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Plotting a function graph/ xcos2x

Graphing y = xcos2x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = x*cos(2*x)
f(x)=xcos⁡(2x)f{\left(x \right)} = x \cos{\left(2 x \right)} 
f = x*cos(2*x)
The graph of the function
02468-8-6-4-2-1010-2020
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:
xcos⁡(2x)=0x \cos{\left(2 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=−3π4x_{2} = - \frac{3 \pi}{4}
x3=−π4x_{3} = - \frac{\pi}{4}
x4=π4x_{4} = \frac{\pi}{4}
x5=3π4x_{5} = \frac{3 \pi}{4}
Numerical solution
x1=55.7632696012188x_{1} = 55.7632696012188
x2=−33.7721210260903x_{2} = -33.7721210260903
x3=3.92699081698724x_{3} = 3.92699081698724
x4=−16.4933614313464x_{4} = -16.4933614313464
x5=−47.9092879672443x_{5} = -47.9092879672443
x6=46.3384916404494x_{6} = 46.3384916404494
x7=16.4933614313464x_{7} = 16.4933614313464
x8=−76.1836218495525x_{8} = -76.1836218495525
x9=90.3207887907066x_{9} = 90.3207887907066
x10=60.4756585816035x_{10} = 60.4756585816035
x11=−99.7455667514759x_{11} = -99.7455667514759
x12=−98.174770424681x_{12} = -98.174770424681
x13=−10.2101761241668x_{13} = -10.2101761241668
x14=−85.6083998103219x_{14} = -85.6083998103219
x15=−2.35619449019234x_{15} = -2.35619449019234
x16=−55.7632696012188x_{16} = -55.7632696012188
x17=63.6172512351933x_{17} = 63.6172512351933
x18=32.2013246992954x_{18} = 32.2013246992954
x19=18.0641577581413x_{19} = 18.0641577581413
x20=−82.4668071567321x_{20} = -82.4668071567321
x21=−91.8915851175014x_{21} = -91.8915851175014
x22=77.7544181763474x_{22} = 77.7544181763474
x23=−90.3207887907066x_{23} = -90.3207887907066
x24=−60.4756585816035x_{24} = -60.4756585816035
x25=−13.3517687777566x_{25} = -13.3517687777566
x26=91.8915851175014x_{26} = 91.8915851175014
x27=73.0420291959627x_{27} = 73.0420291959627
x28=−3.92699081698724x_{28} = -3.92699081698724
x29=−71.4712328691678x_{29} = -71.4712328691678
x30=40.0553063332699x_{30} = 40.0553063332699
x31=−25.9181393921158x_{31} = -25.9181393921158
x32=49.4800842940392x_{32} = 49.4800842940392
x33=33.7721210260903x_{33} = 33.7721210260903
x34=2.35619449019234x_{34} = 2.35619449019234
x35=47.9092879672443x_{35} = 47.9092879672443
x36=99.7455667514759x_{36} = 99.7455667514759
x37=96.6039740978861x_{37} = 96.6039740978861
x38=−11.7809724509617x_{38} = -11.7809724509617
x39=−62.0464549083984x_{39} = -62.0464549083984
x40=−0.785398163397448x_{40} = -0.785398163397448
x41=−18.0641577581413x_{41} = -18.0641577581413
x42=82.4668071567321x_{42} = 82.4668071567321
x43=54.1924732744239x_{43} = 54.1924732744239
x44=5.49778714378214x_{44} = 5.49778714378214
x45=−49.4800842940392x_{45} = -49.4800842940392
x46=7.06858347057703x_{46} = 7.06858347057703
x47=84.037603483527x_{47} = 84.037603483527
x48=88.7499924639117x_{48} = 88.7499924639117
x49=−77.7544181763474x_{49} = -77.7544181763474
x50=−46.3384916404494x_{50} = -46.3384916404494
x51=24.3473430653209x_{51} = 24.3473430653209
x52=−38.484510006475x_{52} = -38.484510006475
x53=19.6349540849362x_{53} = 19.6349540849362
x54=85.6083998103219x_{54} = 85.6083998103219
x55=62.0464549083984x_{55} = 62.0464549083984
x56=−80.8960108299372x_{56} = -80.8960108299372
x57=−57.3340659280137x_{57} = -57.3340659280137
x58=76.1836218495525x_{58} = 76.1836218495525
x59=69.9004365423729x_{59} = 69.9004365423729
x60=8.63937979737193x_{60} = 8.63937979737193
x61=−69.9004365423729x_{61} = -69.9004365423729
x62=68.329640215578x_{62} = 68.329640215578
x63=−63.6172512351933x_{63} = -63.6172512351933
x64=98.174770424681x_{64} = 98.174770424681
x65=41.6261026600648x_{65} = 41.6261026600648
x66=−19.6349540849362x_{66} = -19.6349540849362
x67=−24.3473430653209x_{67} = -24.3473430653209
x68=−93.4623814442964x_{68} = -93.4623814442964
x69=−41.6261026600648x_{69} = -41.6261026600648
x70=−27.4889357189107x_{70} = -27.4889357189107
x71=30.6305283725005x_{71} = 30.6305283725005
x72=0x_{72} = 0
x73=−84.037603483527x_{73} = -84.037603483527
x74=10.2101761241668x_{74} = 10.2101761241668
x75=25.9181393921158x_{75} = 25.9181393921158
x76=74.6128255227576x_{76} = 74.6128255227576
x77=−68.329640215578x_{77} = -68.329640215578
x78=−79.3252145031423x_{78} = -79.3252145031423
x79=−40.0553063332699x_{79} = -40.0553063332699
x80=52.621676947629x_{80} = 52.621676947629
x81=11.7809724509617x_{81} = 11.7809724509617
x82=−54.1924732744239x_{82} = -54.1924732744239
x83=−32.2013246992954x_{83} = -32.2013246992954
x84=27.4889357189107x_{84} = 27.4889357189107
x85=38.484510006475x_{85} = 38.484510006475
x86=−5.49778714378214x_{86} = -5.49778714378214
x87=−35.3429173528852x_{87} = -35.3429173528852
x88=66.7588438887831x_{88} = 66.7588438887831
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x*cos(2*x).
0cos⁡(0⋅2)0 \cos{\left(0 \cdot 2 \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
−2xsin⁡(2x)+cos⁡(2x)=0- 2 x \sin{\left(2 x \right)} + \cos{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=36.1352335301545x_{1} = 36.1352335301545
x2=−15.7238573187731x_{2} = -15.7238573187731
x3=−80.113733189628x_{3} = -80.113733189628
x4=29.8535036526677x_{4} = 29.8535036526677
x5=42.4173943590211x_{5} = 42.4173943590211
x6=−94.2504320905443x_{6} = -94.2504320905443
x7=−97.3919391862849x_{7} = -97.3919391862849
x8=64.4065309145547x_{8} = 64.4065309145547
x9=−103.674968932212x_{9} = -103.674968932212
x10=6.32264361192832x_{10} = 6.32264361192832
x11=37.7057417444241x_{11} = 37.7057417444241
x12=−86.3966915707365x_{12} = -86.3966915707365
x13=−65.9772348386275x_{13} = -65.9772348386275
x14=51.8411010631448x_{14} = 51.8411010631448
x15=−43.9879802762466x_{15} = -43.9879802762466
x16=26.7128952386973x_{16} = 26.7128952386973
x17=−11.0182483639693x_{17} = -11.0182483639693
x18=95.8211849371972x_{18} = 95.8211849371972
x19=0.43016679450969x_{19} = 0.43016679450969
x20=12.5862231633233x_{20} = 12.5862231633233
x21=−39.2762729921215x_{21} = -39.2762729921215
x22=92.6796807176258x_{22} = 92.6796807176258
x23=−51.8411010631448x_{23} = -51.8411010631448
x24=−59.6944483144154x_{24} = -59.6944483144154
x25=−1.71280922974086x_{25} = -1.71280922974086
x26=87.9674362306479x_{26} = 87.9674362306479
x27=81.6844695124177x_{27} = 81.6844695124177
x28=45.5585806972324x_{28} = 45.5585806972324
x29=61.2651372785773x_{29} = 61.2651372785773
x30=−22.0025089604154x_{30} = -22.0025089604154
x31=20.432585165244x_{31} = 20.432585165244
x32=48.6998194395369x_{32} = 48.6998194395369
x33=58.123765151966x_{33} = 58.123765151966
x34=−42.4173943590211x_{34} = -42.4173943590211
x35=94.2504320905443x_{35} = 94.2504320905443
x36=−0.43016679450969x_{36} = -0.43016679450969
x37=15.7238573187731x_{37} = 15.7238573187731
x38=−36.1352335301545x_{38} = -36.1352335301545
x39=34.5647514869476x_{39} = 34.5647514869476
x40=23.5725488683805x_{40} = 23.5725488683805
x41=65.9772348386275x_{41} = 65.9772348386275
x42=−20.432585165244x_{42} = -20.432585165244
x43=7.88564243740794x_{43} = 7.88564243740794
x44=−81.6844695124177x_{44} = -81.6844695124177
x45=−9.45120497843001x_{45} = -9.45120497843001
x46=28.2831721399108x_{46} = 28.2831721399108
x47=72.2600907017656x_{47} = 72.2600907017656
x48=−23.5725488683805x_{48} = -23.5725488683805
x49=3.21864908958597x_{49} = 3.21864908958597
x50=−37.7057417444241x_{50} = -37.7057417444241
x51=1.71280922974086x_{51} = 1.71280922974086
x52=−67.5479430595368x_{52} = -67.5479430595368
x53=22.0025089604154x_{53} = 22.0025089604154
x54=70.6893712463639x_{54} = 70.6893712463639
x55=9.45120497843001x_{55} = 9.45120497843001
x56=−14.154821427226x_{56} = -14.154821427226
x57=−58.123765151966x_{57} = -58.123765151966
x58=−64.4065309145547x_{58} = -64.4065309145547
x59=−89.5381827032021x_{59} = -89.5381827032021
x60=−95.8211849371972x_{60} = -95.8211849371972
x61=−53.4117555918474x_{61} = -53.4117555918474
x62=−6.32264361192832x_{62} = -6.32264361192832
x63=−28.2831721399108x_{63} = -28.2831721399108
x64=80.113733189628x_{64} = 80.113733189628
x65=−17.2932121076445x_{65} = -17.2932121076445
x66=−50.2704553934212x_{66} = -50.2704553934212
x67=−73.8308134276772x_{67} = -73.8308134276772
x68=−87.9674362306479x_{68} = -87.9674362306479
x69=89.5381827032021x_{69} = 89.5381827032021
x70=−83.2552080991765x_{70} = -83.2552080991765
x71=−72.2600907017656x_{71} = -72.2600907017656
x72=114.67031200546x_{72} = 114.67031200546
x73=14.154821427226x_{73} = 14.154821427226
x74=−61.2651372785773x_{74} = -61.2651372785773
x75=50.2704553934212x_{75} = 50.2704553934212
x76=−7.88564243740794x_{76} = -7.88564243740794
x77=−45.5585806972324x_{77} = -45.5585806972324
x78=86.3966915707365x_{78} = 86.3966915707365
x79=−3.21864908958597x_{79} = -3.21864908958597
x80=59.6944483144154x_{80} = 59.6944483144154
x81=−75.4015392197413x_{81} = -75.4015392197413
x82=56.5530882745116x_{82} = 56.5530882745116
x83=73.8308134276772x_{83} = 73.8308134276772
x84=−31.4238815972272x_{84} = -31.4238815972272
x85=−29.8535036526677x_{85} = -29.8535036526677
x86=100.533451628845x_{86} = 100.533451628845
x87=43.9879802762466x_{87} = 43.9879802762466
x88=78.542999266617x_{88} = 78.542999266617
x89=67.5479430595368x_{89} = 67.5479430595368
The values of the extrema at the points:
(36.13523353015448, -36.1317747991247)

(-15.723857318773117, -15.7159136392673)

(-80.11373318962796, 80.112172953406)

(29.85350365266773, -29.8493174201329)

(42.417394359021145, -42.4144477618284)

(-94.25043209054431, -94.2491058646707)

(-97.39193918628494, -97.390655737879)

(64.40653091455466, -64.4045902053056)

(-103.67496893221228, -103.673763262022)

(6.322643611928322, 6.30296564894634)

(37.705741744424074, 37.702427036601)

(-86.39669157073652, 86.3952447924177)

(-65.97723483862752, -65.9753403273413)

(51.84110106314479, -51.8386900171372)

(-43.98798027624661, -43.9851388662124)

(26.71289523869733, -26.7082170799481)

(-11.018248363969283, 11.0069210395792)

(95.82118493719717, -95.8198804506423)

(0.43016679450968986, 0.280548169095523)

(12.586223163323332, 12.5763034089358)

(-39.27627299212146, 39.2730907958671)

(92.67968071762581, -92.6783320156182)

(-51.84110106314479, 51.8386900171372)

(-59.69444831441541, -59.6923544275184)

(-1.7128092297408641, 1.64418569779545)

(87.96743623064788, 87.9660152847086)

(81.68446951241769, 81.6829392767655)

(45.55858069723237, -45.5558372248235)

(61.2651372785773, -61.263097068409)

(-22.002508960415422, -21.9968299895532)

(20.432585165244035, -20.4264702322587)

(48.69981943953688, -48.6972528978117)

(58.12376515196605, -58.1216146879934)

(-42.417394359021145, 42.4144477618284)

(94.25043209054431, 94.2491058646707)

(-0.43016679450968986, -0.280548169095523)

(15.723857318773117, 15.7159136392673)

(-36.13523353015448, 36.1317747991247)

(34.56475148694763, 34.5611356534609)

(23.572548868380515, -23.567247878771)

(65.97723483862752, 65.9753403273413)

(-20.432585165244035, 20.4264702322587)

(7.885642437407941, -7.86983848106687)

(-81.68446951241769, -81.6829392767655)

(-9.451204978430011, -9.43800684898451)

(28.28317213991076, 28.2787535864381)

(72.26009070176562, 72.2583609016736)

(-23.572548868380515, 23.567247878771)

(3.2186490895859734, 3.18050197241693)

(-37.705741744424074, -37.702427036601)

(1.7128092297408641, -1.64418569779545)

(-67.54794305953683, 67.546092598104)

(22.002508960415422, 21.9968299895532)

(70.68937124636392, -70.687603012927)

(9.451204978430011, 9.43800684898451)

(-14.154821427226006, 14.1459987695472)

(-58.12376515196605, 58.1216146879934)

(-64.40653091455466, 64.4045902053056)

(-89.53818270320214, 89.5367866833941)

(-95.82118493719717, 95.8198804506423)

(-53.41175559184737, -53.4094154368825)

(-6.322643611928322, -6.30296564894634)

(-28.28317213991076, -28.2787535864381)

(80.11373318962796, -80.112172953406)

(-17.29321210764446, 17.2859883667942)

(-50.27045539342116, -50.267969027913)

(-73.83081342767719, 73.829120425871)

(-87.96743623064788, -87.9660152847086)

(89.53818270320214, -89.5367866833941)

(-83.2552080991765, 83.2537067322156)

(-72.26009070176562, -72.2583609016736)

(114.67031200545999, -114.669221939379)

(14.154821427226006, -14.1459987695472)

(-61.2651372785773, 61.263097068409)

(50.27045539342116, 50.267969027913)

(-7.885642437407941, 7.86983848106687)

(-45.55858069723237, 45.5558372248235)

(86.39669157073652, -86.3952447924177)

(-3.2186490895859734, -3.18050197241693)

(59.69444831441541, 59.6923544275184)

(-75.40153921974125, -75.3998814833205)

(56.55308827451163, 56.5508780915478)

(73.83081342767719, -73.829120425871)

(-31.423881597227226, -31.4199044860773)

(-29.85350365266773, 29.8493174201329)

(100.53345162884467, 100.532208284673)

(43.98798027624661, 43.9851388662124)

(78.54299926661696, 78.5414078300528)

(67.54794305953683, -67.546092598104)


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=36.1352335301545x_{1} = 36.1352335301545
x2=−15.7238573187731x_{2} = -15.7238573187731
x3=29.8535036526677x_{3} = 29.8535036526677
x4=42.4173943590211x_{4} = 42.4173943590211
x5=−94.2504320905443x_{5} = -94.2504320905443
x6=−97.3919391862849x_{6} = -97.3919391862849
x7=64.4065309145547x_{7} = 64.4065309145547
x8=−103.674968932212x_{8} = -103.674968932212
x9=−65.9772348386275x_{9} = -65.9772348386275
x10=51.8411010631448x_{10} = 51.8411010631448
x11=−43.9879802762466x_{11} = -43.9879802762466
x12=26.7128952386973x_{12} = 26.7128952386973
x13=95.8211849371972x_{13} = 95.8211849371972
x14=92.6796807176258x_{14} = 92.6796807176258
x15=−59.6944483144154x_{15} = -59.6944483144154
x16=45.5585806972324x_{16} = 45.5585806972324
x17=61.2651372785773x_{17} = 61.2651372785773
x18=−22.0025089604154x_{18} = -22.0025089604154
x19=20.432585165244x_{19} = 20.432585165244
x20=48.6998194395369x_{20} = 48.6998194395369
x21=58.123765151966x_{21} = 58.123765151966
x22=−0.43016679450969x_{22} = -0.43016679450969
x23=23.5725488683805x_{23} = 23.5725488683805
x24=7.88564243740794x_{24} = 7.88564243740794
x25=−81.6844695124177x_{25} = -81.6844695124177
x26=−9.45120497843001x_{26} = -9.45120497843001
x27=−37.7057417444241x_{27} = -37.7057417444241
x28=1.71280922974086x_{28} = 1.71280922974086
x29=70.6893712463639x_{29} = 70.6893712463639
x30=−53.4117555918474x_{30} = -53.4117555918474
x31=−6.32264361192832x_{31} = -6.32264361192832
x32=−28.2831721399108x_{32} = -28.2831721399108
x33=80.113733189628x_{33} = 80.113733189628
x34=−50.2704553934212x_{34} = -50.2704553934212
x35=−87.9674362306479x_{35} = -87.9674362306479
x36=89.5381827032021x_{36} = 89.5381827032021
x37=−72.2600907017656x_{37} = -72.2600907017656
x38=114.67031200546x_{38} = 114.67031200546
x39=14.154821427226x_{39} = 14.154821427226
x40=86.3966915707365x_{40} = 86.3966915707365
x41=−3.21864908958597x_{41} = -3.21864908958597
x42=−75.4015392197413x_{42} = -75.4015392197413
x43=73.8308134276772x_{43} = 73.8308134276772
x44=−31.4238815972272x_{44} = -31.4238815972272
x45=67.5479430595368x_{45} = 67.5479430595368
Maxima of the function at points:
x45=−80.113733189628x_{45} = -80.113733189628
x45=6.32264361192832x_{45} = 6.32264361192832
x45=37.7057417444241x_{45} = 37.7057417444241
x45=−86.3966915707365x_{45} = -86.3966915707365
x45=−11.0182483639693x_{45} = -11.0182483639693
x45=0.43016679450969x_{45} = 0.43016679450969
x45=12.5862231633233x_{45} = 12.5862231633233
x45=−39.2762729921215x_{45} = -39.2762729921215
x45=−51.8411010631448x_{45} = -51.8411010631448
x45=−1.71280922974086x_{45} = -1.71280922974086
x45=87.9674362306479x_{45} = 87.9674362306479
x45=81.6844695124177x_{45} = 81.6844695124177
x45=−42.4173943590211x_{45} = -42.4173943590211
x45=94.2504320905443x_{45} = 94.2504320905443
x45=15.7238573187731x_{45} = 15.7238573187731
x45=−36.1352335301545x_{45} = -36.1352335301545
x45=34.5647514869476x_{45} = 34.5647514869476
x45=65.9772348386275x_{45} = 65.9772348386275
x45=−20.432585165244x_{45} = -20.432585165244
x45=28.2831721399108x_{45} = 28.2831721399108
x45=72.2600907017656x_{45} = 72.2600907017656
x45=−23.5725488683805x_{45} = -23.5725488683805
x45=3.21864908958597x_{45} = 3.21864908958597
x45=−67.5479430595368x_{45} = -67.5479430595368
x45=22.0025089604154x_{45} = 22.0025089604154
x45=9.45120497843001x_{45} = 9.45120497843001
x45=−14.154821427226x_{45} = -14.154821427226
x45=−58.123765151966x_{45} = -58.123765151966
x45=−64.4065309145547x_{45} = -64.4065309145547
x45=−89.5381827032021x_{45} = -89.5381827032021
x45=−95.8211849371972x_{45} = -95.8211849371972
x45=−17.2932121076445x_{45} = -17.2932121076445
x45=−73.8308134276772x_{45} = -73.8308134276772
x45=−83.2552080991765x_{45} = -83.2552080991765
x45=−61.2651372785773x_{45} = -61.2651372785773
x45=50.2704553934212x_{45} = 50.2704553934212
x45=−7.88564243740794x_{45} = -7.88564243740794
x45=−45.5585806972324x_{45} = -45.5585806972324
x45=59.6944483144154x_{45} = 59.6944483144154
x45=56.5530882745116x_{45} = 56.5530882745116
x45=−29.8535036526677x_{45} = -29.8535036526677
x45=100.533451628845x_{45} = 100.533451628845
x45=43.9879802762466x_{45} = 43.9879802762466
x45=78.542999266617x_{45} = 78.542999266617
Decreasing at intervals
[114.67031200546,∞)\left[114.67031200546, \infty\right)
Increasing at intervals
(−∞,−103.674968932212]\left(-\infty, -103.674968932212\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
−4(xcos⁡(2x)+sin⁡(2x))=0- 4 \left(x \cos{\left(2 x \right)} + \sin{\left(2 x \right)}\right) = 0
Solve this equation
The roots of this equation
x1=−77.760847792972x_{1} = -77.760847792972
x2=71.4782275499213x_{2} = 71.4782275499213
x3=99.7505790857949x_{3} = 99.7505790857949
x4=30.6468374831214x_{4} = 30.6468374831214
x5=46.3492776216985x_{5} = 46.3492776216985
x6=−46.3492776216985x_{6} = -46.3492776216985
x7=41.6381085824888x_{7} = 41.6381085824888
x8=−10.2587614549708x_{8} = -10.2587614549708
x9=52.6311758774383x_{9} = 52.6311758774383
x10=−35.3570550332742x_{10} = -35.3570550332742
x11=38.4974949445838x_{11} = 38.4974949445838
x12=−27.5071048394191x_{12} = -27.5071048394191
x13=−91.8970257752571x_{13} = -91.8970257752571
x14=−21.2292853858495x_{14} = -21.2292853858495
x15=5.58635293416499x_{15} = 5.58635293416499
x16=−63.6251091208926x_{16} = -63.6251091208926
x17=−32.2168395518658x_{17} = -32.2168395518658
x18=−76.1901839979235x_{18} = -76.1901839979235
x19=−79.3315168346756x_{19} = -79.3315168346756
x20=−60.4839244878466x_{20} = -60.4839244878466
x21=−40.0677825970372x_{21} = -40.0677825970372
x22=0x_{22} = 0
x23=91.8970257752571x_{23} = 91.8970257752571
x24=−47.9197205706165x_{24} = -47.9197205706165
x25=76.1901839979235x_{25} = 76.1901839979235
x26=47.9197205706165x_{26} = 47.9197205706165
x27=66.766332133246x_{27} = 66.766332133246
x28=−90.3263240494369x_{28} = -90.3263240494369
x29=69.9075883539626x_{29} = 69.9075883539626
x30=−13.3890435377793x_{30} = -13.3890435377793
x31=−82.4728694594266x_{31} = -82.4728694594266
x32=8.69662198229738x_{32} = 8.69662198229738
x33=32.2168395518658x_{33} = 32.2168395518658
x34=88.7556256712795x_{34} = 88.7556256712795
x35=−58.9133484807877x_{35} = -58.9133484807877
x36=33.7869153354295x_{36} = 33.7869153354295
x37=−18.0917665453763x_{37} = -18.0917665453763
x38=120.170079673253x_{38} = 120.170079673253
x39=−24.3678503974527x_{39} = -24.3678503974527
x40=−62.0545116429054x_{40} = -62.0545116429054
x41=74.6195257807054x_{41} = 74.6195257807054
x42=4.04808180161146x_{42} = 4.04808180161146
x43=−84.0435524991391x_{43} = -84.0435524991391
x44=19.6603640661261x_{44} = 19.6603640661261
x45=98.1798629425939x_{45} = 98.1798629425939
x46=54.2016970313842x_{46} = 54.2016970313842
x47=68.3369563786298x_{47} = 68.3369563786298
x48=−66.766332133246x_{48} = -66.766332133246
x49=27.5071048394191x_{49} = 27.5071048394191
x50=−19.6603640661261x_{50} = -19.6603640661261
x51=63.6251091208926x_{51} = 63.6251091208926
x52=−99.7505790857949x_{52} = -99.7505790857949
x53=2.54349254705114x_{53} = 2.54349254705114
x54=−41.6381085824888x_{54} = -41.6381085824888
x55=−1.1444648640517x_{55} = -1.1444648640517
x56=−49.4901859325761x_{56} = -49.4901859325761
x57=−4.04808180161146x_{57} = -4.04808180161146
x58=25.9374070267134x_{58} = 25.9374070267134
x59=−68.3369563786298x_{59} = -68.3369563786298
x60=90.3263240494369x_{60} = 90.3263240494369
x61=85.6142396947314x_{61} = 85.6142396947314
x62=−85.6142396947314x_{62} = -85.6142396947314
x63=82.4728694594266x_{63} = 82.4728694594266
x64=−93.4677306800165x_{64} = -93.4677306800165
x65=96.6091494063022x_{65} = 96.6091494063022
x66=55.7722336752062x_{66} = 55.7722336752062
x67=−54.2016970313842x_{67} = -54.2016970313842
x68=11.8231619098018x_{68} = 11.8231619098018
x69=84.0435524991391x_{69} = 84.0435524991391
x70=40.0677825970372x_{70} = 40.0677825970372
x71=−98.1798629425939x_{71} = -98.1798629425939
x72=24.3678503974527x_{72} = 24.3678503974527
x73=−25.9374070267134x_{73} = -25.9374070267134
x74=18.0917665453763x_{74} = 18.0917665453763
x75=49.4901859325761x_{75} = 49.4901859325761
x76=−57.3427845371101x_{76} = -57.3427845371101
x77=62.0545116429054x_{77} = 62.0545116429054
x78=−33.7869153354295x_{78} = -33.7869153354295
x79=−11.8231619098018x_{79} = -11.8231619098018
x80=−69.9075883539626x_{80} = -69.9075883539626
x81=77.760847792972x_{81} = 77.760847792972
x82=−38.4974949445838x_{82} = -38.4974949445838
x83=10.2587614549708x_{83} = 10.2587614549708
x84=−71.4782275499213x_{84} = -71.4782275499213
x85=60.4839244878466x_{85} = 60.4839244878466
x86=−55.7722336752062x_{86} = -55.7722336752062
x87=16.5235843473527x_{87} = 16.5235843473527
x88=−5.58635293416499x_{88} = -5.58635293416499

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
[120.170079673253,∞)\left[120.170079673253, \infty\right)
Convex at the intervals
(−∞,−98.1798629425939]\left(-\infty, -98.1798629425939\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(xcos⁡(2x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x \cos{\left(2 x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(xcos⁡(2x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(x \cos{\left(2 x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x*cos(2*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞cos⁡(2x)=⟨−1,1⟩\lim_{x \to -\infty} \cos{\left(2 x \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
inclined asymptote equation on the left:
y=⟨−1,1⟩xy = \left\langle -1, 1\right\rangle x
lim⁡x→∞cos⁡(2x)=⟨−1,1⟩\lim_{x \to \infty} \cos{\left(2 x \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
inclined asymptote equation on the right:
y=⟨−1,1⟩xy = \left\langle -1, 1\right\rangle x
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:
xcos⁡(2x)=−xcos⁡(2x)x \cos{\left(2 x \right)} = - x \cos{\left(2 x \right)}
- No
xcos⁡(2x)=xcos⁡(2x)x \cos{\left(2 x \right)} = x \cos{\left(2 x \right)}
- Yes
so, the function
is
odd
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