In order to find the extrema, we need to solve the equation
dxdf(x)=0(the derivative equals zero),
and the roots of this equation are the extrema of this function:
dxdf(x)=the first derivative−2xsin(x)cos(x)+cos2(x)=0Solve this equationThe roots of this equation
x1=95.8185759344887x2=37.7123693157661x3=50.2754273458806x4=−0.653271187094403x5=−81.6875298021918x6=94.253084424113x7=12.6060134442754x8=6.36162039206566x9=42.4115008234622x10=4.71238898038469x11=−39.2699081698724x12=−59.6986356231676x13=−7.85398163397448x14=81.6875298021918x15=51.8362787842316x16=67.5442420521806x17=78.5461819355535x18=−84.8288957966139x19=45.553093477052x20=23.5619449019235x21=−36.1283155162826x22=20.4203522483337x23=−23.5619449019235x24=70.6858347057703x25=−83.2522053201295x26=36.1283155162826x27=−80.1106126665397x28=86.3937979737193x29=−22.013857636623x30=56.5575080935408x31=−15.7397193560049x32=−14.1371669411541x33=−42.4115008234622x34=−37.7123693157661x35=−3.29231002128209x36=28.2920048800691x37=92.6769832808989x38=9.4774857054208x39=−64.4026493985908x40=−17.2787595947439x41=65.9810235167388x42=43.9936619344429x43=89.5353906273091x44=22.013857636623x45=58.1194640914112x46=−58.1194640914112x47=−31.43183263459x48=−94.253084424113x49=73.8274273593601x50=48.6946861306418x51=−65.9810235167388x52=−51.8362787842316x53=−12.6060134442754x54=87.970277977177x55=26.7035375555132x56=7.85398163397448x57=−86.3937979737193x58=59.6986356231676x59=−72.26355003974x60=34.5719807601687x61=−45.553093477052x62=−6.36162039206566x63=−75.4048544617952x64=−73.8274273593601x65=72.26355003974x66=100.535938219808x67=−50.2754273458806x68=−67.5442420521806x69=−20.4203522483337x70=−9.4774857054208x71=−53.4164352526291x72=64.4026493985908x73=−61.261056745001x74=80.1106126665397x75=−87.970277977177x76=−95.8185759344887x77=14.1371669411541x78=3.29231002128209x79=−28.2920048800691x80=−97.3945059759883x81=1.5707963267949x82=15.7397193560049x83=−102.101761241668x84=−43.9936619344429x85=−1.5707963267949x86=−29.845130209103x87=29.845130209103x88=−89.5353906273091The 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=95.8185759344887x2=−0.653271187094403x3=−81.6875298021918x4=42.4115008234622x5=4.71238898038469x6=−59.6986356231676x7=51.8362787842316x8=67.5442420521806x9=−84.8288957966139x10=45.553093477052x11=23.5619449019235x12=20.4203522483337x13=70.6858347057703x14=36.1283155162826x15=86.3937979737193x16=−22.013857636623x17=−15.7397193560049x18=−37.7123693157661x19=−3.29231002128209x20=92.6769832808989x21=89.5353906273091x22=58.1194640914112x23=−31.43183263459x24=−94.253084424113x25=73.8274273593601x26=48.6946861306418x27=−65.9810235167388x28=−12.6060134442754x29=26.7035375555132x30=7.85398163397448x31=−72.26355003974x32=−6.36162039206566x33=−75.4048544617952x34=−50.2754273458806x35=−9.4774857054208x36=−53.4164352526291x37=64.4026493985908x38=80.1106126665397x39=−87.970277977177x40=14.1371669411541x41=−28.2920048800691x42=−97.3945059759883x43=1.5707963267949x44=−43.9936619344429x45=29.845130209103Maxima of the function at points:
x45=37.7123693157661x45=50.2754273458806x45=94.253084424113x45=12.6060134442754x45=6.36162039206566x45=−39.2699081698724x45=−7.85398163397448x45=81.6875298021918x45=78.5461819355535x45=−36.1283155162826x45=−23.5619449019235x45=−83.2522053201295x45=−80.1106126665397x45=56.5575080935408x45=−14.1371669411541x45=−42.4115008234622x45=28.2920048800691x45=9.4774857054208x45=−64.4026493985908x45=−17.2787595947439x45=65.9810235167388x45=43.9936619344429x45=22.013857636623x45=−58.1194640914112x45=−51.8362787842316x45=87.970277977177x45=−86.3937979737193x45=59.6986356231676x45=34.5719807601687x45=−45.553093477052x45=−73.8274273593601x45=72.26355003974x45=100.535938219808x45=−67.5442420521806x45=−20.4203522483337x45=−61.261056745001x45=−95.8185759344887x45=3.29231002128209x45=15.7397193560049x45=−102.101761241668x45=−1.5707963267949x45=−29.845130209103x45=−89.5353906273091Decreasing at intervals
[95.8185759344887,∞)Increasing at intervals
(−∞,−97.3945059759883]