Mister Exam

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  • How to use it?

  • Graphing y =:
  • x^4-2x^2+8
  • √(x^2-1)
  • cos(3x) cos(3x)
  • 8/(16-x^2)

  • Identical expressions

  • ((pi+x)*sin(two *x))/x
  • (( Pi plus x) multiply by sinus of (2 multiply by x)) divide by x
  • (( Pi plus x) multiply by sinus of (two multiply by x)) divide by x
  • ((pi+x)sin(2x))/x
  • pi+xsin2x/x
  • ((pi+x)*sin(2*x)) divide by x

  • Similar expressions

  • ((pi-x)*sin(2*x))/x
Plotting a function graph/ ((pi+x)*sin(2*x))/x

Graphing y = ((pi+x)*sin(2*x))/x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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

Analytical solution
x1=−πx_{1} = - \pi
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}
x4=πx_{4} = \pi
Numerical solution
x1=4.71238898038469x_{1} = 4.71238898038469
x2=−59.6902604182061x_{2} = -59.6902604182061
x3=7.85398163397448x_{3} = 7.85398163397448
x4=−21.9911485751286x_{4} = -21.9911485751286
x5=−75.398223686155x_{5} = -75.398223686155
x6=−39.2699081698724x_{6} = -39.2699081698724
x7=50.2654824574367x_{7} = 50.2654824574367
x8=95.8185759344887x_{8} = 95.8185759344887
x9=34.5575191894877x_{9} = 34.5575191894877
x10=−86.3937979737193x_{10} = -86.3937979737193
x11=−45.553093477052x_{11} = -45.553093477052
x12=−204.203522483337x_{12} = -204.203522483337
x13=26.7035375555132x_{13} = 26.7035375555132
x14=−127.234502470387x_{14} = -127.234502470387
x15=−20.4203522483337x_{15} = -20.4203522483337
x16=−64.4026493985908x_{16} = -64.4026493985908
x17=43.9822971502571x_{17} = 43.9822971502571
x18=−43.9822971502571x_{18} = -43.9822971502571
x19=48.6946861306418x_{19} = 48.6946861306418
x20=−58.1194640914112x_{20} = -58.1194640914112
x21=23.5619449019235x_{21} = 23.5619449019235
x22=−80.1106126665397x_{22} = -80.1106126665397
x23=−65.9734457253857x_{23} = -65.9734457253857
x24=−83.2522053201295x_{24} = -83.2522053201295
x25=86.3937979737193x_{25} = 86.3937979737193
x26=45.553093477052x_{26} = 45.553093477052
x27=−3.14159265077471x_{27} = -3.14159265077471
x28=56.5486677646163x_{28} = 56.5486677646163
x29=51.8362787842316x_{29} = 51.8362787842316
x30=−87.9645943005142x_{30} = -87.9645943005142
x31=−14.1371669411541x_{31} = -14.1371669411541
x32=20.4203522483337x_{32} = 20.4203522483337
x33=87.9645943005142x_{33} = 87.9645943005142
x34=59.6902604182061x_{34} = 59.6902604182061
x35=−73.8274273593601x_{35} = -73.8274273593601
x36=−3.1415926230599x_{36} = -3.1415926230599
x37=−81.6814089933346x_{37} = -81.6814089933346
x38=72.2566310325652x_{38} = 72.2566310325652
x39=−9.42477796076938x_{39} = -9.42477796076938
x40=−29.845130209103x_{40} = -29.845130209103
x41=−72.2566310325652x_{41} = -72.2566310325652
x42=81.6814089933346x_{42} = 81.6814089933346
x43=−31.4159265358979x_{43} = -31.4159265358979
x44=−89.5353906273091x_{44} = -89.5353906273091
x45=64.4026493985908x_{45} = 64.4026493985908
x46=−94.2477796076938x_{46} = -94.2477796076938
x47=−50.2654824574367x_{47} = -50.2654824574367
x48=28.2743338823081x_{48} = 28.2743338823081
x49=29.845130209103x_{49} = 29.845130209103
x50=−51.8362787842316x_{50} = -51.8362787842316
x51=92.6769832808989x_{51} = 92.6769832808989
x52=−67.5442420521806x_{52} = -67.5442420521806
x53=100.530964914873x_{53} = 100.530964914873
x54=−53.4070751110265x_{54} = -53.4070751110265
x55=94.2477796076938x_{55} = 94.2477796076938
x56=21.9911485751286x_{56} = 21.9911485751286
x57=−36.1283155162826x_{57} = -36.1283155162826
x58=−15.707963267949x_{58} = -15.707963267949
x59=−7.85398163397448x_{59} = -7.85398163397448
x60=65.9734457253857x_{60} = 65.9734457253857
x61=14.1371669411541x_{61} = 14.1371669411541
x62=−6.28318530717959x_{62} = -6.28318530717959
x63=89.5353906273091x_{63} = 89.5353906273091
x64=80.1106126665397x_{64} = 80.1106126665397
x65=78.5398163397448x_{65} = 78.5398163397448
x66=15.707963267949x_{66} = 15.707963267949
x67=37.6991118430775x_{67} = 37.6991118430775
x68=42.4115008234622x_{68} = 42.4115008234622
x69=−97.3893722612836x_{69} = -97.3893722612836
x70=70.6858347057703x_{70} = 70.6858347057703
x71=36.1283155162826x_{71} = 36.1283155162826
x72=−1.5707963267949x_{72} = -1.5707963267949
x73=−95.8185759344887x_{73} = -95.8185759344887
x74=6.28318530717959x_{74} = 6.28318530717959
x75=−23.5619449019235x_{75} = -23.5619449019235
x76=−28.2743338823081x_{76} = -28.2743338823081
x77=−40.8407044966673x_{77} = -40.8407044966673
x78=67.5442420521806x_{78} = 67.5442420521806
x79=12.5663706143592x_{79} = 12.5663706143592
x80=−17.2787595947439x_{80} = -17.2787595947439
x81=−37.6991118430775x_{81} = -37.6991118430775
x82=73.8274273593601x_{82} = 73.8274273593601
x83=58.1194640914112x_{83} = 58.1194640914112
x84=−61.261056745001x_{84} = -61.261056745001
x85=1.5707963267949x_{85} = 1.5707963267949
x86=−42.4115008234622x_{86} = -42.4115008234622
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to ((pi + x)*sin(2*x))/x.
πsin⁡(0⋅2)0\frac{\pi \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
2(x+π)cos⁡(2x)+sin⁡(2x)x−(x+π)sin⁡(2x)x2=0\frac{2 \left(x + \pi\right) \cos{\left(2 x \right)} + \sin{\left(2 x \right)}}{x} - \frac{\left(x + \pi\right) \sin{\left(2 x \right)}}{x^{2}} = 0
Solve this equation
The roots of this equation
x1=85.6082964375406x_{1} = 85.6082964375406
x2=76.1834918869339x_{2} = 76.1834918869339
x3=−40.0558374988357x_{3} = -40.0558374988357
x4=−27.4901091027858x_{4} = -27.4901091027858
x5=46.3381490906586x_{5} = 46.3381490906586
x6=11.7765020277599x_{6} = 11.7765020277599
x7=−11.7886765310287x_{7} = -11.7886765310287
x8=88.7498961593776x_{8} = 88.7498961593776
x9=33.7714909991948x_{9} = 33.7714909991948
x10=−76.1837629907443x_{10} = -76.1837629907443
x11=−82.4669272166792x_{11} = -82.4669272166792
x12=74.6126901432743x_{12} = 74.6126901432743
x13=−18.0670702786868x_{13} = -18.0670702786868
x14=−84.0377190116962x_{14} = -84.0377190116962
x15=−60.4758850946187x_{15} = -60.4758850946187
x16=8.63165183105941x_{16} = 8.63165183105941
x17=1379.94457267781x_{17} = 1379.94457267781
x18=−16.4969261458718x_{18} = -16.4969261458718
x19=−62.0466697996072x_{19} = -62.0466697996072
x20=69.9002827130884x_{20} = 69.9002827130884
x21=24.3461694681355x_{21} = 24.3461694681355
x22=−24.3488640507118x_{22} = -24.3488640507118
x23=−99.7456482595368x_{23} = -99.7456482595368
x24=−5.55606806484886x_{24} = -5.55606806484886
x25=−4.11405638825337x_{25} = -4.11405638825337
x26=60.4754544373262x_{26} = 60.4754544373262
x27=18.0621070333104x_{27} = 18.0621070333104
x28=−41.626592921105x_{28} = -41.626592921105
x29=−10.2210285897057x_{29} = -10.2210285897057
x30=2.29351925513312x_{30} = 2.29351925513312
x31=98.1746914639647x_{31} = 98.1746914639647
x32=−98.1748546056604x_{32} = -98.1748546056604
x33=−32.2021639665302x_{33} = -32.2021639665302
x34=−2.05016185736717x_{34} = -2.05016185736717
x35=−63.6174553767823x_{35} = -63.6174553767823
x36=−77.754553554868x_{36} = -77.754553554868
x37=66.7586755813446x_{37} = 66.7586755813446
x38=90.3206957514585x_{38} = 90.3206957514585
x39=−91.8916814216247x_{39} = -91.8916814216247
x40=80.8958953011203x_{40} = 80.8958953011203
x41=52.6214092890184x_{41} = 52.6214092890184
x42=10.204409368318x_{42} = 10.204409368318
x43=19.6331976071127x_{43} = 19.6331976071127
x44=63.617066304661x_{44} = 63.617066304661
x45=−19.6373786282684x_{45} = -19.6373786282684
x46=54.1922204947954x_{46} = 54.1922204947954
x47=−57.3343187030552x_{47} = -57.3343187030552
x48=−47.9096541504974x_{48} = -47.9096541504974
x49=−35.3436074269189x_{49} = -35.3436074269189
x50=25.9170965300131x_{50} = 25.9170965300131
x51=32.200634568733x_{51} = 32.200634568733
x52=−46.3388840031421x_{52} = -46.3388840031421
x53=−49.4804268340229x_{53} = -49.4804268340229
x54=3.89840458387666x_{54} = 3.89840458387666
x55=−79.3253444648916x_{55} = -79.3253444648916
x56=99.7454902207637x_{56} = 99.7454902207637
x57=27.488002883706x_{57} = 27.488002883706
x58=96.6038925895545x_{58} = 96.6038925895545
x59=91.8914951802279x_{59} = 91.8914951802279
x60=49.4797826464554x_{60} = 49.4797826464554
x61=−55.7635372545373x_{61} = -55.7635372545373
x62=55.7630304932234x_{62} = 55.7630304932234
x63=16.4909355757335x_{63} = 16.4909355757335
x64=47.9089668434327x_{64} = 47.9089668434327
x65=38.4840197213942x_{65} = 38.4840197213942
x66=62.0462607270261x_{66} = 62.0462607270261
x67=−132.732335274511x_{67} = -132.732335274511
x68=84.0374962811429x_{68} = 84.0374962811429
x69=82.4666959078989x_{69} = 82.4666959078989
x70=41.6256811889882x_{70} = 41.6256811889882
x71=−38.4850874210213x_{71} = -38.4850874210213
x72=40.0548524056392x_{72} = 40.0548524056392
x73=−85.6085110585658x_{73} = -85.6085110585658
x74=−165.71904162811x_{74} = -165.71904162811
x75=68.3294793913564x_{75} = 68.3294793913564
x76=−54.1927571597434x_{76} = -54.1927571597434
x77=−93.4624744831675x_{77} = -93.4624744831675
x78=−90.3208885350595x_{78} = -90.3208885350595
x79=77.7542933118256x_{79} = 77.7542933118256
x80=−69.9006048481523x_{80} = -69.9006048481523
x81=−13.3575240618592x_{81} = -13.3575240618592
x82=44.7673291188129x_{82} = 44.7673291188129
x83=−68.3298165391845x_{83} = -68.3298165391845
x84=−25.9194696918391x_{84} = -25.9194696918391
x85=−33.772880226083x_{85} = -33.772880226083
x86=−71.4713936919086x_{86} = -71.4713936919086
x87=30.6297691007327x_{87} = 30.6297691007327
The values of the extrema at the points:
(85.60829643754059, 0.999999978628136 + 0.0116811105960709*pi)

(76.18349188693392, 0.999999966219436 + 0.0131262028223065*pi)

(-40.05583749883565, 0.999999435726337 - 0.0249651361241768*pi)

(-27.490109102785798, 0.999997246341827 - 0.0363766197726909*pi)

(46.33814909065861, -0.999999765319291 - 0.0215804857324541*pi)

(11.776502027759934, -0.999960030899051 - 0.084911464248205*pi)

(-11.788676531028724, 0.999881296649131 - 0.0848170949484758*pi)

(88.74989615937758, 0.999999981450874 + 0.0112676186083088*pi)

(33.7714909991948, -0.999999206132327 - 0.0296107508595392*pi)

(-76.18376299074434, -0.999999960158328 + 0.0131261560324845*pi)

(-82.46692721667922, -0.999999971171218 + 0.012126072898821*pi)

(74.61269014327426, -0.999999963344791 - 0.0134025453501884*pi)

(-18.06707027868685, 0.999983034496115 - 0.0553483779645095*pi)

(-84.03771901169623, 0.999999973306484 - 0.0118994183215195*pi)

(-60.475885094618725, -0.99999989738371 + 0.0165355148720707*pi)

(8.631651831059406, -0.999880559451095 - 0.115838842787102*pi)

(1379.944572677808, 0.999999999999661 + 0.000724666787202288*pi)

(-16.496926145871825, -0.999974585728353 + 0.0606158127208799*pi)

(-62.04666979960715, 0.999999907643538 - 0.016116898954823*pi)

(69.90028271308837, 0.999999952673103 + 0.0143060931066286*pi)

(24.346169468135457, -0.999997245340557 - 0.0410741101038241*pi)

(-24.348864050711843, 0.999995373210449 - 0.0410694877234412*pi)

(-99.7456482595368, 0.999999986712872 - 0.0100254999006161*pi)

(-5.556068064848859, 0.993214356555512 - 0.178762093077873*pi)

(-4.114056388253371, -0.930825509268131 + 0.226254922495925*pi)

(60.47545443732618, 0.999999916650229 + 0.0165356329432229*pi)

(18.062107033310397, -0.999991589067126 - 0.0553640606393776*pi)

(-41.62659292110495, -0.999999519288263 + 0.0240230931506589*pi)

(-10.221028589705744, -0.999764457230767 + 0.0978144663676701*pi)

(2.2935192551331154, -0.992153911526171 - 0.432590181794043*pi)

(98.17469146396473, 0.999999987530411 + 0.0101859244232763*pi)

(-98.17485460566041, -0.999999985827125 + 0.0101859074795051*pi)

(-32.20216396653017, -0.999998591261348 + 0.0310537699361047*pi)

(-2.0501618573671747, 0.818463147168178 - 0.399218795446352*pi)

(-63.61745537678228, -0.999999916652424 + 0.0157189549743824*pi)

(-77.75455355486802, 0.999999963345312 - 0.0128609826386522*pi)

(66.75867558134459, 0.999999943345213 + 0.014979325677705*pi)

(90.32069575145854, -0.999999982687397 - 0.0110716594283016*pi)

(-91.89168142162467, -0.999999981451032 + 0.0108823776644455*pi)

(80.8958953011203, -0.999999973306185 - 0.0123615663017742*pi)

(52.621409289018374, -0.99999985671774 - 0.0190036692332836*pi)

(10.204409368317986, 0.999933489791241 + 0.0979903347366455*pi)

(19.63319760711271, 0.999993829577657 + 0.0509338239032128*pi)

(63.617066304661, 0.999999931601397 + 0.0157190513440594*pi)

(-19.637378628268372, -0.999988243202298 + 0.0509226950364341*pi)

(54.1922204947954, 0.999999872204922 + 0.01845283073981*pi)

(-57.33431870305522, -0.99999987220956 + 0.0174415584737082*pi)

(-47.90965415049744, -0.999999731819662 + 0.0208726142893536*pi)

(-35.34360742691894, -0.999999047595807 + 0.0282936327216608*pi)

(25.917096530013072, 0.999997824878058 + 0.0385844850992479*pi)

(32.200634568732966, 0.999999047439765 + 0.0310552590292979*pi)

(-46.33888400314214, 0.999999692103051 - 0.0215801418962796*pi)

(-49.48042683402295, 0.999999765332728 - 0.02021000685154*pi)

(3.8984045838766574, 0.998366099685432 + 0.256096071663407*pi)

(-79.3253444648916, -0.999999966219888 + 0.0126063110468115*pi)

(99.74549022076374, -0.9999999882861 - 0.0100255158009934*pi)

(27.488002883705988, -0.999998259637466 - 0.0363794439293454*pi)

(96.60389258955452, -0.999999986712784 - 0.0103515496105476*pi)

(91.89149518022786, 0.999999983822574 + 0.0108823997461491*pi)

(49.47978264645545, -0.999999818017476 - 0.0202102710345902*pi)

(-55.763537254537255, 0.999999856723406 - 0.0179328626905217*pi)

(55.763030493223404, -0.999999885654735 - 0.017933026178989*pi)

(16.490935575733467, 0.999988230472177 + 0.060638659697614*pi)

(47.90896684343273, 0.999999793759002 + 0.0208729150229229*pi)

(38.48401972139424, 0.999999519241118 + 0.0259847990537535*pi)

(62.04626072702611, -0.99999992458719 - 0.0161170054870303*pi)

(-132.7323352745114, -0.999999995830266 + 0.00753395918004538*pi)

(84.03749628114294, -0.999999977015298 - 0.011899449903527*pi)

(82.46669590789888, 0.999999975247394 + 0.0121261069603689*pi)

(41.625681188988175, 0.999999644724284 + 0.0240236223446796*pi)

(-38.48508742102131, -0.999999333184957 + 0.0259840733176752*pi)

(40.054852405639195, -0.999999587899441 - 0.0249657539059775*pi)

(-85.60851105856578, -0.999999975247656 + 0.0116810812719724*pi)

(-165.71904162811032, 0.999999998300409 - 0.00603430956681796*pi)

(68.32947939135637, -0.99999994827114 - 0.0146349709843924*pi)

(-54.19275715974343, -0.999999838818255 + 0.0184526473873726*pi)

(-93.46247448316745, 0.999999982687537 - 0.0106994811363317*pi)

(-90.32088853505948, 0.999999980102128 - 0.011071635767998*pi)

(77.75429331182563, -0.999999968817703 - 0.0128610257546462*pi)

(-69.90060484815228, -0.99999994334633 + 0.0143060270439529*pi)

(-13.357524061859186, -0.99993375414123 + 0.0748592141410713*pi)

(44.76732911881294, 0.999999731802688 + 0.0223377126017207*pi)

(-68.32981653918449, 0.999999937819972 - 0.0146348986206704*pi)

(-25.919469691839065, -0.99999646060738 + 0.038580899705762*pi)

(-33.772880226082954, 0.999998847230964 - 0.0296095222124011*pi)

(-71.47139369190862, 0.999999948272093 - 0.013991611141414*pi)

(30.62976910073273, -0.999998847012987 - 0.0326479394514621*pi)


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=46.3381490906586x_{1} = 46.3381490906586
x2=11.7765020277599x_{2} = 11.7765020277599
x3=33.7714909991948x_{3} = 33.7714909991948
x4=−76.1837629907443x_{4} = -76.1837629907443
x5=−82.4669272166792x_{5} = -82.4669272166792
x6=74.6126901432743x_{6} = 74.6126901432743
x7=−60.4758850946187x_{7} = -60.4758850946187
x8=8.63165183105941x_{8} = 8.63165183105941
x9=−16.4969261458718x_{9} = -16.4969261458718
x10=24.3461694681355x_{10} = 24.3461694681355
x11=−4.11405638825337x_{11} = -4.11405638825337
x12=18.0621070333104x_{12} = 18.0621070333104
x13=−41.626592921105x_{13} = -41.626592921105
x14=−10.2210285897057x_{14} = -10.2210285897057
x15=2.29351925513312x_{15} = 2.29351925513312
x16=−98.1748546056604x_{16} = -98.1748546056604
x17=−32.2021639665302x_{17} = -32.2021639665302
x18=−2.05016185736717x_{18} = -2.05016185736717
x19=−63.6174553767823x_{19} = -63.6174553767823
x20=90.3206957514585x_{20} = 90.3206957514585
x21=−91.8916814216247x_{21} = -91.8916814216247
x22=80.8958953011203x_{22} = 80.8958953011203
x23=52.6214092890184x_{23} = 52.6214092890184
x24=−19.6373786282684x_{24} = -19.6373786282684
x25=−57.3343187030552x_{25} = -57.3343187030552
x26=−47.9096541504974x_{26} = -47.9096541504974
x27=−35.3436074269189x_{27} = -35.3436074269189
x28=−79.3253444648916x_{28} = -79.3253444648916
x29=99.7454902207637x_{29} = 99.7454902207637
x30=27.488002883706x_{30} = 27.488002883706
x31=96.6038925895545x_{31} = 96.6038925895545
x32=49.4797826464554x_{32} = 49.4797826464554
x33=55.7630304932234x_{33} = 55.7630304932234
x34=62.0462607270261x_{34} = 62.0462607270261
x35=−132.732335274511x_{35} = -132.732335274511
x36=84.0374962811429x_{36} = 84.0374962811429
x37=−38.4850874210213x_{37} = -38.4850874210213
x38=40.0548524056392x_{38} = 40.0548524056392
x39=−85.6085110585658x_{39} = -85.6085110585658
x40=68.3294793913564x_{40} = 68.3294793913564
x41=−54.1927571597434x_{41} = -54.1927571597434
x42=77.7542933118256x_{42} = 77.7542933118256
x43=−69.9006048481523x_{43} = -69.9006048481523
x44=−13.3575240618592x_{44} = -13.3575240618592
x45=−25.9194696918391x_{45} = -25.9194696918391
x46=30.6297691007327x_{46} = 30.6297691007327
Maxima of the function at points:
x46=85.6082964375406x_{46} = 85.6082964375406
x46=76.1834918869339x_{46} = 76.1834918869339
x46=−40.0558374988357x_{46} = -40.0558374988357
x46=−27.4901091027858x_{46} = -27.4901091027858
x46=−11.7886765310287x_{46} = -11.7886765310287
x46=88.7498961593776x_{46} = 88.7498961593776
x46=−18.0670702786868x_{46} = -18.0670702786868
x46=−84.0377190116962x_{46} = -84.0377190116962
x46=1379.94457267781x_{46} = 1379.94457267781
x46=−62.0466697996072x_{46} = -62.0466697996072
x46=69.9002827130884x_{46} = 69.9002827130884
x46=−24.3488640507118x_{46} = -24.3488640507118
x46=−99.7456482595368x_{46} = -99.7456482595368
x46=−5.55606806484886x_{46} = -5.55606806484886
x46=60.4754544373262x_{46} = 60.4754544373262
x46=98.1746914639647x_{46} = 98.1746914639647
x46=−77.754553554868x_{46} = -77.754553554868
x46=66.7586755813446x_{46} = 66.7586755813446
x46=10.204409368318x_{46} = 10.204409368318
x46=19.6331976071127x_{46} = 19.6331976071127
x46=63.617066304661x_{46} = 63.617066304661
x46=54.1922204947954x_{46} = 54.1922204947954
x46=25.9170965300131x_{46} = 25.9170965300131
x46=32.200634568733x_{46} = 32.200634568733
x46=−46.3388840031421x_{46} = -46.3388840031421
x46=−49.4804268340229x_{46} = -49.4804268340229
x46=3.89840458387666x_{46} = 3.89840458387666
x46=91.8914951802279x_{46} = 91.8914951802279
x46=−55.7635372545373x_{46} = -55.7635372545373
x46=16.4909355757335x_{46} = 16.4909355757335
x46=47.9089668434327x_{46} = 47.9089668434327
x46=38.4840197213942x_{46} = 38.4840197213942
x46=82.4666959078989x_{46} = 82.4666959078989
x46=41.6256811889882x_{46} = 41.6256811889882
x46=−165.71904162811x_{46} = -165.71904162811
x46=−93.4624744831675x_{46} = -93.4624744831675
x46=−90.3208885350595x_{46} = -90.3208885350595
x46=44.7673291188129x_{46} = 44.7673291188129
x46=−68.3298165391845x_{46} = -68.3298165391845
x46=−33.772880226083x_{46} = -33.772880226083
x46=−71.4713936919086x_{46} = -71.4713936919086
Decreasing at intervals
[99.7454902207637,∞)\left[99.7454902207637, \infty\right)
Increasing at intervals
(−∞,−132.732335274511]\left(-\infty, -132.732335274511\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(−2(x+π)sin⁡(2x)+2cos⁡(2x)−2(x+π)cos⁡(2x)+sin⁡(2x)x+(x+π)sin⁡(2x)x2)x=0\frac{2 \left(- 2 \left(x + \pi\right) \sin{\left(2 x \right)} + 2 \cos{\left(2 x \right)} - \frac{2 \left(x + \pi\right) \cos{\left(2 x \right)} + \sin{\left(2 x \right)}}{x} + \frac{\left(x + \pi\right) \sin{\left(2 x \right)}}{x^{2}}\right)}{x} = 0
Solve this equation
The roots of this equation
x1=−87.9648048213831x_{1} = -87.9648048213831
x2=−7.89551442298467x_{2} = -7.89551442298467
x3=−21.9949358402698x_{3} = -21.9949358402698
x4=−51.8369010677348x_{4} = -51.8369010677348
x5=87.9643982956046x_{5} = 87.9643982956046
x6=−39.2710152365393x_{6} = -39.2710152365393
x7=−64.4030475271897x_{7} = -64.4030475271897
x8=−45.5539064838015x_{8} = -45.5539064838015
x9=−36.1296334177803x_{9} = -36.1296334177803
x10=−2.50547193301345x_{10} = -2.50547193301345
x11=−75.3985120069398x_{11} = -75.3985120069398
x12=−67.5446031465002x_{12} = -67.5446031465002
x13=−114.66825468406x_{13} = -114.66825468406
x14=72.2563427050213x_{14} = 72.2563427050213
x15=23.5594476117201x_{15} = 23.5594476117201
x16=59.6898415808367x_{16} = 59.6898415808367
x17=−53.4076602195951x_{17} = -53.4076602195951
x18=−80.1108674138632x_{18} = -80.1108674138632
x19=80.1103771410977x_{19} = 80.1103771410977
x20=12.5583993914354x_{20} = 12.5583993914354
x21=−15.7159072810519x_{21} = -15.7159072810519
x22=37.698091539784x_{22} = 37.698091539784
x23=−3.62364587330008x_{23} = -3.62364587330008
x24=89.5352013249796x_{24} = 89.5352013249796
x25=−42.4124438981567x_{25} = -42.4124438981567
x26=−31.4176946071907x_{26} = -31.4176946071907
x27=−62.8322718919733x_{27} = -62.8322718919733
x28=64.4022882924056x_{28} = 64.4022882924056
x29=−14.1472470132127x_{29} = -14.1472470132127
x30=−58.1199556768211x_{30} = -58.1199556768211
x31=42.4106877264177x_{31} = 42.4106877264177
x32=70.6855336997594x_{32} = 70.6855336997594
x33=73.8271509256817x_{33} = 73.8271509256817
x34=−29.8471007522921x_{34} = -29.8471007522921
x35=−20.4248009329191x_{35} = -20.4248009329191
x36=95.8184102766651x_{36} = 95.8184102766651
x37=20.417086080627x_{37} = 20.417086080627
x38=−94.2479625433243x_{38} = -94.2479625433243
x39=−61.261497913127x_{39} = -61.261497913127
x40=6.25638099879256x_{40} = 6.25638099879256
x41=−43.9831715735826x_{41} = -43.9831715735826
x42=−9.45102274681725x_{42} = -9.45102274681725
x43=21.9883054965914x_{43} = 21.9883054965914
x44=94.2476084724309x_{44} = 94.2476084724309
x45=−155.508902646452x_{45} = -155.508902646452
x46=50.2648973092021x_{46} = 50.2648973092021
x47=78.5395714837451x_{47} = 78.5395714837451
x48=15.7026532181978x_{48} = 15.7026532181978
x49=43.9815392318454x_{49} = 43.9815392318454
x50=−83.2524408414953x_{50} = -83.2524408414953
x51=−72.2569455638147x_{51} = -72.2569455638147
x52=−120.951427400091x_{52} = -120.951427400091
x53=48.694063800867x_{53} = 48.694063800867
x54=−6.35848862596169x_{54} = -6.35848862596169
x55=58.1190229037068x_{55} = 58.1190229037068
x56=−23.5652081467641x_{56} = -23.5652081467641
x57=−81.6816538448424x_{57} = -81.6816538448424
x58=−59.6907257723168x_{58} = -59.6907257723168
x59=34.5563133498413x_{59} = 34.5563133498413
x60=56.548202388132x_{60} = 56.548202388132
x61=100.530814199299x_{61} = 100.530814199299
x62=−65.9738246590155x_{62} = -65.9738246590155
x63=28.2725651805871x_{63} = 28.2725651805871
x64=45.552385303081x_{64} = 45.552385303081
x65=−50.2661455758163x_{65} = -50.2661455758163
x66=4.66902364156826x_{66} = 4.66902364156826
x67=−73.8277283578441x_{67} = -73.8277283578441
x68=−89.5355936944075x_{68} = -89.5355936944075
x69=7.83568514770558x_{69} = 7.83568514770558
x70=67.5439130450252x_{70} = 67.5439130450252
x71=−95.8187528214518x_{71} = -95.8187528214518
x72=40.8398299650088x_{72} = 40.8398299650088
x73=92.6768063919433x_{73} = 92.6768063919433
x74=−86.3940163664614x_{74} = -86.3940163664614
x75=−17.2851820485436x_{75} = -17.2851820485436
x76=−37.7003174408347x_{76} = -37.7003174408347
x77=29.8435344284095x_{77} = 29.8435344284095
x78=26.7015661854223x_{78} = 26.7015661854223
x79=−97.3895433947121x_{79} = -97.3895433947121
x80=36.1272082541825x_{80} = 36.1272082541825
x81=−28.2765438134118x_{81} = -28.2765438134118
x82=65.9731012289065x_{82} = 65.9731012289065
x83=86.3935949038073x_{83} = 86.3935949038073
x84=81.6811822750425x_{84} = 81.6811822750425
x85=51.8357275812562x_{85} = 51.8357275812562
x86=14.1307285336934x_{86} = 14.1307285336934
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(−2(x+π)sin⁡(2x)+2cos⁡(2x)−2(x+π)cos⁡(2x)+sin⁡(2x)x+(x+π)sin⁡(2x)x2)x)=−8π3\lim_{x \to 0^-}\left(\frac{2 \left(- 2 \left(x + \pi\right) \sin{\left(2 x \right)} + 2 \cos{\left(2 x \right)} - \frac{2 \left(x + \pi\right) \cos{\left(2 x \right)} + \sin{\left(2 x \right)}}{x} + \frac{\left(x + \pi\right) \sin{\left(2 x \right)}}{x^{2}}\right)}{x}\right) = - \frac{8 \pi}{3}
lim⁡x→0+(2(−2(x+π)sin⁡(2x)+2cos⁡(2x)−2(x+π)cos⁡(2x)+sin⁡(2x)x+(x+π)sin⁡(2x)x2)x)=−8π3\lim_{x \to 0^+}\left(\frac{2 \left(- 2 \left(x + \pi\right) \sin{\left(2 x \right)} + 2 \cos{\left(2 x \right)} - \frac{2 \left(x + \pi\right) \cos{\left(2 x \right)} + \sin{\left(2 x \right)}}{x} + \frac{\left(x + \pi\right) \sin{\left(2 x \right)}}{x^{2}}\right)}{x}\right) = - \frac{8 \pi}{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
[95.8184102766651,∞)\left[95.8184102766651, \infty\right)
Convex at the intervals
(−∞,−155.508902646452]\left(-\infty, -155.508902646452\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→−∞((x+π)sin⁡(2x)x)=⟨−1,1⟩\lim_{x \to -\infty}\left(\frac{\left(x + \pi\right) \sin{\left(2 x \right)}}{x}\right) = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞((x+π)sin⁡(2x)x)=⟨−1,1⟩\lim_{x \to \infty}\left(\frac{\left(x + \pi\right) \sin{\left(2 x \right)}}{x}\right) = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of ((pi + x)*sin(2*x))/x, divided by x at x->+oo and x ->-oo
lim⁡x→−∞((x+π)sin⁡(2x)x2)=0\lim_{x \to -\infty}\left(\frac{\left(x + \pi\right) \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→∞((x+π)sin⁡(2x)x2)=0\lim_{x \to \infty}\left(\frac{\left(x + \pi\right) \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:
(x+π)sin⁡(2x)x=(π−x)sin⁡(2x)x\frac{\left(x + \pi\right) \sin{\left(2 x \right)}}{x} = \frac{\left(\pi - x\right) \sin{\left(2 x \right)}}{x}
- No
(x+π)sin⁡(2x)x=−(π−x)sin⁡(2x)x\frac{\left(x + \pi\right) \sin{\left(2 x \right)}}{x} = - \frac{\left(\pi - x\right) \sin{\left(2 x \right)}}{x}
- No
so, the function
not is
neither even, nor odd
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