Mister Exam

Other calculators

  • How to use it?

  • Graphing y =:
  • -x^2-2x+1
  • (x+1)(x-2)^2
  • 6x-2x^2
  • 9^(1/(x-3))

  • Identical expressions

  • (sin(x)*cos(x))/(x+ one)
  • ( sinus of (x) multiply by co sinus of e of (x)) divide by (x plus 1)
  • ( sinus of (x) multiply by co sinus of e of (x)) divide by (x plus one)
  • (sin(x)cos(x))/(x+1)
  • sinxcosx/x+1
  • (sin(x)*cos(x)) divide by (x+1)

  • Similar expressions

  • (sin(x)*cos(x))/(x-1)
  • (sinx*cosx)/(x+1)
Plotting a function graph/ (sin(x)*cos(x))/(x+1)

Graphing y = (sin(x)*cos(x))/(x+1)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       sin(x)*cos(x)
f(x) = -------------
           x + 1
f(x)=sin⁡(x)cos⁡(x)x+1f{\left(x \right)} = \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x + 1} 
f = (sin(x)*cos(x))/(x + 1)
The graph of the function
-6.0-5.5-5.0-4.5-4.0-3.5-3.0-2.5-2.0-1.5-1.0-0.50.0-25002500
The domain of the function
The points at which the function is not precisely defined:
x1=−1x_{1} = -1
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⁡(x)cos⁡(x)x+1=0\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x + 1} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}
Numerical solution
x1=48.6946861306418x_{1} = 48.6946861306418
x2=81.6814089933346x_{2} = 81.6814089933346
x3=−14.1371669411541x_{3} = -14.1371669411541
x4=−86.3937979737193x_{4} = -86.3937979737193
x5=−1.5707963267949x_{5} = -1.5707963267949
x6=23.5619449019235x_{6} = 23.5619449019235
x7=59.6902604182061x_{7} = 59.6902604182061
x8=73.8274273593601x_{8} = 73.8274273593601
x9=4.71238898038469x_{9} = 4.71238898038469
x10=34.5575191894877x_{10} = 34.5575191894877
x11=−21.9911485751286x_{11} = -21.9911485751286
x12=−20.4203522483337x_{12} = -20.4203522483337
x13=15.707963267949x_{13} = 15.707963267949
x14=−95.8185759344887x_{14} = -95.8185759344887
x15=26.7035375555132x_{15} = 26.7035375555132
x16=−81.6814089933346x_{16} = -81.6814089933346
x17=20.4203522483337x_{17} = 20.4203522483337
x18=−94.2477796076938x_{18} = -94.2477796076938
x19=67.5442420521806x_{19} = 67.5442420521806
x20=−59.6902604182061x_{20} = -59.6902604182061
x21=36.1283155162826x_{21} = 36.1283155162826
x22=−43.9822971502571x_{22} = -43.9822971502571
x23=−370.707933123596x_{23} = -370.707933123596
x24=58.1194640914112x_{24} = 58.1194640914112
x25=−29.845130209103x_{25} = -29.845130209103
x26=−31.4159265358979x_{26} = -31.4159265358979
x27=−292.168116783851x_{27} = -292.168116783851
x28=12.5663706143592x_{28} = 12.5663706143592
x29=43.9822971502571x_{29} = 43.9822971502571
x30=7.85398163397448x_{30} = 7.85398163397448
x31=−15.707963267949x_{31} = -15.707963267949
x32=0x_{32} = 0
x33=6346.01716025138x_{33} = 6346.01716025138
x34=89.5353906273091x_{34} = 89.5353906273091
x35=−65.9734457253857x_{35} = -65.9734457253857
x36=−28.2743338823081x_{36} = -28.2743338823081
x37=51.8362787842316x_{37} = 51.8362787842316
x38=70.6858347057703x_{38} = 70.6858347057703
x39=−50.2654824574367x_{39} = -50.2654824574367
x40=80.1106126665397x_{40} = 80.1106126665397
x41=−75.398223686155x_{41} = -75.398223686155
x42=45.553093477052x_{42} = 45.553093477052
x43=14.1371669411541x_{43} = 14.1371669411541
x44=28.2743338823081x_{44} = 28.2743338823081
x45=65.9734457253857x_{45} = 65.9734457253857
x46=−67.5442420521806x_{46} = -67.5442420521806
x47=3702.36694225557x_{47} = 3702.36694225557
x48=42.4115008234622x_{48} = 42.4115008234622
x49=−45.553093477052x_{49} = -45.553093477052
x50=−58.1194640914112x_{50} = -58.1194640914112
x51=−87.9645943005142x_{51} = -87.9645943005142
x52=−6.28318530717959x_{52} = -6.28318530717959
x53=−83.2522053201295x_{53} = -83.2522053201295
x54=−97.3893722612836x_{54} = -97.3893722612836
x55=94.2477796076938x_{55} = 94.2477796076938
x56=−17.2787595947439x_{56} = -17.2787595947439
x57=95.8185759344887x_{57} = 95.8185759344887
x58=−39.2699081698724x_{58} = -39.2699081698724
x59=72.2566310325652x_{59} = 72.2566310325652
x60=−36.1283155162826x_{60} = -36.1283155162826
x61=1.5707963267949x_{61} = 1.5707963267949
x62=−9.42477796076938x_{62} = -9.42477796076938
x63=−64.4026493985908x_{63} = -64.4026493985908
x64=56.5486677646163x_{64} = 56.5486677646163
x65=92.6769832808989x_{65} = 92.6769832808989
x66=−51.8362787842316x_{66} = -51.8362787842316
x67=100.530964914873x_{67} = 100.530964914873
x68=−89.5353906273091x_{68} = -89.5353906273091
x69=6.28318530717959x_{69} = 6.28318530717959
x70=−61.261056745001x_{70} = -61.261056745001
x71=−53.4070751110265x_{71} = -53.4070751110265
x72=−73.8274273593601x_{72} = -73.8274273593601
x73=21.9911485751286x_{73} = 21.9911485751286
x74=29.845130209103x_{74} = 29.845130209103
x75=87.9645943005142x_{75} = 87.9645943005142
x76=−72.2566310325652x_{76} = -72.2566310325652
x77=37.6991118430775x_{77} = 37.6991118430775
x78=50.2654824574367x_{78} = 50.2654824574367
x79=86.3937979737193x_{79} = 86.3937979737193
x80=64.4026493985908x_{80} = 64.4026493985908
x81=−37.6991118430775x_{81} = -37.6991118430775
x82=−23.5619449019235x_{82} = -23.5619449019235
x83=78.5398163397448x_{83} = 78.5398163397448
x84=−42.4115008234622x_{84} = -42.4115008234622
x85=−109.955742875643x_{85} = -109.955742875643
x86=−10.9955742875643x_{86} = -10.9955742875643
x87=−80.1106126665397x_{87} = -80.1106126665397
x88=−7.85398163397448x_{88} = -7.85398163397448
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (sin(x)*cos(x))/(x + 1).
sin⁡(0)cos⁡(0)1\frac{\sin{\left(0 \right)} \cos{\left(0 \right)}}{1}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
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
−sin⁡2(x)+cos⁡2(x)x+1−sin⁡(x)cos⁡(x)(x+1)2=0\frac{- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{x + 1} - \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\left(x + 1\right)^{2}} = 0
Solve this equation
The roots of this equation
x1=−80.8928816808707x_{1} = -80.8928816808707
x2=−99.7430349489701x_{2} = -99.7430349489701
x3=−63.6132585554971x_{3} = -63.6132585554971
x4=−47.9039581285518x_{4} = -47.9039581285518
x5=−98.172197695036x_{5} = -98.172197695036
x6=84.0346635398793x_{6} = 84.0346635398793
x7=46.3332101330021x_{7} = 46.3332101330021
x8=38.478177732588x_{8} = 38.478177732588
x9=19.6228339741551x_{9} = 19.6228339741551
x10=82.4638118824473x_{10} = 82.4638118824473
x11=16.4790625040945x_{11} = 16.4790625040945
x12=41.6202371710741x_{12} = 41.6202371710741
x13=−46.33297711484x_{13} = -46.33297711484
x14=66.7551541995631x_{14} = 66.7551541995631
x15=−57.3296278828154x_{15} = -57.3296278828154
x16=62.0424894121024x_{16} = 62.0424894121024
x17=33.7649303669424x_{17} = 33.7649303669424
x18=8.61339783522102x_{18} = 8.61339783522102
x19=−5.44173882723211x_{19} = -5.44173882723211
x20=18.0510381254578x_{20} = 18.0510381254578
x21=−33.76449140679x_{21} = -33.76449140679
x22=−62.042359483332x_{22} = -62.042359483332
x23=98.1722495795572x_{23} = 98.1722495795572
x24=32.1937937404494x_{24} = 32.1937937404494
x25=55.7588651168628x_{25} = 55.7588651168628
x26=−25.9081038458568x_{26} = -25.9081038458568
x27=24.3374775388136x_{27} = 24.3374775388136
x28=−82.4637383451864x_{28} = -82.4637383451864
x29=2.28057021563236x_{29} = 2.28057021563236
x30=−60.471454983256x_{30} = -60.471454983256
x31=85.6055131901373x_{31} = 85.6055131901373
x32=74.6095191089778x_{32} = 74.6095191089778
x33=40.049216384194x_{33} = 40.049216384194
x34=−27.4794955733025x_{34} = -27.4794955733025
x35=−11.7577501031099x_{35} = -11.7577501031099
x36=187.70883626286x_{36} = 187.70883626286
x37=25.9088498373569x_{37} = 25.9088498373569
x38=27.4801585795776x_{38} = 27.4801585795776
x39=−13.3315066037056x_{39} = -13.3315066037056
x40=63.6133821448946x_{40} = 63.6133821448946
x41=99.743085211956x_{41} = 99.743085211956
x42=44.762232510841x_{42} = 44.762232510841
x43=−19.6215319912886x_{43} = -19.6215319912886
x44=−3.83985112537054x_{44} = -3.83985112537054
x45=60.4715917520353x_{45} = 60.4715917520353
x46=−16.4772142695041x_{46} = -16.4772142695041
x47=11.7613921271159x_{47} = 11.7613921271159
x48=−49.4749271716005x_{48} = -49.4749271716005
x49=0.637196330969125x_{49} = 0.637196330969125
x50=30.6226232987428x_{50} = 30.6226232987428
x51=−38.4778397936073x_{51} = -38.4778397936073
x52=68.3260341292281x_{52} = 68.3260341292281
x53=−79.322022596248x_{53} = -79.322022596248
x54=−90.317989831739x_{54} = -90.317989831739
x55=−41.6199483594113x_{55} = -41.6199483594113
x56=−84.0345927265613x_{56} = -84.0345927265613
x57=−85.6054449521595x_{57} = -85.6054449521595
x58=88.7472068907202x_{58} = 88.7472068907202
x59=76.1803827297937x_{59} = 76.1803827297937
x60=−69.8968079909424x_{60} = -69.8968079909424
x61=22.7660290738215x_{61} = 22.7660290738215
x62=96.6014126819581x_{62} = 96.6014126819581
x63=54.1879434234347x_{63} = 54.1879434234347
x64=−93.4596775892801x_{64} = -93.4596775892801
x65=−54.1877730844831x_{65} = -54.1877730844831
x66=3.87589679173726x_{66} = 3.87589679173726
x67=−68.3259270041136x_{67} = -68.3259270041136
x68=91.8888937560759x_{68} = 91.8888937560759
x69=69.8969103540797x_{69} = 69.8969103540797
x70=10.1878453044909x_{70} = 10.1878453044909
x71=−10.1829786980484x_{71} = -10.1829786980484
x72=−40.0489044548563x_{72} = -40.0489044548563
x73=−77.7511609427056x_{73} = -77.7511609427056
x74=−71.4676852032561x_{74} = -71.4676852032561
x75=−18.0494987719381x_{75} = -18.0494987719381
x76=52.6170143837707x_{76} = 52.6170143837707
x77=90.3180511337085x_{77} = 90.3180511337085
x78=−24.3366319328506x_{78} = -24.3366319328506
x79=−76.1802965592094x_{79} = -76.1802965592094
x80=47.9041761074919x_{80} = 47.9041761074919
x81=−55.7587042438767x_{81} = -55.7587042438767
x82=77.7512436659628x_{82} = 77.7512436659628
x83=−35.3356368025816x_{83} = -35.3356368025816
x84=−44.7619828411789x_{84} = -44.7619828411789
x85=−91.8888345323567x_{85} = -91.8888345323567
x86=−2.15134433588925x_{86} = -2.15134433588925
x87=−32.1933108467876x_{87} = -32.1933108467876
The values of the extrema at the points:
(-80.89288168087066, -0.00625825728073336)

(-99.74303494897006, -0.00506358337322892)

(-63.61325855549712, 0.00798527452696611)

(-47.90395812855178, 0.0106594755120283)

(-98.172197695036, 0.00514543658512232)

(84.03466353987932, -0.00587985341438749)

(46.33321013300211, -0.0105628184636869)

(38.47817773258804, 0.0126642092304746)

(19.622833974155125, 0.0242378477533134)

(82.46381188244735, 0.00599051273937728)

(16.479062504094543, 0.0285939565978739)

(41.620237171074066, 0.011730708921835)

(-46.33297711483998, -0.0110288276247335)

(66.75515419956312, 0.00737931146584952)

(-57.32962788281537, 0.00887597384673406)

(62.04248941210236, -0.00793090944828029)

(33.76493036694237, -0.0143808225713452)

(8.61339783522102, -0.0519405415328574)

(-5.441738827232107, -0.111862019279856)

(18.05103812545779, -0.0262362545191566)

(-33.76449140678997, -0.0152586464362218)

(-62.04235948333203, -0.00819075854569188)

(98.17224957955716, 0.00504166888895891)

(32.19379374044943, 0.0150613482038312)

(55.758865116862815, -0.0088088547861832)

(-25.908103845856804, 0.0200697449426835)

(24.337477538813616, -0.0197297727754678)

(-82.46373834518636, 0.00613758455702892)

(2.2805702156323617, -0.150672544934396)

(-60.471454983255995, 0.00840709765728415)

(85.60551319013732, 0.00577320829852196)

(74.60951910897779, -0.00661277936376119)

(40.04921638419398, -0.0121795970061747)

(-27.479495573302465, -0.0188791695413614)

(-11.757750103109899, -0.0464279997867718)

(187.70883626286036, -0.00264957515947118)

(25.908849837356932, 0.0185780406697524)

(27.480158579577623, -0.0175533771136332)

(-13.331506603705582, 0.0405132573447961)

(63.61338214489455, 0.00773810337143086)

(99.74308521195597, -0.0049630586647441)

(44.762232510840974, 0.0109253882656147)

(-19.621531991288602, 0.0268409633779861)

(-3.8398511253705365, 0.173398481994393)

(60.47159175203532, 0.00813356945441207)

(-16.47721426950408, 0.0322887105600465)

(11.761392127115943, -0.0391506392825053)

(-49.4749271716005, -0.0103140620006304)

(0.637196330969125, 0.292082592805874)

(30.622623298742763, -0.015809488870913)

(-38.47783979360729, 0.0133400300463799)

(68.32603412922812, -0.00721211017589145)

(-79.32202259624798, 0.00638377042868822)

(-90.317989831739, -0.00559788869847007)

(-41.619948359411254, 0.0123082905137621)

(-84.03459272656133, -0.00602147754269582)

(-85.6054449521595, 0.00590968192557079)

(88.74720689072021, 0.00557111756370173)

(76.18038272979369, 0.00647819420500065)

(-69.89680799094245, 0.00725703916120515)

(22.766029073821525, 0.0210337781400242)

(96.60141268195807, -0.00512280944940889)

(54.18794342343466, 0.00905957804504919)

(-93.45967758928009, -0.00540768367597939)

(-54.18777308448308, 0.00940024143486964)

(3.8758967917372615, 0.102010295291554)

(-68.3259270041136, -0.00742635466561701)

(91.88889375607587, 0.00538269685829326)

(69.89691035407965, 0.00705231812713097)

(10.187845304490947, 0.0446467933531117)

(-10.182978698048352, 0.0543680322906561)

(-40.048904454856334, -0.0128034069967485)

(-77.75116094270562, -0.00651442120647222)

(-71.46768520325605, -0.0070952722893795)

(-18.049498771938094, -0.0293137725748063)

(52.61701438377071, -0.00932499256816396)

(90.31805113370845, -0.00547528663879163)

(-24.336631932850587, -0.021420626484021)

(-76.18029655920942, 0.00665053166012875)

(47.90417610749193, 0.0102235414010152)

(-55.7587042438767, -0.0091305878135904)

(77.75124366596276, -0.00634897811440877)

(-35.33563680258161, 0.0145605860581051)

(-44.76198284117895, 0.0114246964043575)

(-91.8888345323567, 0.0055011425426788)

(-2.1513443358892483, -0.398334456870323)

(-32.19331084678758, 0.0160270188053562)


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=−80.8928816808707x_{1} = -80.8928816808707
x2=−99.7430349489701x_{2} = -99.7430349489701
x3=84.0346635398793x_{3} = 84.0346635398793
x4=46.3332101330021x_{4} = 46.3332101330021
x5=−46.33297711484x_{5} = -46.33297711484
x6=62.0424894121024x_{6} = 62.0424894121024
x7=33.7649303669424x_{7} = 33.7649303669424
x8=8.61339783522102x_{8} = 8.61339783522102
x9=−5.44173882723211x_{9} = -5.44173882723211
x10=18.0510381254578x_{10} = 18.0510381254578
x11=−33.76449140679x_{11} = -33.76449140679
x12=−62.042359483332x_{12} = -62.042359483332
x13=55.7588651168628x_{13} = 55.7588651168628
x14=24.3374775388136x_{14} = 24.3374775388136
x15=2.28057021563236x_{15} = 2.28057021563236
x16=74.6095191089778x_{16} = 74.6095191089778
x17=40.049216384194x_{17} = 40.049216384194
x18=−27.4794955733025x_{18} = -27.4794955733025
x19=−11.7577501031099x_{19} = -11.7577501031099
x20=187.70883626286x_{20} = 187.70883626286
x21=27.4801585795776x_{21} = 27.4801585795776
x22=99.743085211956x_{22} = 99.743085211956
x23=11.7613921271159x_{23} = 11.7613921271159
x24=−49.4749271716005x_{24} = -49.4749271716005
x25=30.6226232987428x_{25} = 30.6226232987428
x26=68.3260341292281x_{26} = 68.3260341292281
x27=−90.317989831739x_{27} = -90.317989831739
x28=−84.0345927265613x_{28} = -84.0345927265613
x29=96.6014126819581x_{29} = 96.6014126819581
x30=−93.4596775892801x_{30} = -93.4596775892801
x31=−68.3259270041136x_{31} = -68.3259270041136
x32=−40.0489044548563x_{32} = -40.0489044548563
x33=−77.7511609427056x_{33} = -77.7511609427056
x34=−71.4676852032561x_{34} = -71.4676852032561
x35=−18.0494987719381x_{35} = -18.0494987719381
x36=52.6170143837707x_{36} = 52.6170143837707
x37=90.3180511337085x_{37} = 90.3180511337085
x38=−24.3366319328506x_{38} = -24.3366319328506
x39=−55.7587042438767x_{39} = -55.7587042438767
x40=77.7512436659628x_{40} = 77.7512436659628
x41=−2.15134433588925x_{41} = -2.15134433588925
Maxima of the function at points:
x41=−63.6132585554971x_{41} = -63.6132585554971
x41=−47.9039581285518x_{41} = -47.9039581285518
x41=−98.172197695036x_{41} = -98.172197695036
x41=38.478177732588x_{41} = 38.478177732588
x41=19.6228339741551x_{41} = 19.6228339741551
x41=82.4638118824473x_{41} = 82.4638118824473
x41=16.4790625040945x_{41} = 16.4790625040945
x41=41.6202371710741x_{41} = 41.6202371710741
x41=66.7551541995631x_{41} = 66.7551541995631
x41=−57.3296278828154x_{41} = -57.3296278828154
x41=98.1722495795572x_{41} = 98.1722495795572
x41=32.1937937404494x_{41} = 32.1937937404494
x41=−25.9081038458568x_{41} = -25.9081038458568
x41=−82.4637383451864x_{41} = -82.4637383451864
x41=−60.471454983256x_{41} = -60.471454983256
x41=85.6055131901373x_{41} = 85.6055131901373
x41=25.9088498373569x_{41} = 25.9088498373569
x41=−13.3315066037056x_{41} = -13.3315066037056
x41=63.6133821448946x_{41} = 63.6133821448946
x41=44.762232510841x_{41} = 44.762232510841
x41=−19.6215319912886x_{41} = -19.6215319912886
x41=−3.83985112537054x_{41} = -3.83985112537054
x41=60.4715917520353x_{41} = 60.4715917520353
x41=−16.4772142695041x_{41} = -16.4772142695041
x41=0.637196330969125x_{41} = 0.637196330969125
x41=−38.4778397936073x_{41} = -38.4778397936073
x41=−79.322022596248x_{41} = -79.322022596248
x41=−41.6199483594113x_{41} = -41.6199483594113
x41=−85.6054449521595x_{41} = -85.6054449521595
x41=88.7472068907202x_{41} = 88.7472068907202
x41=76.1803827297937x_{41} = 76.1803827297937
x41=−69.8968079909424x_{41} = -69.8968079909424
x41=22.7660290738215x_{41} = 22.7660290738215
x41=54.1879434234347x_{41} = 54.1879434234347
x41=−54.1877730844831x_{41} = -54.1877730844831
x41=3.87589679173726x_{41} = 3.87589679173726
x41=91.8888937560759x_{41} = 91.8888937560759
x41=69.8969103540797x_{41} = 69.8969103540797
x41=10.1878453044909x_{41} = 10.1878453044909
x41=−10.1829786980484x_{41} = -10.1829786980484
x41=−76.1802965592094x_{41} = -76.1802965592094
x41=47.9041761074919x_{41} = 47.9041761074919
x41=−35.3356368025816x_{41} = -35.3356368025816
x41=−44.7619828411789x_{41} = -44.7619828411789
x41=−91.8888345323567x_{41} = -91.8888345323567
x41=−32.1933108467876x_{41} = -32.1933108467876
Decreasing at intervals
[187.70883626286,∞)\left[187.70883626286, \infty\right)
Increasing at intervals
(−∞,−99.7430349489701]\left(-\infty, -99.7430349489701\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⁡(x)cos⁡(x)+sin⁡2(x)−cos⁡2(x)x+1+sin⁡(x)cos⁡(x)(x+1)2)x+1=0\frac{2 \left(- 2 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{x + 1} + \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\left(x + 1\right)^{2}}\right)}{x + 1} = 0
Solve this equation
The roots of this equation
x1=4.62300432054552x_{1} = 4.62300432054552
x2=695.862055267776x_{2} = 695.862055267776
x3=−20.3945604100041x_{3} = -20.3945604100041
x4=64.3950032565303x_{4} = 64.3950032565303
x5=−94.2424171385515x_{5} = -94.2424171385515
x6=−15.6738627393008x_{6} = -15.6738627393008
x7=28.2572407694131x_{7} = 28.2572407694131
x8=23.5415656686274x_{8} = 23.5415656686274
x9=−31.3994759185537x_{9} = -31.3994759185537
x10=−17.2479671086262x_{10} = -17.2479671086262
x11=37.6861858939753x_{11} = 37.6861858939753
x12=58.1110050261765x_{12} = 58.1110050261765
x13=83.24627019992x_{13} = 83.24627019992
x14=−87.9588443273669x_{14} = -87.9588443273669
x15=−7.77996936241974x_{15} = -7.77996936241974
x16=−14.0989589564501x_{16} = -14.0989589564501
x17=−72.2496132202303x_{17} = -72.2496132202303
x18=92.6716453849436x_{18} = 92.6716453849436
x19=−75.3915022859684x_{19} = -75.3915022859684
x20=42.3999790613261x_{20} = 42.3999790613261
x21=−89.5297426877915x_{21} = -89.5297426877915
x22=−28.2559851737141x_{22} = -28.2559851737141
x23=−21.9672928762567x_{23} = -21.9672928762567
x24=−65.9657490582878x_{24} = -65.9657490582878
x25=43.971178001924x_{25} = 43.971178001924
x26=48.6846219752982x_{26} = 48.6846219752982
x27=−86.3879422121676x_{27} = -86.3879422121676
x28=65.965978970039x_{28} = 65.965978970039
x29=−9.36486234975728x_{29} = -9.36486234975728
x30=−29.82778235232x_{30} = -29.82778235232
x31=−23.539754601238x_{31} = -23.539754601238
x32=78.533529517364x_{32} = 78.533529517364
x33=31.4004922222866x_{33} = 31.4004922222866
x34=−67.5367271225522x_{34} = -67.5367271225522
x35=20.3969759591125x_{35} = 20.3969759591125
x36=17.2513506676003x_{36} = 17.2513506676003
x37=12.5293804122395x_{37} = 12.5293804122395
x38=95.8134112689315x_{38} = 95.8134112689315
x39=−43.97066025381x_{39} = -43.97066025381
x40=45.5423497465057x_{40} = 45.5423497465057
x41=−4.57056886322693x_{41} = -4.57056886322693
x42=15.6779656482954x_{42} = 15.6779656482954
x43=81.6753610948482x_{43} = 81.6753610948482
x44=−58.1107087185846x_{44} = -58.1107087185846
x45=67.53694645812x_{45} = 67.53694645812
x46=−80.1042917286259x_{46} = -80.1042917286259
x47=−97.3841845951523x_{47} = -97.3841845951523
x48=89.5298674758724x_{48} = 89.5298674758724
x49=−50.2553305746095x_{49} = -50.2553305746095
x50=−81.6752111440063x_{50} = -81.6752111440063
x51=73.8207445217556x_{51} = 73.8207445217556
x52=26.6854736175187x_{52} = 26.6854736175187
x53=70.678858922031x_{53} = 70.678858922031
x54=59.6820203723766x_{54} = 59.6820203723766
x55=−32.9710811801128x_{55} = -32.9710811801128
x56=14.1040390604271x_{56} = 14.1040390604271
x57=56.5399777166298x_{57} = 56.5399777166298
x58=−83.246125856646x_{58} = -83.246125856646
x59=50.2557268311419x_{59} = 50.2557268311419
x60=34.543450044636x_{60} = 34.543450044636
x61=−36.1140742873842x_{61} = -36.1140742873842
x62=−5271.59237785768x_{62} = -5271.59237785768
x63=−59.6817394676977x_{63} = -59.6817394676977
x64=94.2425297556831x_{64} = 94.2425297556831
x65=−62.8237652143331x_{65} = -62.8237652143331
x66=−73.8205609509248x_{66} = -73.8205609509248
x67=−6.18618465671037x_{67} = -6.18618465671037
x68=−12.5229245089216x_{68} = -12.5229245089216
x69=−64.3947619845809x_{69} = -64.3947619845809
x70=−51.8264407501884x_{70} = -51.8264407501884
x71=21.969373585987x_{71} = 21.969373585987
x72=87.9589736137015x_{72} = 87.9589736137015
x73=−45.5418671403736x_{73} = -45.5418671403736
x74=36.1148421872068x_{74} = 36.1148421872068
x75=−2.86125249133971x_{75} = -2.86125249133971
x76=−53.3975320976564x_{76} = -53.3975320976564
x77=80.104447620322x_{77} = 80.104447620322
x78=−42.3994221859646x_{78} = -42.3994221859646
x79=7.79702228789617x_{79} = 7.79702228789617
x80=6.21365157224329x_{80} = 6.21365157224329
x81=72.2498048637954x_{81} = 72.2498048637954
x82=51.8268133285914x_{82} = 51.8268133285914
x83=86.3880762444708x_{83} = 86.3880762444708
x84=−95.8133023150968x_{84} = -95.8133023150968
x85=−61.2527579887425x_{85} = -61.2527579887425
x86=29.8289088223657x_{86} = 29.8289088223657
x87=−39.2568371226105x_{87} = -39.2568371226105
x88=100.526039990347x_{88} = 100.526039990347
x89=1.35219346515022x_{89} = 1.35219346515022
x90=−100.525941014496x_{90} = -100.525941014496
x91=−37.6854807853566x_{91} = -37.6854807853566
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=−1x_{1} = -1

lim⁡x→−1−(2(−2sin⁡(x)cos⁡(x)+sin⁡2(x)−cos⁡2(x)x+1+sin⁡(x)cos⁡(x)(x+1)2)x+1)=∞\lim_{x \to -1^-}\left(\frac{2 \left(- 2 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{x + 1} + \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\left(x + 1\right)^{2}}\right)}{x + 1}\right) = \infty
lim⁡x→−1+(2(−2sin⁡(x)cos⁡(x)+sin⁡2(x)−cos⁡2(x)x+1+sin⁡(x)cos⁡(x)(x+1)2)x+1)=−∞\lim_{x \to -1^+}\left(\frac{2 \left(- 2 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{x + 1} + \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\left(x + 1\right)^{2}}\right)}{x + 1}\right) = -\infty
- the limits are not equal, so
x1=−1x_{1} = -1
- is an inflection 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
[695.862055267776,∞)\left[695.862055267776, \infty\right)
Convex at the intervals
(−∞,−5271.59237785768]\left(-\infty, -5271.59237785768\right]
Vertical asymptotes
Have:
x1=−1x_{1} = -1
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(sin⁡(x)cos⁡(x)x+1)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x + 1}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(sin⁡(x)cos⁡(x)x+1)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x + 1}\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(x)*cos(x))/(x + 1), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x)cos⁡(x)x(x+1))=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x \left(x + 1\right)}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(x)cos⁡(x)x(x+1))=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x \left(x + 1\right)}\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⁡(x)cos⁡(x)x+1=−sin⁡(x)cos⁡(x)1−x\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x + 1} = - \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{1 - x}
- No
sin⁡(x)cos⁡(x)x+1=sin⁡(x)cos⁡(x)1−x\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x + 1} = \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{1 - x}
- No
so, the function
not is
neither even, nor odd
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