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Trazado el gráfico de la función/ arctgx/(x^2+12)

Gráfico de la función y = arctgx/(x^2+12)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       acot(x)
f(x) = -------
        2     
       x  + 12
f(x)=acot⁡(x)x2+12f{\left(x \right)} = \frac{\operatorname{acot}{\left(x \right)}}{x^{2} + 12} 
f = acot(x)/(x^2 + 12)
Gráfico de la función
02468-8-6-4-2-10100.25-0.25
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
acot⁡(x)x2+12=0\frac{\operatorname{acot}{\left(x \right)}}{x^{2} + 12} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=19603.5808654486x_{1} = 19603.5808654486
x2=−23702.9156045585x_{2} = -23702.9156045585
x3=31453.8086005791x_{3} = 31453.8086005791
x4=−11871.9979442106x_{4} = -11871.9979442106
x5=26373.3545985656x_{5} = 26373.3545985656
x6=−22856.6306589527x_{6} = -22856.6306589527
x7=35688.6447108881x_{7} = 35688.6447108881
x8=20449.4025454201x_{8} = 20449.4025454201
x9=−38945.8498812287x_{9} = -38945.8498812287
x10=−19472.636226644x_{10} = -19472.636226644
x11=−33863.4777311176x_{11} = -33863.4777311176
x12=16222.1498374366x_{12} = 16222.1498374366
x13=10318.7636465647x_{13} = 10318.7636465647
x14=−10188.5747062869x_{14} = -10188.5747062869
x15=36535.700510522x_{15} = 36535.700510522
x16=−30475.8321516995x_{16} = -30475.8321516995
x17=22987.6541069539x_{17} = 22987.6541069539
x18=33994.6150746654x_{18} = 33994.6150746654
x19=−12714.8770466922x_{19} = -12714.8770466922
x20=12002.459801906x_{20} = 12002.459801906
x21=−13558.342355548x_{21} = -13558.342355548
x22=24680.3427118023x_{22} = 24680.3427118023
x23=32300.7090366626x_{23} = 32300.7090366626
x24=−33016.5130408165x_{24} = -33016.5130408165
x25=−39792.9926786493x_{25} = -39792.9926786493
x26=−20318.4344919678x_{26} = -20318.4344919678
x27=9478.4414407088x_{27} = 9478.4414407088
x28=33147.6454227736x_{28} = 33147.6454227736
x29=−22010.4470710218x_{29} = -22010.4470710218
x30=29760.1278403165x_{30} = 29760.1278403165
x31=−18627.0017567296x_{31} = -18627.0017567296
x32=22141.4541563996x_{32} = 22141.4541563996
x33=40771.32125966x_{33} = 40771.32125966
x34=−24549.2914465586x_{34} = -24549.2914465586
x35=−36404.5502983023x_{35} = -36404.5502983023
x36=−26242.280702424x_{36} = -26242.280702424
x37=17912.4418387747x_{37} = 17912.4418387747
x38=13688.9829123395x_{38} = 13688.9829123395
x39=14532.9990407553x_{39} = 14532.9990407553
x40=−42334.5318497325x_{40} = -42334.5318497325
x41=42465.7036086828x_{41} = 42465.7036086828
x42=25526.8122576948x_{42} = 25526.8122576948
x43=−28782.2538673002x_{43} = -28782.2538673002
x44=−37251.6272276598x_{44} = -37251.6272276598
x45=−14402.2915089098x_{45} = -14402.2915089098
x46=−27935.5388044178x_{46} = -27935.5388044178
x47=39924.156373672x_{47} = 39924.156373672
x48=−17781.5543719122x_{48} = -17781.5543719122
x49=15377.40838734x_{49} = 15377.40838734
x50=−16936.3219853541x_{50} = -16936.3219853541
x51=27219.9629534748x_{51} = 27219.9629534748
x52=−38098.7275980981x_{52} = -38098.7275980981
x53=17067.1742874566x_{53} = 17067.1742874566
x54=−40640.1547084183x_{54} = -40640.1547084183
x55=28066.6313575868x_{55} = 28066.6313575868
x56=28913.3545430823x_{56} = 28913.3545430823
x57=−35557.4984838541x_{57} = -35557.4984838541
x58=−21164.3769687586x_{58} = -21164.3769687586
x59=−41487.3347931834x_{59} = -41487.3347931834
x60=21295.3657012151x_{60} = 21295.3657012151
x61=30606.9470953203x_{61} = 30606.9470953203
x62=37382.7811569958x_{62} = 37382.7811569958
x63=−27088.8792933523x_{63} = -27088.8792933523
x64=11160.1790436696x_{64} = 11160.1790436696
x65=−34710.4736212948x_{65} = -34710.4736212948
x66=−9348.4462585132x_{66} = -9348.4462585132
x67=39077.0105320832x_{67} = 39077.0105320832
x68=34841.6155686792x_{68} = 34841.6155686792
x69=−25395.7491138888x_{69} = -25395.7491138888
x70=23833.953702267x_{70} = 23833.953702267
x71=41618.5040278486x_{71} = 41618.5040278486
x72=−32169.5820120744x_{72} = -32169.5820120744
x73=−15246.6445957806x_{73} = -15246.6445957806
x74=−29629.0197259321x_{74} = -29629.0197259321
x75=−31322.6873723474x_{75} = -31322.6873723474
x76=38229.8849999843x_{76} = 38229.8849999843
x77=12845.4370221647x_{77} = 12845.4370221647
x78=−16091.3383375545x_{78} = -16091.3383375545
x79=18757.9197416093x_{79} = 18757.9197416093
x80=−11029.8382893323x_{80} = -11029.8382893323
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en acot(x)/(x^2 + 12).
acot⁡(0)02+12\frac{\operatorname{acot}{\left(0 \right)}}{0^{2} + 12}
Resultado:
f(0)=π24f{\left(0 \right)} = \frac{\pi}{24}
Punto:
(0, pi/24)
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
−2xacot⁡(x)(x2+12)2−1(x2+1)(x2+12)=0- \frac{2 x \operatorname{acot}{\left(x \right)}}{\left(x^{2} + 12\right)^{2}} - \frac{1}{\left(x^{2} + 1\right) \left(x^{2} + 12\right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−4360.12657634809x_{1} = -4360.12657634809
x2=−8060.14976299884x_{2} = -8060.14976299884
x3=−9585.29845337467x_{3} = -9585.29845337467
x4=2232.19615359684x_{4} = 2232.19615359684
x5=−3708.66495835059x_{5} = -3708.66495835059
x6=2228.22064612239x_{6} = 2228.22064612239
x7=5698.64631137486x_{7} = 5698.64631137486
x8=−10021.13030227x_{8} = -10021.13030227
x9=4393.77640207287x_{9} = 4393.77640207287
x10=4611.11519703223x_{10} = 4611.11519703223
x11=6133.93878270854x_{11} = 6133.93878270854
x12=6787.08144708697x_{12} = 6787.08144708697
x13=3308.52226956504x_{13} = 3308.52226956504
x14=−5447.35794262058x_{14} = -5447.35794262058
x15=8529.5950365092x_{15} = 8529.5950365092
x16=1799.46311093739x_{16} = 1799.46311093739
x17=−5012.30301905213x_{17} = -5012.30301905213
x18=2443.57156345037x_{18} = 2443.57156345037
x19=−2410.19055521587x_{19} = -2410.19055521587
x20=−7624.49165612708x_{20} = -7624.49165612708
x21=−5664.94792508273x_{21} = -5664.94792508273
x22=7440.41162911171x_{22} = 7440.41162911171
x23=10926.6181406867x_{23} = 10926.6181406867
x24=−10456.9878754167x_{24} = -10456.9878754167
x25=−9803.21094796885x_{25} = -9803.21094796885
x26=−8931.60714549778x_{26} = -8931.60714549778
x27=−10892.8680863725x_{27} = -10892.8680863725
x28=−6753.36209877807x_{28} = -6753.36209877807
x29=6351.62860216233x_{29} = 6351.62860216233
x30=−9149.49600262774x_{30} = -9149.49600262774
x31=7658.22174077254x_{31} = 7658.22174077254
x32=−9367.39329693573x_{32} = -9367.39329693573
x33=8965.34784579568x_{33} = 8965.34784579568
x34=7222.61720395967x_{34} = 7222.61720395967
x35=2875.49617961712x_{35} = 2875.49617961712
x36=−3058.37290655895x_{36} = -3058.37290655895
x37=5263.49405044165x_{37} = 5263.49405044165
x38=−5882.57343170759x_{38} = -5882.57343170759
x39=2659.36286508512x_{39} = 2659.36286508512
x40=7876.0462386779x_{40} = 7876.0462386779
x41=−3274.9641438687x_{41} = -3274.9641438687
x42=−2194.91880172545x_{42} = -2194.91880172545
x43=−6971.1175151447x_{43} = -6971.1175151447
x44=4176.51369520699x_{44} = 4176.51369520699
x45=−1766.41238180394x_{45} = -1766.41238180394
x46=3742.26936893857x_{46} = 3742.26936893857
x47=−2842.00680980548x_{47} = -2842.00680980548
x48=−8713.7273580381x_{48} = -8713.7273580381
x49=5045.98181329941x_{49} = 5045.98181329941
x50=3091.90025757797x_{50} = 3091.90025757797
x51=−1980.25355880048x_{51} = -1980.25355880048
x52=2013.44968090002x_{52} = 2013.44968090002
x53=−8495.8573375382x_{53} = -8495.8573375382
x54=9183.23804574845x_{54} = 9183.23804574845
x55=4828.51983103122x_{55} = 4828.51983103122
x56=8093.88396230433x_{56} = 8093.88396230433
x57=−3925.71750319763x_{57} = -3925.71750319763
x58=5916.2769517365x_{58} = 5916.2769517365
x59=6569.34363131717x_{59} = 6569.34363131717
x60=7004.83992698629x_{60} = 7004.83992698629
x61=−8277.99785459491x_{61} = -8277.99785459491
x62=−2625.92110341184x_{62} = -2625.92110341184
x63=8747.46661379663x_{63} = 8747.46661379663
x64=1375.27106210798x_{64} = 1375.27106210798
x65=−6100.23066632744x_{65} = -6100.23066632744
x66=−4577.45433935364x_{66} = -4577.45433935364
x67=−5229.80791014177x_{67} = -5229.80791014177
x68=8311.73387260043x_{68} = 8311.73387260043
x69=3959.33956225455x_{69} = 3959.33956225455
x70=10272.8036025903x_{70} = 10272.8036025903
x71=−4142.87666974265x_{71} = -4142.87666974265
x72=−1553.72836418746x_{72} = -1553.72836418746
x73=−7406.68387842315x_{73} = -7406.68387842315
x74=10708.6745866586x_{74} = 10708.6745866586
x75=−7188.89200183468x_{75} = -7188.89200183468
x76=−3491.7388369394x_{76} = -3491.7388369394
x77=−1342.73562537559x_{77} = -1342.73562537559
x78=−10674.9253243983x_{78} = -10674.9253243983
x79=−10239.05607852x_{79} = -10239.05607852
x80=−4794.8493990003x_{80} = -4794.8493990003
x81=10490.7362956907x_{81} = 10490.7362956907
x82=5481.05057077953x_{82} = 5481.05057077953
x83=9619.04291324894x_{83} = 9619.04291324894
x84=3525.32223851161x_{84} = 3525.32223851161
x85=−6317.91635442995x_{85} = -6317.91635442995
x86=10054.8768711649x_{86} = 10054.8768711649
x87=−7842.31401126387x_{87} = -7842.31401126387
x88=9836.95649742935x_{88} = 9836.95649742935
x89=9401.13659048815x_{89} = 9401.13659048815
x90=1586.57194298976x_{90} = 1586.57194298976
x91=−6535.62765662719x_{91} = -6535.62765662719
Signos de extremos en los puntos:
(-4360.1265763480915, -1.20643082912952e-11)

(-8060.149762998841, -1.9097239007911e-12)

(-9585.298453374668, -1.13548923722249e-12)

(2232.196153596841, 8.99087290840881e-11)

(-3708.6649583505905, -1.96040954358131e-11)

(2228.2206461223905, 9.03908227166096e-11)

(5698.646311374858, 5.40361906866529e-12)

(-10021.13030227002, -9.93687482614462e-13)

(4393.77640207287, 1.17892415604732e-11)

(4611.115197032229, 1.01995692694973e-11)

(6133.938782708542, 4.33292872918898e-12)

(6787.081447086972, 3.1985320076139e-12)

(3308.522269565041, 2.76119656242492e-11)

(-5447.357942620581, -6.18645814721903e-12)

(8529.595036509203, 1.61144203807887e-12)

(1799.463110937393, 1.71620633901507e-10)

(-5012.303019052133, -7.94123104204964e-12)

(2443.571563450374, 6.85368145608751e-11)

(-2410.1905552158696, -7.14241264017831e-11)

(-7624.491656127081, -2.25614614056736e-12)

(-5664.947925082734, -5.50062536519614e-12)

(7440.411629111706, 2.4277782404329e-12)

(10926.61814068673, 7.66553829825206e-13)

(-10456.98787541667, -8.74540925179594e-13)

(-9803.210947968846, -1.0614386568335e-12)

(-8931.607145497775, -1.403495729115e-12)

(-10892.868086372458, -7.73701113524408e-13)

(-6753.36209877807, -3.24668215594303e-12)

(6351.628602162328, 3.90251499131334e-12)

(-9149.496002627737, -1.30559482742556e-12)

(7658.221740772542, 2.22646615451123e-12)

(-9367.393296935734, -1.2165884094823e-12)

(8965.347845795677, 1.38770930450604e-12)

(7222.617203959666, 2.65409279639476e-12)

(2875.496179617116, 4.20591960888666e-11)

(-3058.3729065589478, -3.4956513810457e-11)

(5263.494050441651, 6.85768287055537e-12)

(-5882.573431707593, -4.91244582289644e-12)

(2659.362865085115, 5.31699733328918e-11)

(7876.046238677896, 2.04679874620175e-12)

(-3274.9641438687017, -2.84695008114753e-11)

(-2194.9188017254487, -9.45678492708277e-11)

(-6971.117515144699, -2.95183907969577e-12)

(4176.5136952069915, 1.37264414968015e-11)

(-1766.4123818039427, -1.81435381616251e-10)

(3742.2693689385724, 1.90807078936892e-11)

(-2842.0068098054826, -4.35636233207266e-11)

(-8713.727358038097, -1.51142996342871e-12)

(5045.981813299411, 7.78328166042216e-12)

(3091.9002575779677, 3.38316370774726e-11)

(-1980.2535588004805, -1.28776384194482e-10)

(2013.449680900016, 1.22511353374969e-10)

(-8495.8573375382, -1.63071585001502e-12)

(9183.238045748449, 1.29125617100504e-12)

(4828.519831031217, 8.88296011286378e-12)

(8093.883962304328, 1.88594488478865e-12)

(-3925.717503197635, -1.65288454726794e-11)

(5916.276951736497, 4.82896835285547e-12)

(6569.343631317167, 3.52723127704661e-12)

(7004.839926986288, 2.90941212768906e-12)

(-8277.997854594912, -1.76288504400633e-12)

(-2625.921103411845, -5.52273497614387e-11)

(8747.46661379663, 1.49400843410532e-12)

(1375.2710621079846, 3.84443261947859e-10)

(-6100.230666327439, -4.40515388835828e-12)

(-4577.4543393536405, -1.04262391389471e-11)

(-5229.807910141773, -6.99105294795593e-12)

(8311.733872600435, 1.74150623725827e-12)

(3959.339562254549, 1.61113306217276e-11)

(10272.803602590348, 9.22429096145035e-13)

(-4142.876669742651, -1.40635082824767e-11)

(-1553.728364187457, -2.66607583751994e-10)

(-7406.6838784231495, -2.461095549377e-12)

(10708.674586658624, 8.14315663845166e-13)

(-7188.892001834681, -2.69162168476282e-12)

(-3491.7388369394016, -2.34895285625075e-11)

(-1342.7356253755852, -4.13071749373307e-10)

(-10674.925324398342, -8.22063595276857e-13)

(-10239.056078519952, -9.31580059987144e-13)

(-4794.849399000296, -9.07141120814539e-12)

(10490.736295690673, 8.66127923502696e-13)

(5481.050570779533, 6.07307150260347e-12)

(9619.042913248937, 1.12358091924907e-12)

(3525.322238511607, 2.28245964190028e-11)

(-6317.916354429945, -3.96532009064325e-12)

(10054.87687116495, 9.83715867804223e-13)

(-7842.314011263869, -2.07332426134036e-12)

(9836.956497429353, 1.05055233478143e-12)

(9401.136590488146, 1.2035353517068e-12)

(1586.5719429897629, 2.50390912190865e-10)

(-6535.627656627194, -3.58210217098777e-12)


Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
La función no tiene puntos mínimos
La función no tiene puntos máximos
Decrece en todo el eje numérico
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
2(2x(x2+1)(x2+12)+x(x2+1)2+(4x2x2+12−1)acot⁡(x)x2+12)x2+12=0\frac{2 \left(\frac{2 x}{\left(x^{2} + 1\right) \left(x^{2} + 12\right)} + \frac{x}{\left(x^{2} + 1\right)^{2}} + \frac{\left(\frac{4 x^{2}}{x^{2} + 12} - 1\right) \operatorname{acot}{\left(x \right)}}{x^{2} + 12}\right)}{x^{2} + 12} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=3338.48766975712x_{1} = 3338.48766975712
x2=−1481.26045746958x_{2} = -1481.26045746958
x3=−1573.10591607386x_{3} = -1573.10591607386
x4=−1848.98384753028x_{4} = -1848.98384753028
x5=−3231.90031340261x_{5} = -3231.90031340261
x6=3061.62655148954x_{6} = 3061.62655148954
x7=−1297.81269373678x_{7} = -1297.81269373678
x8=765.493362989691x_{8} = 765.493362989691
x9=1220.41969311912x_{9} = 1220.41969311912
x10=4354.07019237528x_{10} = 4354.07019237528
x11=−4247.42351744386x_{11} = -4247.42351744386
x12=675.667524146536x_{12} = 675.667524146536
x13=2231.62473356863x_{13} = 2231.62473356863
x14=−3416.49374864545x_{14} = -3416.49374864545
x15=3523.09585895905x_{15} = 3523.09585895905
x16=3615.40861964748x_{16} = 3615.40861964748
x17=3430.78872026696x_{17} = 3430.78872026696
x18=−2955.06662502193x_{18} = -2955.06662502193
x19=−2217.35380707213x_{19} = -2217.35380707213
x20=−3508.7999775096x_{20} = -3508.7999775096
x21=2047.3729999172x_{21} = 2047.3729999172
x22=−3601.11189728098x_{22} = -3601.11189728098
x23=586.638596567495x_{23} = 586.638596567495
x24=−841.842427298942x_{24} = -841.842427298942
x25=−2401.68838043373x_{25} = -2401.68838043373
x26=2139.48631016284x_{26} = 2139.48631016284
x27=2600.3761940483x_{27} = 2600.3761940483
x28=−3047.33608094518x_{28} = -3047.33608094518
x29=−2309.51111231931x_{29} = -2309.51111231931
x30=2508.16302198485x_{30} = 2508.16302198485
x31=4631.1238948429x_{31} = 4631.1238948429
x32=1587.33593829719x_{32} = 1587.33593829719
x33=−3878.07773910118x_{33} = -3878.07773910118
x34=−1941.03007400262x_{34} = -1941.03007400262
x35=3246.19322658204x_{35} = 3246.19322658204
x36=4077.04338971139x_{36} = 4077.04338971139
x37=2784.84310226124x_{37} = 2784.84310226124
x38=−572.944582157458x_{38} = -572.944582157458
x39=−3785.75114477467x_{39} = -3785.75114477467
x40=−2770.5572481958x_{40} = -2770.5572481958
x41=−399.913698984701x_{41} = -399.913698984701
x42=1863.23660118329x_{42} = 1863.23660118329
x43=0.106351110720591x_{43} = 0.106351110720591
x44=855.867898527652x_{44} = 855.867898527652
x45=−4155.08168792801x_{45} = -4155.08168792801
x46=4446.41868592013x_{46} = 4446.41868592013
x47=2969.35569961203x_{47} = 2969.35569961203
x48=1312.00413492169x_{48} = 1312.00413492169
x49=946.635183927018x_{49} = 946.635183927018
x50=1037.69275026019x_{50} = 1037.69275026019
x51=4169.38233747283x_{51} = 4169.38233747283
x52=−751.540800724547x_{52} = -751.540800724547
x53=−1206.24727257418x_{53} = -1206.24727257418
x54=1403.6975592915x_{54} = 1403.6975592915
x55=−2493.88339931342x_{55} = -2493.88339931342
x56=−1665.01457630092x_{56} = -1665.01457630092
x57=−2033.1099338936x_{57} = -2033.1099338936
x58=−4524.46746902514x_{58} = -4524.46746902514
x59=−4616.82099183234x_{59} = -4616.82099183234
x60=1495.47996253163x_{60} = 1495.47996253163
x61=2323.78528518779x_{61} = 2323.78528518779
x62=−485.371407291448x_{62} = -485.371407291448
x63=1955.28834869394x_{63} = 1955.28834869394
x64=−4062.74328521871x_{64} = -4062.74328521871
x65=1771.22288149234x_{65} = 1771.22288149234
x66=−4432.11660030056x_{66} = -4432.11660030056
x67=1128.97067485777x_{67} = 1128.97067485777
x68=−1023.57461242716x_{68} = -1023.57461242716
x69=−2678.31936094924x_{69} = -2678.31936094924
x70=−4339.76855506379x_{70} = -4339.76855506379
x71=−932.556716589136x_{71} = -932.556716589136
x72=2692.60335036032x_{72} = 2692.60335036032
x73=−661.818880945226x_{73} = -661.818880945226
x74=−2586.09427165635x_{74} = -2586.09427165635
x75=3707.72658252995x_{75} = 3707.72658252995
x76=498.823042219015x_{76} = 498.823042219015
x77=−3693.42908137384x_{77} = -3693.42908137384
x78=−1389.49070871297x_{78} = -1389.49070871297
x79=−3139.61422504539x_{79} = -3139.61422504539
x80=3153.90597059664x_{80} = 3153.90597059664
x81=−2125.21905958223x_{81} = -2125.21905958223
x82=−1114.82210027979x_{82} = -1114.82210027979
x83=4261.7246769455x_{83} = 4261.7246769455
x84=2877.0942388847x_{84} = 2877.0942388847
x85=3984.70806940041x_{85} = 3984.70806940041
x86=−2862.80669697552x_{86} = -2862.80669697552
x87=−3324.19368460896x_{87} = -3324.19368460896
x88=412.961919477159x_{88} = 412.961919477159
x89=2415.96543445005x_{89} = 2415.96543445005
x90=1679.25343260401x_{90} = 1679.25343260401
x91=4538.76997582979x_{91} = 4538.76997582979
x92=−3970.40854834927x_{92} = -3970.40854834927
x93=−1756.97653420511x_{93} = -1756.97653420511
x94=3892.3766346318x_{94} = 3892.3766346318
x95=3800.04936856228x_{95} = 3800.04936856228

Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
[0.106351110720591,∞)\left[0.106351110720591, \infty\right)
Convexa en los intervalos
(−∞,0.106351110720591]\left(-\infty, 0.106351110720591\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(acot⁡(x)x2+12)=0\lim_{x \to -\infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{x^{2} + 12}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(acot⁡(x)x2+12)=0\lim_{x \to \infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{x^{2} + 12}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función acot(x)/(x^2 + 12), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(acot⁡(x)x(x2+12))=0\lim_{x \to -\infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{x \left(x^{2} + 12\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(acot⁡(x)x(x2+12))=0\lim_{x \to \infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{x \left(x^{2} + 12\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
acot⁡(x)x2+12=−acot⁡(x)x2+12\frac{\operatorname{acot}{\left(x \right)}}{x^{2} + 12} = - \frac{\operatorname{acot}{\left(x \right)}}{x^{2} + 12}
- No
acot⁡(x)x2+12=acot⁡(x)x2+12\frac{\operatorname{acot}{\left(x \right)}}{x^{2} + 12} = \frac{\operatorname{acot}{\left(x \right)}}{x^{2} + 12}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор