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−(tan2(x)+4)2x(2tan2(x)+2)sin(x)tan(x)+tan2(x)+4xcos(x)+sin(x)=0Solve this equationThe roots of this equation
x1=454.703366375065x2=14.1371669411541x3=69.9460058299699x4=−80.1106126665397x5=−7.85398163397448x6=11.7561606649435x7=−90.2804556957995x8=−55.7243617726486x9=10.2725607549115x10=67.5442420521806x11=60.5216851226385x12=−76.2289489691309x13=80.1106126665397x14=−49.4416503630997x15=62.0071694687546x16=77.7144563954775x17=−29.845130209103x18=−19.6879744764593x19=−62.0071694687546x20=70.6858347057703x21=33.735647763708x22=2.40467079662226x23=−69.9460058299699x24=68.2900468139916x25=24.3132699931952x26=0x27=46.3003430565376x28=−85.65343015591x29=−5.49385370718343x30=76.2289489691309x31=45.553093477052x32=−54.238892641524x33=−10.2725607549115x34=−83.997442131108x35=82.5119289102023x36=−99.7050166950322x37=58.1194640914112x38=−7.13941996437343x39=38.5324692900873x40=4.01872293357174x41=−63.663110358029x42=−91.9364513523705x43=83.997442131108x44=−11.7561606649435x45=−51.8362787842316x46=−73.8274273593601x47=−47.9562027081837x48=4.71238898038469x49=18.0330900698586x50=−95.8185759344887x51=−25.9686624297459x52=−98.2194935076171x53=1.5707963267949x54=16.5483332246309x55=54.238892641524x56=90.2804556957995x57=92.6769832808989x58=73.0874722075759x59=25.9686624297459x60=−40.0178633797347x61=−68.2900468139916x62=−58.865755516954x63=55.7243617726486x64=−18.0330900698586x65=32.2503175803598x66=−33.735647763708x67=−32.2503175803598x68=5.49385370718343x69=27.453878484238x70=40.0178633797347x71=87.138945811754x72=36.1283155162826x73=−93.4219711563462x74=−4.01872293357174x75=99.7050166950322x76=−16.5483332246309x77=0.997817540674201x78=91.9364513523705x79=−36.8767140943411x80=−27.453878484238x81=−77.7144563954775x82=98.2194935076171x83=−71.4315058489604x84=−41.6736614988808x85=47.9562027081837x86=−46.3003430565376The values of the extrema at the points:
(454.7033663750651, 64.582180890724)
(14.137166941154069, 4.29347676987172e-30)
(69.9460058299699, 9.93432901594521)
(-80.11061266653972, -1.92283264304371e-27)
(-7.853981633974483, 7.36192861774987e-31)
(11.756160664943506, -1.66847511356391)
(-90.28045569579949, 12.8225080937804)
(-55.72436177264862, -7.91435195205804)
(10.27256075491152, -1.45763299397885)
(67.54424205218055, -1.3132184568469e-27)
(60.52168512263849, -8.59574594780144)
(-76.2289489691309, 10.8267221900235)
(80.11061266653972, -1.92283264304371e-27)
(-49.441650363099725, -7.02197513700988)
(62.0071694687546, -8.80673558504463)
(77.71445639547751, 11.0377136292074)
(-29.845130209103036, -1.12127665170554e-29)
(-19.687974476459267, 2.79557707373584)
(-62.0071694687546, -8.80673558504463)
(70.68583470577035, 6.77618297499813e-29)
(33.73564776370796, 4.7910880047921)
(2.4046707966222645, 0.335020556538555)
(-69.9460058299699, 9.93432901594521)
(68.29004681399161, -9.69912415120835)
(24.313269993195227, -3.45264222691292)
(0, 0)
(46.30034305653763, 6.57579015556219)
(-85.65343015590999, -12.1653166830211)
(-5.493853707183429, -0.777532226748693)
(76.2289489691309, 10.8267221900235)
(45.553093477052, 1.74530768724744e-35)
(-54.238892641523954, -7.70336350275712)
(-10.27256075491152, -1.45763299397885)
(-83.99744213110796, 11.9301098752818)
(82.51192891020227, 11.7191179837181)
(-99.70501669503224, -14.1611083032433)
(58.119464091411174, 1.39112146798308e-29)
(-7.139419964373433, 1.012038154624)
(38.53246929008733, 5.47244715183819)
(4.018722933571741, -0.567362924067202)
(-63.66311035802902, 9.04193923563947)
(-91.93645135237047, -13.0577154126865)
(83.99744213110796, 11.9301098752818)
(-11.756160664943506, -1.66847511356391)
(-51.83627878423159, 3.09398107171563e-30)
(-73.82742735936014, -4.43565443427593e-28)
(-47.956202708183724, -6.81098836459494)
(4.71238898038469, -1.59017658143397e-31)
(18.033090069858616, -2.56044421510836)
(-95.81857593448869, 3.676520165044e-28)
(-25.968662429745905, 3.68780913408032)
(-98.21949350761713, -13.9501156336138)
(1.5707963267948966, 5.8895428941999e-33)
(16.54833322463088, -2.34951143571411)
(54.238892641523954, -7.70336350275712)
(90.28045569579949, 12.8225080937804)
(92.6769832808989, -2.69152684487792e-27)
(73.08747220757594, -10.3805252333763)
(25.968662429745905, 3.68780913408032)
(-40.01786337973469, 5.68342974137886)
(-68.29004681399161, -9.69912415120835)
(-58.86575551695403, 8.36054305299913)
(55.72436177264862, -7.91435195205804)
(-18.033090069858616, -2.56044421510836)
(32.2503175803598, 4.58011040647008)
(-33.73564776370796, 4.7910880047921)
(-32.2503175803598, 4.58011040647008)
(5.493853707183429, -0.777532226748693)
(27.453878484238032, 3.89877781647259)
(40.01786337973469, 5.68342974137886)
(87.13894581175403, -12.3763087646776)
(36.12831551628262, -3.66424875021481e-28)
(-93.42197115634625, -13.2687078181799)
(-4.018722933571741, -0.567362924067202)
(99.70501669503224, -14.1611083032433)
(-16.54833322463088, -2.34951143571411)
(0.9978175406742009, 0.130960306067674)
(91.93645135237047, -13.0577154126865)
(-36.87671409434108, -5.23725594007668)
(-27.453878484238032, 3.89877781647259)
(-77.71445639547751, 11.0377136292074)
(98.21949350761713, -13.9501156336138)
(-71.43150584896037, 10.1453198768279)
(-41.67366149888085, -5.91862382854916)
(47.956202708183724, -6.81098836459494)
(-46.30034305653763, 6.57579015556219)
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=14.1371669411541x2=−7.85398163397448x3=11.7561606649435x4=−55.7243617726486x5=10.2725607549115x6=60.5216851226385x7=−49.4416503630997x8=62.0071694687546x9=−62.0071694687546x10=70.6858347057703x11=68.2900468139916x12=24.3132699931952x13=0x14=−85.65343015591x15=−5.49385370718343x16=45.553093477052x17=−54.238892641524x18=−10.2725607549115x19=−99.7050166950322x20=58.1194640914112x21=4.01872293357174x22=−91.9364513523705x23=−11.7561606649435x24=−51.8362787842316x25=−47.9562027081837x26=18.0330900698586x27=−95.8185759344887x28=−98.2194935076171x29=1.5707963267949x30=16.5483332246309x31=54.238892641524x32=73.0874722075759x33=−68.2900468139916x34=55.7243617726486x35=−18.0330900698586x36=5.49385370718343x37=87.138945811754x38=−93.4219711563462x39=−4.01872293357174x40=99.7050166950322x41=−16.5483332246309x42=91.9364513523705x43=−36.8767140943411x44=98.2194935076171x45=−41.6736614988808x46=47.9562027081837Maxima of the function at points:
x46=454.703366375065x46=69.9460058299699x46=−80.1106126665397x46=−90.2804556957995x46=67.5442420521806x46=−76.2289489691309x46=80.1106126665397x46=77.7144563954775x46=−29.845130209103x46=−19.6879744764593x46=33.735647763708x46=2.40467079662226x46=−69.9460058299699x46=46.3003430565376x46=76.2289489691309x46=−83.997442131108x46=82.5119289102023x46=−7.13941996437343x46=38.5324692900873x46=−63.663110358029x46=83.997442131108x46=−73.8274273593601x46=4.71238898038469x46=−25.9686624297459x46=90.2804556957995x46=92.6769832808989x46=25.9686624297459x46=−40.0178633797347x46=−58.865755516954x46=32.2503175803598x46=−33.735647763708x46=−32.2503175803598x46=27.453878484238x46=40.0178633797347x46=36.1283155162826x46=0.997817540674201x46=−27.453878484238x46=−77.7144563954775x46=−71.4315058489604x46=−46.3003430565376Decreasing at intervals
[99.7050166950322,∞)Increasing at intervals
(−∞,−99.7050166950322]