=sin⁡(7x)6x\frac{\sin{\left(7 x \right)}}{6 x} = \frac{\sin{\left(7 x \right)}}{6 x}
- No
sin⁡(7x)6x=−sin⁡(7x)6x\frac{\sin{\left(7 x \right)}}{6 x} = - \frac{\sin{\left(7 x \right)}}{6 x}
- No
so, the function
not is
neither even, nor odd
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