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cos(x)^2*x
Plotting a function graph/ cos(x)^2*x

You entered:

cos(x)^2*x

What you mean?

cos(x)^2*x
 
cos(x)^(2*x)
 
cos(x)^(2*x)

Graphing y = cos(x)^2*x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          2     
f(x) = cos (x)*x
f(x)=xcos⁡2(x)f{\left(x \right)} = x \cos^{2}{\left(x \right)} 
f = x*cos(x)^2
The graph of the function
0-60-40-2020406080-2020
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⁡2(x)=0x \cos^{2}{\left(x \right)} = 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=3π2x_{3} = \frac{3 \pi}{2}
Numerical solution
x1=98.9601686087366x_{1} = 98.9601686087366
x2=−61.2610566539601x_{2} = -61.2610566539601
x3=4.71238887959385x_{3} = 4.71238887959385
x4=32.9867229629465x_{4} = 32.9867229629465
x5=86.3937978884654x_{5} = 86.3937978884654
x6=−10.9955745631002x_{6} = -10.9955745631002
x7=−10.9955743263138x_{7} = -10.9955743263138
x8=83.2522052483519x_{8} = 83.2522052483519
x9=−20.4203520386574x_{9} = -20.4203520386574
x10=−1.57079644421114x_{10} = -1.57079644421114
x11=−95.818575868229x_{11} = -95.818575868229
x12=−73.8274272802265x_{12} = -73.8274272802265
x13=−29.8451300972962x_{13} = -29.8451300972962
x14=36.1283155438697x_{14} = 36.1283155438697
x15=−58.1194639996716x_{15} = -58.1194639996716
x16=4.71238879747405x_{16} = 4.71238879747405
x17=0x_{17} = 0
x18=−80.1106125797813x_{18} = -80.1106125797813
x19=−17.2787598166076x_{19} = -17.2787598166076
x20=61.2610566964313x_{20} = 61.2610566964313
x21=10.995574454305x_{21} = 10.995574454305
x22=10.9955740709476x_{22} = 10.9955740709476
x23=−7.85398150362569x_{23} = -7.85398150362569
x24=51.8362789005928x_{24} = 51.8362789005928
x25=26.7035375075089x_{25} = 26.7035375075089
x26=−45.5530935889655x_{26} = -45.5530935889655
x27=−54.9778713319267x_{27} = -54.9778713319267
x28=7.85398174387101x_{28} = 7.85398174387101
x29=1.57079662189054x_{29} = 1.57079662189054
x30=−92.6769832051665x_{30} = -92.6769832051665
x31=23.5619451288484x_{31} = 23.5619451288484
x32=−23.5619450101619x_{32} = -23.5619450101619
x33=39.2699081548191x_{33} = 39.2699081548191
x34=−67.544242168036x_{34} = -67.544242168036
x35=95.818575867218x_{35} = 95.818575867218
x36=14.1371671090135x_{36} = 14.1371671090135
x37=54.9778715082452x_{37} = 54.9778715082452
x38=73.8274274800118x_{38} = 73.8274274800118
x39=64.4026493089789x_{39} = 64.4026493089789
x40=20.4203521508117x_{40} = 20.4203521508117
x41=−76.969019889145x_{41} = -76.969019889145
x42=76.9690200583129x_{42} = 76.9690200583129
x43=29.8451303212946x_{43} = 29.8451303212946
x44=−70.685834665631x_{44} = -70.685834665631
x45=20.420352470754x_{45} = 20.420352470754
x46=−48.6946861271829x_{46} = -48.6946861271829
x47=−89.5353907471387x_{47} = -89.5353907471387
x48=−14.1371668409786x_{48} = -14.1371668409786
x49=−51.8362786901255x_{49} = -51.8362786901255
x50=−26.7035375987313x_{50} = -26.7035375987313
x51=42.4115007296533x_{51} = 42.4115007296533
x52=−32.9867227850364x_{52} = -32.9867227850364
x53=26.7035373507946x_{53} = 26.7035373507946
x54=−36.1283154198435x_{54} = -36.1283154198435
x55=−4.71238918991607x_{55} = -4.71238918991607
x56=−4.71238880469592x_{56} = -4.71238880469592
x57=17.2787596657011x_{57} = 17.2787596657011
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x)^2*x.
cos⁡2(0)0\cos^{2}{\left(0 \right)} 0
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⁡(x)cos⁡(x)+cos⁡2(x)=0- 2 x \sin{\left(x \right)} \cos{\left(x \right)} + \cos^{2}{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=95.8185759344887x_{1} = 95.8185759344887
x2=37.7123693157661x_{2} = 37.7123693157661
x3=50.2754273458806x_{3} = 50.2754273458806
x4=−0.653271187094403x_{4} = -0.653271187094403
x5=−81.6875298021918x_{5} = -81.6875298021918
x6=94.253084424113x_{6} = 94.253084424113
x7=12.6060134442754x_{7} = 12.6060134442754
x8=6.36162039206566x_{8} = 6.36162039206566
x9=42.4115008234622x_{9} = 42.4115008234622
x10=4.71238898038469x_{10} = 4.71238898038469
x11=−39.2699081698724x_{11} = -39.2699081698724
x12=−59.6986356231676x_{12} = -59.6986356231676
x13=−7.85398163397448x_{13} = -7.85398163397448
x14=81.6875298021918x_{14} = 81.6875298021918
x15=51.8362787842316x_{15} = 51.8362787842316
x16=67.5442420521806x_{16} = 67.5442420521806
x17=78.5461819355535x_{17} = 78.5461819355535
x18=−84.8288957966139x_{18} = -84.8288957966139
x19=45.553093477052x_{19} = 45.553093477052
x20=23.5619449019235x_{20} = 23.5619449019235
x21=−36.1283155162826x_{21} = -36.1283155162826
x22=20.4203522483337x_{22} = 20.4203522483337
x23=−23.5619449019235x_{23} = -23.5619449019235
x24=70.6858347057703x_{24} = 70.6858347057703
x25=−83.2522053201295x_{25} = -83.2522053201295
x26=36.1283155162826x_{26} = 36.1283155162826
x27=−80.1106126665397x_{27} = -80.1106126665397
x28=86.3937979737193x_{28} = 86.3937979737193
x29=−22.013857636623x_{29} = -22.013857636623
x30=56.5575080935408x_{30} = 56.5575080935408
x31=−15.7397193560049x_{31} = -15.7397193560049
x32=−14.1371669411541x_{32} = -14.1371669411541
x33=−42.4115008234622x_{33} = -42.4115008234622
x34=−37.7123693157661x_{34} = -37.7123693157661
x35=−3.29231002128209x_{35} = -3.29231002128209
x36=28.2920048800691x_{36} = 28.2920048800691
x37=92.6769832808989x_{37} = 92.6769832808989
x38=9.4774857054208x_{38} = 9.4774857054208
x39=−64.4026493985908x_{39} = -64.4026493985908
x40=−17.2787595947439x_{40} = -17.2787595947439
x41=65.9810235167388x_{41} = 65.9810235167388
x42=43.9936619344429x_{42} = 43.9936619344429
x43=89.5353906273091x_{43} = 89.5353906273091
x44=22.013857636623x_{44} = 22.013857636623
x45=58.1194640914112x_{45} = 58.1194640914112
x46=−58.1194640914112x_{46} = -58.1194640914112
x47=−31.43183263459x_{47} = -31.43183263459
x48=−94.253084424113x_{48} = -94.253084424113
x49=73.8274273593601x_{49} = 73.8274273593601
x50=48.6946861306418x_{50} = 48.6946861306418
x51=−65.9810235167388x_{51} = -65.9810235167388
x52=−51.8362787842316x_{52} = -51.8362787842316
x53=−12.6060134442754x_{53} = -12.6060134442754
x54=87.970277977177x_{54} = 87.970277977177
x55=26.7035375555132x_{55} = 26.7035375555132
x56=7.85398163397448x_{56} = 7.85398163397448
x57=−86.3937979737193x_{57} = -86.3937979737193
x58=59.6986356231676x_{58} = 59.6986356231676
x59=−72.26355003974x_{59} = -72.26355003974
x60=34.5719807601687x_{60} = 34.5719807601687
x61=−45.553093477052x_{61} = -45.553093477052
x62=−6.36162039206566x_{62} = -6.36162039206566
x63=−75.4048544617952x_{63} = -75.4048544617952
x64=−73.8274273593601x_{64} = -73.8274273593601
x65=72.26355003974x_{65} = 72.26355003974
x66=100.535938219808x_{66} = 100.535938219808
x67=−50.2754273458806x_{67} = -50.2754273458806
x68=−67.5442420521806x_{68} = -67.5442420521806
x69=−20.4203522483337x_{69} = -20.4203522483337
x70=−9.4774857054208x_{70} = -9.4774857054208
x71=−53.4164352526291x_{71} = -53.4164352526291
x72=64.4026493985908x_{72} = 64.4026493985908
x73=−61.261056745001x_{73} = -61.261056745001
x74=80.1106126665397x_{74} = 80.1106126665397
x75=−87.970277977177x_{75} = -87.970277977177
x76=−95.8185759344887x_{76} = -95.8185759344887
x77=14.1371669411541x_{77} = 14.1371669411541
x78=3.29231002128209x_{78} = 3.29231002128209
x79=−28.2920048800691x_{79} = -28.2920048800691
x80=−97.3945059759883x_{80} = -97.3945059759883
x81=1.5707963267949x_{81} = 1.5707963267949
x82=15.7397193560049x_{82} = 15.7397193560049
x83=−102.101761241668x_{83} = -102.101761241668
x84=−43.9936619344429x_{84} = -43.9936619344429
x85=−1.5707963267949x_{85} = -1.5707963267949
x86=−29.845130209103x_{86} = -29.845130209103
x87=29.845130209103x_{87} = 29.845130209103
x88=−89.5353906273091x_{88} = -89.5353906273091
The values of the extrema at the points:
(95.8185759344887, 3.676520165044e-28)

(37.7123693157661, 37.7057413561082)

(50.2754273458806, 50.2704552295047)

(-0.653271187094403, -0.411949279841571)

(-81.6875298021918, -81.6844694741999)

(94.253084424113, 94.2504320656642)

(12.6060134442754, 12.5862127897398)

(6.36162039206566, 6.32256349768101)

(42.4115008234622, 4.98859428281591e-28)

(4.71238898038469, 1.59017658143397e-31)

(-39.2699081698724, -2.36773935254175e-30)

(-59.6986356231676, -59.6944482165077)

(-7.85398163397448, -7.36192861774987e-31)

(81.6875298021918, 81.6844694741999)

(51.8362787842316, 3.09398107171563e-30)

(67.5442420521806, 1.3132184568469e-27)

(78.5461819355535, 78.5429992236281)

(-84.8288957966139, -84.8259487900249)

(45.553093477052, 1.74530768724744e-35)

(23.5619449019235, 1.73402701495236e-29)

(-36.1283155162826, -3.66424875021481e-28)

(20.4203522483337, 1.96251734458305e-29)

(-23.5619449019235, -1.73402701495236e-29)

(70.6858347057703, 6.77618297499812e-29)

(-83.2522053201295, -1.7964453843451e-28)

(36.1283155162826, 3.66424875021481e-28)

(-80.1106126665397, -1.92283264304371e-27)

(86.3937979737193, 3.32328051180095e-28)

(-22.013857636623, -22.002507009172)

(56.5575080935408, 56.5530881593697)

(-15.7397193560049, -15.7238519846239)

(-14.1371669411541, -4.29347676987172e-30)

(-42.4115008234622, -4.98859428281591e-28)

(-37.7123693157661, -37.7057413561082)

(-3.29231002128209, -3.21808738200779)

(28.2920048800691, 28.2831712204135)

(92.6769832808989, 2.69152684487792e-27)

(9.4774857054208, 9.45118061522278)

(-64.4026493985908, -2.61429235475567e-27)

(-17.2787595947439, -2.10139136502906e-29)

(65.9810235167388, 65.9772347661069)

(43.9936619344429, 43.9879800316228)

(89.5353906273091, 2.60267852044683e-27)

(22.013857636623, 22.002507009172)

(58.1194640914112, 1.39112146798308e-29)

(-58.1194640914112, -1.39112146798308e-29)

(-31.43183263459, -31.4238809266115)

(-94.253084424113, -94.2504320656642)

(73.8274273593601, 4.43565443427593e-28)

(48.6946861306418, 5.73178094238607e-28)

(-65.9810235167388, -65.9772347661069)

(-51.8362787842316, -3.09398107171563e-30)

(-12.6060134442754, -12.5862127897398)

(87.970277977177, 87.9674362000474)

(26.7035375555132, 1.44419018202913e-29)

(7.85398163397448, 7.36192861774987e-31)

(-86.3937979737193, -3.32328051180095e-28)

(59.6986356231676, 59.6944482165077)

(-72.26355003974, -72.260090646562)

(34.5719807601687, 34.564750982936)

(-45.553093477052, -1.74530768724744e-35)

(-6.36162039206566, -6.32256349768101)

(-75.4048544617952, -75.4015391711531)

(-73.8274273593601, -4.43565443427593e-28)

(72.26355003974, 72.260090646562)

(100.535938219808, 100.533451608344)

(-50.2754273458806, -50.2704552295047)

(-67.5442420521806, -1.3132184568469e-27)

(-20.4203522483337, -1.96251734458305e-29)

(-9.4774857054208, -9.45118061522278)

(-53.4164352526291, -53.4117554551774)

(64.4026493985908, 2.61429235475567e-27)

(-61.261056745001, -5.29879683037424e-28)

(80.1106126665397, 1.92283264304371e-27)

(-87.970277977177, -87.9674362000474)

(-95.8185759344887, -3.676520165044e-28)

(14.1371669411541, 4.29347676987172e-30)

(3.29231002128209, 3.21808738200779)

(-28.2920048800691, -28.2831712204135)

(-97.3945059759883, -97.3919391637355)

(1.5707963267949, 5.8895428941999e-33)

(15.7397193560049, 15.7238519846239)

(-102.101761241668, -2.45314668072882e-27)

(-43.9936619344429, -43.9879800316228)

(-1.5707963267949, -5.8895428941999e-33)

(-29.845130209103, -1.12127665170554e-29)

(29.845130209103, 1.12127665170554e-29)

(-89.5353906273091, -2.60267852044683e-27)


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=95.8185759344887x_{1} = 95.8185759344887
x2=−0.653271187094403x_{2} = -0.653271187094403
x3=−81.6875298021918x_{3} = -81.6875298021918
x4=42.4115008234622x_{4} = 42.4115008234622
x5=4.71238898038469x_{5} = 4.71238898038469
x6=−59.6986356231676x_{6} = -59.6986356231676
x7=51.8362787842316x_{7} = 51.8362787842316
x8=67.5442420521806x_{8} = 67.5442420521806
x9=−84.8288957966139x_{9} = -84.8288957966139
x10=45.553093477052x_{10} = 45.553093477052
x11=23.5619449019235x_{11} = 23.5619449019235
x12=20.4203522483337x_{12} = 20.4203522483337
x13=70.6858347057703x_{13} = 70.6858347057703
x14=36.1283155162826x_{14} = 36.1283155162826
x15=86.3937979737193x_{15} = 86.3937979737193
x16=−22.013857636623x_{16} = -22.013857636623
x17=−15.7397193560049x_{17} = -15.7397193560049
x18=−37.7123693157661x_{18} = -37.7123693157661
x19=−3.29231002128209x_{19} = -3.29231002128209
x20=92.6769832808989x_{20} = 92.6769832808989
x21=89.5353906273091x_{21} = 89.5353906273091
x22=58.1194640914112x_{22} = 58.1194640914112
x23=−31.43183263459x_{23} = -31.43183263459
x24=−94.253084424113x_{24} = -94.253084424113
x25=73.8274273593601x_{25} = 73.8274273593601
x26=48.6946861306418x_{26} = 48.6946861306418
x27=−65.9810235167388x_{27} = -65.9810235167388
x28=−12.6060134442754x_{28} = -12.6060134442754
x29=26.7035375555132x_{29} = 26.7035375555132
x30=7.85398163397448x_{30} = 7.85398163397448
x31=−72.26355003974x_{31} = -72.26355003974
x32=−6.36162039206566x_{32} = -6.36162039206566
x33=−75.4048544617952x_{33} = -75.4048544617952
x34=−50.2754273458806x_{34} = -50.2754273458806
x35=−9.4774857054208x_{35} = -9.4774857054208
x36=−53.4164352526291x_{36} = -53.4164352526291
x37=64.4026493985908x_{37} = 64.4026493985908
x38=80.1106126665397x_{38} = 80.1106126665397
x39=−87.970277977177x_{39} = -87.970277977177
x40=14.1371669411541x_{40} = 14.1371669411541
x41=−28.2920048800691x_{41} = -28.2920048800691
x42=−97.3945059759883x_{42} = -97.3945059759883
x43=1.5707963267949x_{43} = 1.5707963267949
x44=−43.9936619344429x_{44} = -43.9936619344429
x45=29.845130209103x_{45} = 29.845130209103
Maxima of the function at points:
x45=37.7123693157661x_{45} = 37.7123693157661
x45=50.2754273458806x_{45} = 50.2754273458806
x45=94.253084424113x_{45} = 94.253084424113
x45=12.6060134442754x_{45} = 12.6060134442754
x45=6.36162039206566x_{45} = 6.36162039206566
x45=−39.2699081698724x_{45} = -39.2699081698724
x45=−7.85398163397448x_{45} = -7.85398163397448
x45=81.6875298021918x_{45} = 81.6875298021918
x45=78.5461819355535x_{45} = 78.5461819355535
x45=−36.1283155162826x_{45} = -36.1283155162826
x45=−23.5619449019235x_{45} = -23.5619449019235
x45=−83.2522053201295x_{45} = -83.2522053201295
x45=−80.1106126665397x_{45} = -80.1106126665397
x45=56.5575080935408x_{45} = 56.5575080935408
x45=−14.1371669411541x_{45} = -14.1371669411541
x45=−42.4115008234622x_{45} = -42.4115008234622
x45=28.2920048800691x_{45} = 28.2920048800691
x45=9.4774857054208x_{45} = 9.4774857054208
x45=−64.4026493985908x_{45} = -64.4026493985908
x45=−17.2787595947439x_{45} = -17.2787595947439
x45=65.9810235167388x_{45} = 65.9810235167388
x45=43.9936619344429x_{45} = 43.9936619344429
x45=22.013857636623x_{45} = 22.013857636623
x45=−58.1194640914112x_{45} = -58.1194640914112
x45=−51.8362787842316x_{45} = -51.8362787842316
x45=87.970277977177x_{45} = 87.970277977177
x45=−86.3937979737193x_{45} = -86.3937979737193
x45=59.6986356231676x_{45} = 59.6986356231676
x45=34.5719807601687x_{45} = 34.5719807601687
x45=−45.553093477052x_{45} = -45.553093477052
x45=−73.8274273593601x_{45} = -73.8274273593601
x45=72.26355003974x_{45} = 72.26355003974
x45=100.535938219808x_{45} = 100.535938219808
x45=−67.5442420521806x_{45} = -67.5442420521806
x45=−20.4203522483337x_{45} = -20.4203522483337
x45=−61.261056745001x_{45} = -61.261056745001
x45=−95.8185759344887x_{45} = -95.8185759344887
x45=3.29231002128209x_{45} = 3.29231002128209
x45=15.7397193560049x_{45} = 15.7397193560049
x45=−102.101761241668x_{45} = -102.101761241668
x45=−1.5707963267949x_{45} = -1.5707963267949
x45=−29.845130209103x_{45} = -29.845130209103
x45=−89.5353906273091x_{45} = -89.5353906273091
Decreasing at intervals
[95.8185759344887,∞)\left[95.8185759344887, \infty\right)
Increasing at intervals
(−∞,−97.3945059759883]\left(-\infty, -97.3945059759883\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(x(sin⁡2(x)−cos⁡2(x))−2sin⁡(x)cos⁡(x))=02 \left(x \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) - 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) = 0
Solve this equation
The roots of this equation
x1=−71.4782275499213x_{1} = -71.4782275499213
x2=32.2168395518658x_{2} = 32.2168395518658
x3=60.4839244878466x_{3} = 60.4839244878466
x4=69.9075883539626x_{4} = 69.9075883539626
x5=−49.4901859325761x_{5} = -49.4901859325761
x6=38.4974949445838x_{6} = 38.4974949445838
x7=85.6142396947314x_{7} = 85.6142396947314
x8=19.6603640661261x_{8} = 19.6603640661261
x9=−35.3570550332742x_{9} = -35.3570550332742
x10=−98.1798629425939x_{10} = -98.1798629425939
x11=−77.760847792972x_{11} = -77.760847792972
x12=−90.3263240494369x_{12} = -90.3263240494369
x13=74.6195257807054x_{13} = 74.6195257807054
x14=90.3263240494369x_{14} = 90.3263240494369
x15=11.8231619098018x_{15} = 11.8231619098018
x16=−10.2587614549708x_{16} = -10.2587614549708
x17=49.4901859325761x_{17} = 49.4901859325761
x18=−1.1444648640517x_{18} = -1.1444648640517
x19=−18.0917665453763x_{19} = -18.0917665453763
x20=18.0917665453763x_{20} = 18.0917665453763
x21=55.7722336752062x_{21} = 55.7722336752062
x22=91.8970257752571x_{22} = 91.8970257752571
x23=68.3369563786298x_{23} = 68.3369563786298
x24=5.58635293416499x_{24} = 5.58635293416499
x25=27.5071048394191x_{25} = 27.5071048394191
x26=40.0677825970372x_{26} = 40.0677825970372
x27=−25.9374070267134x_{27} = -25.9374070267134
x28=−19.6603640661261x_{28} = -19.6603640661261
x29=−62.0545116429054x_{29} = -62.0545116429054
x30=63.6251091208926x_{30} = 63.6251091208926
x31=−47.9197205706165x_{31} = -47.9197205706165
x32=−76.1901839979235x_{32} = -76.1901839979235
x33=47.9197205706165x_{33} = 47.9197205706165
x34=120.170079673253x_{34} = 120.170079673253
x35=82.4728694594266x_{35} = 82.4728694594266
x36=−4.04808180161146x_{36} = -4.04808180161146
x37=8.69662198229738x_{37} = 8.69662198229738
x38=2.54349254705114x_{38} = 2.54349254705114
x39=41.6381085824888x_{39} = 41.6381085824888
x40=−54.2016970313842x_{40} = -54.2016970313842
x41=96.6091494063022x_{41} = 96.6091494063022
x42=−60.4839244878466x_{42} = -60.4839244878466
x43=−79.3315168346756x_{43} = -79.3315168346756
x44=−38.4974949445838x_{44} = -38.4974949445838
x45=−58.9133484807877x_{45} = -58.9133484807877
x46=16.5235843473527x_{46} = 16.5235843473527
x47=76.1901839979235x_{47} = 76.1901839979235
x48=−69.9075883539626x_{48} = -69.9075883539626
x49=46.3492776216985x_{49} = 46.3492776216985
x50=−32.2168395518658x_{50} = -32.2168395518658
x51=−82.4728694594266x_{51} = -82.4728694594266
x52=10.2587614549708x_{52} = 10.2587614549708
x53=−46.3492776216985x_{53} = -46.3492776216985
x54=−68.3369563786298x_{54} = -68.3369563786298
x55=88.7556256712795x_{55} = 88.7556256712795
x56=54.2016970313842x_{56} = 54.2016970313842
x57=62.0545116429054x_{57} = 62.0545116429054
x58=−41.6381085824888x_{58} = -41.6381085824888
x59=−63.6251091208926x_{59} = -63.6251091208926
x60=25.9374070267134x_{60} = 25.9374070267134
x61=−27.5071048394191x_{61} = -27.5071048394191
x62=−84.0435524991391x_{62} = -84.0435524991391
x63=−55.7722336752062x_{63} = -55.7722336752062
x64=33.7869153354295x_{64} = 33.7869153354295
x65=84.0435524991391x_{65} = 84.0435524991391
x66=−11.8231619098018x_{66} = -11.8231619098018
x67=−99.7505790857949x_{67} = -99.7505790857949
x68=−93.4677306800165x_{68} = -93.4677306800165
x69=0x_{69} = 0
x70=−85.6142396947314x_{70} = -85.6142396947314
x71=77.760847792972x_{71} = 77.760847792972
x72=−40.0677825970372x_{72} = -40.0677825970372
x73=71.4782275499213x_{73} = 71.4782275499213
x74=24.3678503974527x_{74} = 24.3678503974527
x75=−57.3427845371101x_{75} = -57.3427845371101
x76=−5.58635293416499x_{76} = -5.58635293416499
x77=30.6468374831214x_{77} = 30.6468374831214
x78=−33.7869153354295x_{78} = -33.7869153354295
x79=−21.2292853858495x_{79} = -21.2292853858495
x80=99.7505790857949x_{80} = 99.7505790857949
x81=−66.766332133246x_{81} = -66.766332133246
x82=66.766332133246x_{82} = 66.766332133246
x83=52.6311758774383x_{83} = 52.6311758774383
x84=98.1798629425939x_{84} = 98.1798629425939
x85=−24.3678503974527x_{85} = -24.3678503974527
x86=4.04808180161146x_{86} = 4.04808180161146
x87=−91.8970257752571x_{87} = -91.8970257752571
x88=−13.3890435377793x_{88} = -13.3890435377793

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⁡2(x))=−∞sign⁡(⟨0,1⟩)\lim_{x \to -\infty}\left(x \cos^{2}{\left(x \right)}\right) = - \infty \operatorname{sign}{\left(\left\langle 0, 1\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=−∞sign⁡(⟨0,1⟩)y = - \infty \operatorname{sign}{\left(\left\langle 0, 1\right\rangle \right)}
lim⁡x→∞(xcos⁡2(x))=∞sign⁡(⟨0,1⟩)\lim_{x \to \infty}\left(x \cos^{2}{\left(x \right)}\right) = \infty \operatorname{sign}{\left(\left\langle 0, 1\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=∞sign⁡(⟨0,1⟩)y = \infty \operatorname{sign}{\left(\left\langle 0, 1\right\rangle \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x)^2*x, divided by x at x->+oo and x ->-oo
lim⁡x→−∞cos⁡2(x)=⟨0,1⟩\lim_{x \to -\infty} \cos^{2}{\left(x \right)} = \left\langle 0, 1\right\rangle
Let's take the limit
so,
inclined asymptote equation on the left:
y=x⟨0,1⟩y = x \left\langle 0, 1\right\rangle
lim⁡x→∞cos⁡2(x)=⟨0,1⟩\lim_{x \to \infty} \cos^{2}{\left(x \right)} = \left\langle 0, 1\right\rangle
Let's take the limit
so,
inclined asymptote equation on the right:
y=x⟨0,1⟩y = x \left\langle 0, 1\right\rangle
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⁡2(x)=−xcos⁡2(x)x \cos^{2}{\left(x \right)} = - x \cos^{2}{\left(x \right)}
- No
xcos⁡2(x)=xcos⁡2(x)x \cos^{2}{\left(x \right)} = x \cos^{2}{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = cos(x)^2*x
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