Mister Exam

Other calculators

  • How to use it?

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

  • Identical expressions

  • (two ^arcctgx)*sin(5x)
  • (2 to the power of arcctgx) multiply by sinus of (5x)
  • (two to the power of arcctgx) multiply by sinus of (5x)
  • (2arcctgx)*sin(5x)
  • 2arcctgx*sin5x
  • (2^arcctgx)sin(5x)
  • (2arcctgx)sin(5x)
  • 2arcctgxsin5x
  • 2^arcctgxsin5x
Plotting a function graph/ (2^arcctgx)*sin(5x)

Graphing y = (2^arcctgx)*sin(5x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
        acot(x)         
f(x) = 2       *sin(5*x)
f(x)=2acot⁡(x)sin⁡(5x)f{\left(x \right)} = 2^{\operatorname{acot}{\left(x \right)}} \sin{\left(5 x \right)} 
f = 2^acot(x)*sin(5*x)
The graph of the function
02468-8-6-4-2-10105-5
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:
2acot⁡(x)sin⁡(5x)=02^{\operatorname{acot}{\left(x \right)}} \sin{\left(5 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=−4π5x_{2} = - \frac{4 \pi}{5}
x3=−3π5x_{3} = - \frac{3 \pi}{5}
x4=−2π5x_{4} = - \frac{2 \pi}{5}
x5=−π5x_{5} = - \frac{\pi}{5}
x6=π5x_{6} = \frac{\pi}{5}
x7=2π5x_{7} = \frac{2 \pi}{5}
x8=3π5x_{8} = \frac{3 \pi}{5}
x9=4π5x_{9} = \frac{4 \pi}{5}
x10=πx_{10} = \pi
Numerical solution
x1=−71.6283125018473x_{1} = -71.6283125018473
x2=−57.8053048260522x_{2} = -57.8053048260522
x3=−86.0796387083603x_{3} = -86.0796387083603
x4=−10.0530964914873x_{4} = -10.0530964914873
x5=89.2212313619501x_{5} = 89.2212313619501
x6=−69.7433569096934x_{6} = -69.7433569096934
x7=−11.9380520836412x_{7} = -11.9380520836412
x8=33.9292006587698x_{8} = 33.9292006587698
x9=38.3274303737955x_{9} = 38.3274303737955
x10=42.0973415581032x_{10} = 42.0973415581032
x11=−21.9911485751286x_{11} = -21.9911485751286
x12=40.2123859659494x_{12} = 40.2123859659494
x13=−49.6371639267187x_{13} = -49.6371639267187
x14=−81.6814089933346x_{14} = -81.6814089933346
x15=55.9203492338983x_{15} = 55.9203492338983
x16=96.1327351998477x_{16} = 96.1327351998477
x17=−96.1327351998477x_{17} = -96.1327351998477
x18=−59.6902604182061x_{18} = -59.6902604182061
x19=−43.9822971502571x_{19} = -43.9822971502571
x20=64.0884901332318x_{20} = 64.0884901332318
x21=99.9026463841554x_{21} = 99.9026463841554
x22=52.1504380495906x_{22} = 52.1504380495906
x23=43.9822971502571x_{23} = 43.9822971502571
x24=18.2212373908208x_{24} = 18.2212373908208
x25=16.3362817986669x_{25} = 16.3362817986669
x26=1.88495559215388x_{26} = 1.88495559215388
x27=−15.707963267949x_{27} = -15.707963267949
x28=48.3805268652828x_{28} = 48.3805268652828
x29=−64.0884901332318x_{29} = -64.0884901332318
x30=0x_{30} = 0
x31=−54.0353936417444x_{31} = -54.0353936417444
x32=−47.7522083345649x_{32} = -47.7522083345649
x33=−65.9734457253857x_{33} = -65.9734457253857
x34=−32.0442450666159x_{34} = -32.0442450666159
x35=70.3716754404114x_{35} = 70.3716754404114
x36=11.9380520836412x_{36} = 11.9380520836412
x37=705.601709996268x_{37} = 705.601709996268
x38=49.0088453960008x_{38} = 49.0088453960008
x39=30.159289474462x_{39} = 30.159289474462
x40=10.0530964914873x_{40} = 10.0530964914873
x41=−23.8761041672824x_{41} = -23.8761041672824
x42=92.3628240155399x_{42} = 92.3628240155399
x43=−93.6194610769758x_{43} = -93.6194610769758
x44=−35.8141562509236x_{44} = -35.8141562509236
x45=74.1415866247191x_{45} = 74.1415866247191
x46=−3.76991118430775x_{46} = -3.76991118430775
x47=26.3893782901543x_{47} = 26.3893782901543
x48=28.2743338823081x_{48} = 28.2743338823081
x49=−8.79645943005142x_{49} = -8.79645943005142
x50=65.9734457253857x_{50} = 65.9734457253857
x51=84.1946831162065x_{51} = 84.1946831162065
x52=67.8584013175395x_{52} = 67.8584013175395
x53=20.1061929829747x_{53} = 20.1061929829747
x54=−91.734505484822x_{54} = -91.734505484822
x55=82.3097275240526x_{55} = 82.3097275240526
x56=−87.9645943005142x_{56} = -87.9645943005142
x57=−33.9292006587698x_{57} = -33.9292006587698
x58=−20.1061929829747x_{58} = -20.1061929829747
x59=−77.9114978090269x_{59} = -77.9114978090269
x60=−5.65486677646163x_{60} = -5.65486677646163
x61=94.2477796076938x_{61} = 94.2477796076938
x62=−67.8584013175395x_{62} = -67.8584013175395
x63=98.0176907920015x_{63} = 98.0176907920015
x64=32.0442450666159x_{64} = 32.0442450666159
x65=−37.0707933123596x_{65} = -37.0707933123596
x66=72.2566310325652x_{66} = 72.2566310325652
x67=−76.026542216873x_{67} = -76.026542216873
x68=−25.7610597594363x_{68} = -25.7610597594363
x69=−98.0176907920015x_{69} = -98.0176907920015
x70=−89.8495498926681x_{70} = -89.8495498926681
x71=−13.8230076757951x_{71} = -13.8230076757951
x72=−45.867252742411x_{72} = -45.867252742411
x73=−74.1415866247191x_{73} = -74.1415866247191
x74=8.16814089933346x_{74} = 8.16814089933346
x75=6.28318530717959x_{75} = 6.28318530717959
x76=23.8761041672824x_{76} = 23.8761041672824
x77=21.9911485751286x_{77} = 21.9911485751286
x78=87.9645943005142x_{78} = 87.9645943005142
x79=54.0353936417444x_{79} = 54.0353936417444
x80=50.2654824574367x_{80} = 50.2654824574367
x81=−1.88495559215388x_{81} = -1.88495559215388
x82=−55.9203492338983x_{82} = -55.9203492338983
x83=76.026542216873x_{83} = 76.026542216873
x84=−99.9026463841554x_{84} = -99.9026463841554
x85=62.2035345410779x_{85} = 62.2035345410779
x86=79.7964534011807x_{86} = 79.7964534011807
x87=45.867252742411x_{87} = 45.867252742411
x88=−79.7964534011807x_{88} = -79.7964534011807
x89=−37.6991118430775x_{89} = -37.6991118430775
x90=−52.7787565803085x_{90} = -52.7787565803085
x91=5.02654824574367x_{91} = 5.02654824574367
x92=60.318578948924x_{92} = 60.318578948924
x93=77.9114978090269x_{93} = 77.9114978090269
x94=−27.6460153515902x_{94} = -27.6460153515902
x95=86.0796387083603x_{95} = 86.0796387083603
x96=−42.0973415581032x_{96} = -42.0973415581032
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2^acot(x)*sin(5*x).
2acot⁡(0)sin⁡(0⋅5)2^{\operatorname{acot}{\left(0 \right)}} \sin{\left(0 \cdot 5 \right)}
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
5⋅2acot⁡(x)cos⁡(5x)−2acot⁡(x)log⁡(2)sin⁡(5x)x2+1=05 \cdot 2^{\operatorname{acot}{\left(x \right)}} \cos{\left(5 x \right)} - \frac{2^{\operatorname{acot}{\left(x \right)}} \log{\left(2 \right)} \sin{\left(5 x \right)}}{x^{2} + 1} = 0
Solve this equation
The roots of this equation
x1=12.2520278702219x_{1} = 12.2520278702219
x2=−19.792104315915x_{2} = -19.792104315915
x3=14.1370289022608x_{3} = 14.1370289022608
x4=−7.85442388901902x_{4} = -7.85442388901902
x5=81.9955641354391x_{5} = 81.9955641354391
x6=−70.0575218229581x_{6} = -70.0575218229581
x7=21.0486083395372x_{7} = 21.0486083395372
x8=−71.9424771230855x_{8} = -71.9424771230855
x9=34.243336299603x_{9} = 34.243336299603
x10=−60.0044273819258x_{10} = -60.0044273819258
x11=76.3406967256093x_{11} = 76.3406967256093
x12=−43.6681524169898x_{12} = -43.6681524169898
x13=36.1282942907824x_{13} = 36.1282942907824
x14=2.19434789945819x_{14} = 2.19434789945819
x15=4.08250110523468x_{15} = 4.08250110523468
x16=48.066355604513x_{16} = 48.066355604513
x17=7.22514196736028x_{17} = 7.22514196736028
x18=24.1902161322856x_{18} = 24.1902161322856
x19=−50.5796525561773x_{19} = -50.5796525561773
x20=54.3495435239968x_{20} = 54.3495435239968
x21=−103.358400898199x_{21} = -103.358400898199
x22=−61.8893825124058x_{22} = -61.8893825124058
x23=56.2344997344481x_{23} = 56.2344997344481
x24=88.9070685894122x_{24} = 88.9070685894122
x25=−29.8451613012199x_{25} = -29.8451613012199
x26=−38.0132902824965x_{26} = -38.0132902824965
x27=−14.137304974683x_{27} = -14.137304974683
x28=39.8982092943127x_{28} = 39.8982092943127
x29=−95.8185789540142x_{29} = -95.8185789540142
x30=0.288704878724942x_{30} = 0.288704878724942
x31=92.0486614782868x_{31} = 92.0486614782868
x32=93.9336172004219x_{32} = 93.9336172004219
x33=−36.128336741733x_{33} = -36.128336741733
x34=80.1106083469975x_{34} = 80.1106083469975
x35=46.1813990136254x_{35} = 46.1813990136254
x36=−11.624096505875x_{36} = -11.624096505875
x37=65.0309613747656x_{37} = 65.0309613747656
x38=41.7831664206562x_{38} = 41.7831664206562
x39=78.2256525441992x_{39} = 78.2256525441992
x40=−21.6770481889691x_{40} = -21.6770481889691
x41=−39.8982441068377x_{41} = -39.8982441068377
x42=5.9682689261015x_{42} = 5.9682689261015
x43=−49.9513342996013x_{43} = -49.9513342996013
x44=−97.703534430795x_{44} = -97.703534430795
x45=−63.7743376832009x_{45} = -63.7743376832009
x46=−9.73922648158337x_{46} = -9.73922648158337
x47=−5.9697827840821x_{47} = -5.9697827840821
x48=32.3583778776069x_{48} = 32.3583778776069
x49=58.1194558857455x_{49} = 58.1194558857455
x50=17.9069919293091x_{50} = 17.9069919293091
x51=71.9424664113254x_{51} = 71.9424664113254
x52=−92.0486680220746x_{52} = -92.0486680220746
x53=100.216802889184x_{53} = 100.216802889184
x54=90.163705747916x_{54} = 90.163705747916
x55=−381.703507601456x_{55} = -381.703507601456
x56=−0.956925542610824x_{56} = -0.956925542610824
x57=−53.7212439801803x_{57} = -53.7212439801803
x58=−53.0929256780349x_{58} = -53.0929256780349
x59=−69.4292033949059x_{59} = -69.4292033949059
x60=−85.7654832117861x_{60} = -85.7654832117861
x61=−73.8274324452886x_{61} = -73.8274324452886
x62=−93.9336234842473x_{62} = -93.9336234842473
x63=68.1725546184145x_{63} = 68.1725546184145
x64=−26.0752597431184x_{64} = -26.0752597431184
x65=16.0220149456067x_{65} = 16.0220149456067
x66=83.8805199107995x_{66} = 83.8805199107995
x67=−80.1106169860811x_{67} = -80.1106169860811
x68=−16.0222301181307x_{68} = -16.0222301181307
x69=49.9513120845443x_{69} = 49.9513120845443
x70=26.0751783062182x_{70} = 26.0751783062182
x71=−76.9690246922423x_{71} = -76.9690246922423
x72=70.0575105271449x_{72} = 70.0575105271449
x73=38.0132519343378x_{73} = 38.0132519343378
x74=−65.6592928897506x_{74} = -65.6592928897506
x75=−51.8362890989184x_{75} = -51.8362890989184
x76=27.9601391966996x_{76} = 27.9601391966996
x77=10.367000159236x_{77} = 10.367000159236
x78=−83.8805277908947x_{78} = -83.8805277908947
x79=98.3318471901991x_{79} = 98.3318471901991
x80=66.2875986823035x_{80} = 66.2875986823035
x81=88.2787500085987x_{81} = 88.2787500085987
x82=−87.6504386436056x_{82} = -87.6504386436056
x83=−31.7301133125533x_{83} = -31.7301133125533
x84=−75.7123877873973x_{84} = -75.7123877873973
x85=−48.0663795953227x_{85} = -48.0663795953227
x86=61.8893680390286x_{86} = 61.8893680390286
x87=−81.9955723819473x_{87} = -81.9955723819473
x88=−58.1194722970723x_{88} = -58.1194722970723
x89=−41.7831981648082x_{89} = -41.7831981648082
x90=22.3052522245671x_{90} = 22.3052522245671
x91=72.5707850343669x_{91} = 72.5707850343669
x92=44.2964422926316x_{92} = 44.2964422926316
x93=−4.08563752177751x_{93} = -4.08563752177751
x94=−27.9602100370195x_{94} = -27.9602100370195
x95=−17.9071643199601x_{95} = -17.9071643199601
x96=−33.6150659084925x_{96} = -33.6150659084925
x97=60.0044119852004x_{97} = 60.0044119852004
x98=−0.338901775254251x_{98} = -0.338901775254251
The values of the extrema at the points:
(12.252027870221896, -1.05807215521277)

(-19.792104315915044, 0.965613425069861)

(14.13702890226082, 1.05016662476151)

(-7.854423889019023, -0.915962849245845)

(81.99556413543907, 1.00848888033338)

(-70.05752182295805, 0.990155476880893)

(21.04860833953717, -1.03345338269864)

(-71.9424771230855, -0.990412137904529)

(34.24333629960299, 1.02044219108894)

(-60.004427381925794, 0.988515919196205)

(76.34069672560932, -1.00912047570885)

(-43.66815241698978, 0.984254981133367)

(36.128294290782385, -1.01936594620173)

(2.194347899458193, -1.3446097808291)

(4.082501105234678, 1.18113485380654)

(48.066355604513014, 1.01452299791628)

(7.225141967360282, -1.10001785215559)

(24.190216132285567, 1.02905169778141)

(-50.57965255617731, -0.986391161113446)

(54.349543523996836, 1.01283371940802)

(-103.35840089819906, -0.99331639538537)

(-61.88938251240584, -0.988863671662468)

(56.23449973444807, -1.01240097538847)

(88.90706858941216, -1.00782645016034)

(-29.84516130121995, 0.977051317723769)

(-38.01329028249655, -0.98193502580247)

(-14.137304974682984, -0.952230296384834)

(39.898209294312686, -1.01752097179299)

(-95.81857895401419, -0.9927924088039)

(0.288704878724942, 2.42505555198911)

(92.0486614782868, 1.00755835037893)

(93.9336172004219, -1.00740612914916)

(-36.128336741733015, 0.981001979700351)

(80.11060834699747, -1.00868946344935)

(46.18139901362537, -1.01512005004677)

(-11.624096505874975, -0.94225036255822)

(65.03096137476564, -1.01071488274638)

(41.78316642065618, 1.01672429070752)

(78.22565254419919, 1.00889975450513)

(-21.677048188969135, -0.968551641768527)

(-39.89824410683771, 0.982780734054391)

(5.968268926101499, -1.1219437698845)

(-49.95133429960127, 0.986221210678361)

(-97.70353443079502, 0.992930959572939)

(-63.77433768320094, 0.989190984254599)

(-9.739226481583373, 0.931533462178175)

(-5.969782784082104, 0.891323032921505)

(32.35837787760692, -1.02164505316667)

(58.11945588574545, 1.01199645997223)

(17.906991929309118, 1.03942526502092)

(71.94246641132544, 1.00968067983586)

(-92.04866802207461, -0.992498349987912)

(100.21680288918364, -1.00694021952127)

(90.16370574791597, -1.00771695995662)

(-381.7035076014561, 0.99818572108688)

(-0.9569255426108237, 0.569918099668127)

(-53.72124398018026, 0.987181687879379)

(-53.09292567803487, -0.987031015269614)

(-69.42920339490585, -0.990066842329396)

(-85.76548321178613, -0.991951043994481)

(-73.82743244528864, 0.990655756825763)

(-93.9336234842473, 0.992648318599545)

(68.17255461841451, 1.01021866807885)

(-26.075259743118448, 0.973780304431222)

(16.022014945606745, -1.04415295032884)

(83.88051991079955, -1.008297348118)

(-80.11061698608106, 0.991385393326929)

(-16.022230118130654, 0.957714374177887)

(49.95131208454428, -1.01397130000539)

(26.075178306218156, -1.02692572123226)

(-76.96902469224233, -0.99103539606095)

(70.05751052714491, -1.00994240212415)

(38.01325193433779, 1.01839733069293)

(-65.65929288975057, -0.989499607464085)

(-51.836289098918435, -0.98671878927279)

(27.96013919669959, 1.02508954569211)

(10.36700015923604, 1.06892544311491)

(-83.88052779089466, 0.991770931714354)

(98.33184719019911, 1.00707371920741)

(66.28759868230352, -1.01051072450733)

(88.2787500085987, 1.00788236863702)

(-87.65043864360563, 0.992123441171333)

(-31.73011331255335, -0.978398848786504)

(-75.7123877873973, -0.990887303041292)

(-48.066379595322694, -0.985684903793344)

(61.88936803902863, 1.01126174412703)

(-81.99557238194731, -0.991582574608515)

(-58.11947229707227, -0.988145750737867)

(-41.78319816480822, -0.983550816517019)

(22.30525222456711, -1.03154190750363)

(72.57078503436693, -1.00959647424186)

(44.29644229263163, 1.01576828779445)

(-4.085637521777506, -0.846695432297942)

(-27.960210037019475, -0.975524563260983)

(-17.90716431996009, -0.96207030884588)

(-33.615065908492504, 0.979596954857131)

(60.00441198520036, -1.01161749857782)

(-0.3389017752542513, -0.418961352736205)


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=12.2520278702219x_{1} = 12.2520278702219
x2=−7.85442388901902x_{2} = -7.85442388901902
x3=21.0486083395372x_{3} = 21.0486083395372
x4=−71.9424771230855x_{4} = -71.9424771230855
x5=76.3406967256093x_{5} = 76.3406967256093
x6=36.1282942907824x_{6} = 36.1282942907824
x7=2.19434789945819x_{7} = 2.19434789945819
x8=7.22514196736028x_{8} = 7.22514196736028
x9=−50.5796525561773x_{9} = -50.5796525561773
x10=−103.358400898199x_{10} = -103.358400898199
x11=−61.8893825124058x_{11} = -61.8893825124058
x12=56.2344997344481x_{12} = 56.2344997344481
x13=88.9070685894122x_{13} = 88.9070685894122
x14=−38.0132902824965x_{14} = -38.0132902824965
x15=−14.137304974683x_{15} = -14.137304974683
x16=39.8982092943127x_{16} = 39.8982092943127
x17=−95.8185789540142x_{17} = -95.8185789540142
x18=93.9336172004219x_{18} = 93.9336172004219
x19=80.1106083469975x_{19} = 80.1106083469975
x20=46.1813990136254x_{20} = 46.1813990136254
x21=−11.624096505875x_{21} = -11.624096505875
x22=65.0309613747656x_{22} = 65.0309613747656
x23=−21.6770481889691x_{23} = -21.6770481889691
x24=5.9682689261015x_{24} = 5.9682689261015
x25=32.3583778776069x_{25} = 32.3583778776069
x26=−92.0486680220746x_{26} = -92.0486680220746
x27=100.216802889184x_{27} = 100.216802889184
x28=90.163705747916x_{28} = 90.163705747916
x29=−53.0929256780349x_{29} = -53.0929256780349
x30=−69.4292033949059x_{30} = -69.4292033949059
x31=−85.7654832117861x_{31} = -85.7654832117861
x32=16.0220149456067x_{32} = 16.0220149456067
x33=83.8805199107995x_{33} = 83.8805199107995
x34=49.9513120845443x_{34} = 49.9513120845443
x35=26.0751783062182x_{35} = 26.0751783062182
x36=−76.9690246922423x_{36} = -76.9690246922423
x37=70.0575105271449x_{37} = 70.0575105271449
x38=−65.6592928897506x_{38} = -65.6592928897506
x39=−51.8362890989184x_{39} = -51.8362890989184
x40=66.2875986823035x_{40} = 66.2875986823035
x41=−31.7301133125533x_{41} = -31.7301133125533
x42=−75.7123877873973x_{42} = -75.7123877873973
x43=−48.0663795953227x_{43} = -48.0663795953227
x44=−81.9955723819473x_{44} = -81.9955723819473
x45=−58.1194722970723x_{45} = -58.1194722970723
x46=−41.7831981648082x_{46} = -41.7831981648082
x47=22.3052522245671x_{47} = 22.3052522245671
x48=72.5707850343669x_{48} = 72.5707850343669
x49=−4.08563752177751x_{49} = -4.08563752177751
x50=−27.9602100370195x_{50} = -27.9602100370195
x51=−17.9071643199601x_{51} = -17.9071643199601
x52=60.0044119852004x_{52} = 60.0044119852004
x53=−0.338901775254251x_{53} = -0.338901775254251
Maxima of the function at points:
x53=−19.792104315915x_{53} = -19.792104315915
x53=14.1370289022608x_{53} = 14.1370289022608
x53=81.9955641354391x_{53} = 81.9955641354391
x53=−70.0575218229581x_{53} = -70.0575218229581
x53=34.243336299603x_{53} = 34.243336299603
x53=−60.0044273819258x_{53} = -60.0044273819258
x53=−43.6681524169898x_{53} = -43.6681524169898
x53=4.08250110523468x_{53} = 4.08250110523468
x53=48.066355604513x_{53} = 48.066355604513
x53=24.1902161322856x_{53} = 24.1902161322856
x53=54.3495435239968x_{53} = 54.3495435239968
x53=−29.8451613012199x_{53} = -29.8451613012199
x53=0.288704878724942x_{53} = 0.288704878724942
x53=92.0486614782868x_{53} = 92.0486614782868
x53=−36.128336741733x_{53} = -36.128336741733
x53=41.7831664206562x_{53} = 41.7831664206562
x53=78.2256525441992x_{53} = 78.2256525441992
x53=−39.8982441068377x_{53} = -39.8982441068377
x53=−49.9513342996013x_{53} = -49.9513342996013
x53=−97.703534430795x_{53} = -97.703534430795
x53=−63.7743376832009x_{53} = -63.7743376832009
x53=−9.73922648158337x_{53} = -9.73922648158337
x53=−5.9697827840821x_{53} = -5.9697827840821
x53=58.1194558857455x_{53} = 58.1194558857455
x53=17.9069919293091x_{53} = 17.9069919293091
x53=71.9424664113254x_{53} = 71.9424664113254
x53=−381.703507601456x_{53} = -381.703507601456
x53=−0.956925542610824x_{53} = -0.956925542610824
x53=−53.7212439801803x_{53} = -53.7212439801803
x53=−73.8274324452886x_{53} = -73.8274324452886
x53=−93.9336234842473x_{53} = -93.9336234842473
x53=68.1725546184145x_{53} = 68.1725546184145
x53=−26.0752597431184x_{53} = -26.0752597431184
x53=−80.1106169860811x_{53} = -80.1106169860811
x53=−16.0222301181307x_{53} = -16.0222301181307
x53=38.0132519343378x_{53} = 38.0132519343378
x53=27.9601391966996x_{53} = 27.9601391966996
x53=10.367000159236x_{53} = 10.367000159236
x53=−83.8805277908947x_{53} = -83.8805277908947
x53=98.3318471901991x_{53} = 98.3318471901991
x53=88.2787500085987x_{53} = 88.2787500085987
x53=−87.6504386436056x_{53} = -87.6504386436056
x53=61.8893680390286x_{53} = 61.8893680390286
x53=44.2964422926316x_{53} = 44.2964422926316
x53=−33.6150659084925x_{53} = -33.6150659084925
Decreasing at intervals
[100.216802889184,∞)\left[100.216802889184, \infty\right)
Increasing at intervals
(−∞,−103.358400898199]\left(-\infty, -103.358400898199\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
2acot⁡(x)((2x+log⁡(2))log⁡(2)sin⁡(5x)(x2+1)2−25sin⁡(5x)−10log⁡(2)cos⁡(5x)x2+1)=02^{\operatorname{acot}{\left(x \right)}} \left(\frac{\left(2 x + \log{\left(2 \right)}\right) \log{\left(2 \right)} \sin{\left(5 x \right)}}{\left(x^{2} + 1\right)^{2}} - 25 \sin{\left(5 x \right)} - \frac{10 \log{\left(2 \right)} \cos{\left(5 x \right)}}{x^{2} + 1}\right) = 0
Solve this equation
The roots of this equation
x1=96.1327292001973x_{1} = 96.1327292001973
x2=65.9734329880793x_{2} = 65.9734329880793
x3=18.2210708711803x_{3} = 18.2210708711803
x4=6.28181452808594x_{4} = 6.28181452808594
x5=−32.0442990164888x_{5} = -32.0442990164888
x6=−96.1327411994958x_{6} = -96.1327411994958
x7=−91.7345120735002x_{7} = -91.7345120735002
x8=1.87259178730817x_{8} = 1.87259178730817
x9=16.3360747839943x_{9} = 16.3360747839943
x10=70.3716642452029x_{10} = 70.3716642452029
x11=74.1415765388605x_{11} = 74.1415765388605
x12=−10.0536397039186x_{12} = -10.0536397039186
x13=−69.7433683074733x_{13} = -69.7433683074733
x14=−45.8672790876934x_{14} = -45.8672790876934
x15=−21.9912629987655x_{15} = -21.9912629987655
x16=−87.9646014659632x_{16} = -87.9646014659632
x17=−11.9384384237795x_{17} = -11.9384384237795
x18=−52.7787764797446x_{18} = -52.7787764797446
x19=−25.7611431908976x_{19} = -25.7611431908976
x20=−27.6460878083804x_{20} = -27.6460878083804
x21=92.3628175161854x_{21} = 92.3628175161854
x22=−54.0354126267023x_{22} = -54.0354126267023
x23=20.1060561497343x_{23} = 20.1060561497343
x24=99.902640828721x_{24} = 99.902640828721
x25=−15.7081870888228x_{25} = -15.7081870888228
x26=−20.1063298106417x_{26} = -20.1063298106417
x27=52.1504176678918x_{27} = 52.1504176678918
x28=−3.77354645984327x_{28} = -3.77354645984327
x29=10.0525531046131x_{29} = 10.0525531046131
x30=40.2123516948228x_{30} = 40.2123516948228
x31=−1.89695122318583x_{31} = -1.89695122318583
x32=−35.8141994491553x_{32} = -35.8141994491553
x33=−86.0796461910176x_{33} = -86.0796461910176
x34=21.9910341479267x_{34} = 21.9910341479267
x35=−93.6194674030392x_{35} = -93.6194674030392
x36=45.8672263970378x_{36} = 45.8672263970378
x37=−74.1415967105695x_{37} = -74.1415967105695
x38=−98.0176965631366x_{38} = -98.0176965631366
x39=50.2654605190619x_{39} = 50.2654605190619
x40=−57.8053214161799x_{40} = -57.8053214161799
x41=−59.690275977371x_{41} = -59.690275977371
x42=86.0796312256991x_{42} = 86.0796312256991
x43=−79.796462108409x_{43} = -79.796462108409
x44=−13.8232963554913x_{44} = -13.8232963554913
x45=77.9114886754486x_{45} = 77.9114886754486
x46=−42.0973728304264x_{46} = -42.0973728304264
x47=87.9645871350617x_{47} = 87.9645871350617
x48=−0.054777546278755x_{48} = -0.054777546278755
x49=48.3805031848631x_{49} = 48.3805031848631
x50=26.3892987772048x_{50} = 26.3892987772048
x51=−38.9557854206669x_{51} = -38.9557854206669
x52=55.9203315068164x_{52} = 55.9203315068164
x53=−67.8584133571841x_{53} = -67.8584133571841
x54=89.2212243968948x_{54} = 89.2212243968948
x55=38.3273926511814x_{55} = 38.3273926511814
x56=6.91036625904162x_{56} = 6.91036625904162
x57=72.256620413726x_{57} = 72.256620413726
x58=−76.0265518088842x_{58} = -76.0265518088842
x59=−81.6814173033875x_{59} = -81.6814173033875
x60=60.318563712108x_{60} = 60.318563712108
x61=28.2742646050075x_{61} = 28.2742646050075
x62=−23.8762012679149x_{62} = -23.8762012679149
x63=23.8760070642847x_{63} = 23.8760070642847
x64=33.9291525313096x_{64} = 33.9291525313096
x65=30.1592285770751x_{65} = 30.1592285770751
x66=32.0441911161986x_{66} = 32.0441911161986
x67=−42.725690448636x_{67} = -42.725690448636
x68=64.088476635838x_{68} = 64.088476635838
x69=−47.7522326419055x_{69} = -47.7522326419055
x70=−108.070792030952x_{70} = -108.070792030952
x71=−64.0885036306085x_{71} = -64.0885036306085
x72=94.2477733656848x_{72} = 94.2477733656848
x73=67.8583892778822x_{73} = 67.8583892778822
x74=98.0176850208645x_{74} = 98.0176850208645
x75=54.0353746567466x_{75} = 54.0353746567466
x76=−33.9292487858207x_{76} = -33.9292487858207
x77=−49.6371864237238x_{77} = -49.6371864237238
x78=−37.6991508324632x_{78} = -37.6991508324632
x79=−99.902651939588x_{79} = -99.902651939588
x80=8.16732180093718x_{80} = 8.16732180093718
x81=442.964563873559x_{81} = 442.964563873559
x82=43.9822684995048x_{82} = 43.9822684995048
x83=−71.628323307753x_{83} = -71.628323307753
x84=62.2035202134797x_{84} = 62.2035202134797
x85=76.0265326248546x_{85} = 76.0265326248546
x86=−77.9115069425987x_{86} = -77.9115069425987
x87=−43.9823258008975x_{87} = -43.9823258008975
x88=11.9376656689947x_{88} = 11.9376656689947
x89=21.3627088019739x_{89} = 21.3627088019739
x90=−89.8495567606595x_{90} = -89.8495567606595
x91=−55.9203669609465x_{91} = -55.9203669609465
x92=84.1946752947979x_{92} = 84.1946752947979
x93=82.3097193403629x_{93} = 82.3097193403629
x94=42.0973102856408x_{94} = 42.0973102856408
x95=−65.9734584626773x_{95} = -65.9734584626773
x96=−5.65654681650801x_{96} = -5.65654681650801

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
[442.964563873559,∞)\left[442.964563873559, \infty\right)
Convex at the intervals
(−∞,−99.902651939588]\left(-\infty, -99.902651939588\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(2acot⁡(x)sin⁡(5x))=⟨−1,1⟩\lim_{x \to -\infty}\left(2^{\operatorname{acot}{\left(x \right)}} \sin{\left(5 x \right)}\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→∞(2acot⁡(x)sin⁡(5x))=⟨−1,1⟩\lim_{x \to \infty}\left(2^{\operatorname{acot}{\left(x \right)}} \sin{\left(5 x \right)}\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 2^acot(x)*sin(5*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2acot⁡(x)sin⁡(5x)x)=0\lim_{x \to -\infty}\left(\frac{2^{\operatorname{acot}{\left(x \right)}} \sin{\left(5 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2acot⁡(x)sin⁡(5x)x)=0\lim_{x \to \infty}\left(\frac{2^{\operatorname{acot}{\left(x \right)}} \sin{\left(5 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:
2acot⁡(x)sin⁡(5x)=−2−acot⁡(x)sin⁡(5x)2^{\operatorname{acot}{\left(x \right)}} \sin{\left(5 x \right)} = - 2^{- \operatorname{acot}{\left(x \right)}} \sin{\left(5 x \right)}
- No
2acot⁡(x)sin⁡(5x)=2−acot⁡(x)sin⁡(5x)2^{\operatorname{acot}{\left(x \right)}} \sin{\left(5 x \right)} = 2^{- \operatorname{acot}{\left(x \right)}} \sin{\left(5 x \right)}
- No
so, the function
not is
neither even, nor odd
    © Mister Exam – Calculator