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Trazado el gráfico de la función/ (x-atan(x))/x^4

Gráfico de la función y = (x-atan(x))/x^4

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       x - atan(x)
f(x) = -----------
             4    
            x
f(x)=x−atan⁡(x)x4f{\left(x \right)} = \frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}} 
f = (x - atan(x))/x^4
Gráfico de la función
02468-8-6-4-2-1010-1010
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
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:
x−atan⁡(x)x4=0\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−39994.3117715111x_{1} = -39994.3117715111
x2=18936.7305243798x_{2} = 18936.7305243798
x3=−14568.2175547177x_{3} = -14568.2175547177
x4=−32366.1463157406x_{4} = -32366.1463157406
x5=−19653.0118392567x_{5} = -19653.0118392567
x6=−38299.15446228x_{6} = -38299.15446228
x7=−35756.4276394746x_{7} = -35756.4276394746
x8=17241.7646307201x_{8} = 17241.7646307201
x9=38430.3834054348x_{9} = 38430.3834054348
x10=−11178.7812896696x_{10} = -11178.7812896696
x11=31649.807545843x_{11} = 31649.807545843
x12=40125.5410566807x_{12} = 40125.5410566807
x13=−39146.7325605076x_{13} = -39146.7325605076
x14=−12026.0877876276x_{14} = -12026.0877876276
x15=13852.0131898669x_{15} = 13852.0131898669
x16=36735.2304672236x_{16} = 36735.2304672236
x17=−22195.5529146244x_{17} = -22195.5529146244
x18=29954.6819417028x_{18} = 29954.6819417028
x19=32497.3736075588x_{19} = 32497.3736075588
x20=21479.2517369837x_{20} = 21479.2517369837
x21=−31518.5805692067x_{21} = -31518.5805692067
x22=26564.4626731453x_{22} = 26564.4626731453
x23=−23043.0800289223x_{23} = -23043.0800289223
x24=33344.9416202645x_{24} = 33344.9416202645
x25=−16263.0906158625x_{25} = -16263.0906158625
x26=−13720.812492127x_{26} = -13720.812492127
x27=−18805.5146223174x_{27} = -18805.5146223174
x28=29107.1227812346x_{28} = 29107.1227812346
x29=−24738.1502236852x_{29} = -24738.1502236852
x30=−36604.001914841x_{30} = -36604.001914841
x31=−42537.0554132388x_{31} = -42537.0554132388
x32=25716.9160556673x_{32} = 25716.9160556673
x33=−15415.6447618528x_{33} = -15415.6447618528
x34=37582.8063071541x_{34} = 37582.8063071541
x35=−21348.0319743947x_{35} = -21348.0319743947
x36=−28975.8969224047x_{36} = -28975.8969224047
x37=19784.2291965083x_{37} = 19784.2291965083
x38=18089.2418399584x_{38} = 18089.2418399584
x39=39277.9616802381x_{39} = 39277.9616802381
x40=−41689.4732592161x_{40} = -41689.4732592161
x41=−20500.5179804253x_{41} = -20500.5179804253
x42=−26433.2382714306x_{42} = -26433.2382714306
x43=11309.9654284321x_{43} = 11309.9654284321
x44=−27280.7879435868x_{44} = -27280.7879435868
x45=−25585.6922392123x_{45} = -25585.6922392123
x46=−23890.6126554485x_{46} = -23890.6126554485
x47=−9484.33077449154x_{47} = -9484.33077449154
x48=10462.7001393807x_{48} = 10462.7001393807
x49=−10331.5244397661x_{49} = -10331.5244397661
x50=−17958.0276051904x_{50} = -17958.0276051904
x51=−17110.5523186578x_{51} = -17110.5523186578
x52=13004.6302650024x_{52} = 13004.6302650024
x53=27412.0128766577x_{53} = 27412.0128766577
x54=30802.243597088x_{54} = 30802.243597088
x55=−29823.4556782561x_{55} = -29823.4556782561
x56=23174.301683587x_{56} = 23174.301683587
x57=40973.1214661016x_{57} = 40973.1214661016
x58=16394.3006948607x_{58} = 16394.3006948607
x59=42668.2851366246x_{59} = 42668.2851366246
x60=−40841.8920257127x_{60} = -40841.8920257127
x61=20631.7366152646x_{61} = 20631.7366152646
x62=−34061.2835852816x_{62} = -34061.2835852816
x63=12157.2786320315x_{63} = 12157.2786320315
x64=28259.5663415576x_{64} = 28259.5663415576
x65=14699.4219330538x_{65} = 14699.4219330538
x66=−30671.016962217x_{66} = -30671.016962217
x67=34192.5114381302x_{67} = 34192.5114381302
x68=24021.8351081285x_{68} = 24021.8351081285
x69=22326.773677669x_{69} = 22326.773677669
x70=15546.8522268746x_{70} = 15546.8522268746
x71=−28128.3409246126x_{71} = -28128.3409246126
x72=−33213.7140371608x_{72} = -33213.7140371608
x73=24869.3733934327x_{73} = 24869.3733934327
x74=−37451.5775527345x_{74} = -37451.5775527345
x75=35887.6559752384x_{75} = 35887.6559752384
x76=35040.0829295059x_{76} = 35040.0829295059
x77=−12873.4340044398x_{77} = -12873.4340044398
x78=41820.7028454293x_{78} = 41820.7028454293
x79=−34908.8548263803x_{79} = -34908.8548263803
x80=9615.4956486422x_{80} = 9615.4956486422
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 (x - atan(x))/x^4.
(−1)atan⁡(0)04\frac{\left(-1\right) \operatorname{atan}{\left(0 \right)}}{0^{4}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
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
1−1x2+1x4−4(x−atan⁡(x))x5=0\frac{1 - \frac{1}{x^{2} + 1}}{x^{4}} - \frac{4 \left(x - \operatorname{atan}{\left(x \right)}\right)}{x^{5}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−9635.16086967068x_{1} = -9635.16086967068
x2=7924.45270508001x_{2} = 7924.45270508001
x3=−5710.16615325996x_{3} = -5710.16615325996
x4=−8762.91883789065x_{4} = -8762.91883789065
x5=−6146.25953865561x_{5} = -6146.25953865561
x6=7052.23069237461x_{6} = 7052.23069237461
x7=6180.02544917054x_{7} = 6180.02544917054
x8=4217.67951297655x_{8} = 4217.67951297655
x9=−5056.04145920651x_{9} = -5056.04145920651
x10=9450.86753715877x_{10} = 9450.86753715877
x11=8142.51019295217x_{11} = 8142.51019295217
x12=5525.88734327328x_{12} = 5525.88734327328
x13=−2440.01361836832x_{13} = -2440.01361836832
x14=−8108.74284034062x_{14} = -8108.74284034062
x15=−4401.94262668798x_{15} = -4401.94262668798
x16=9886.99043828033x_{16} = 9886.99043828033
x17=−9417.09968254908x_{17} = -9417.09968254908
x18=−7236.5180052863x_{18} = -7236.5180052863
x19=5961.97762857279x_{19} = 5961.97762857279
x20=8578.6271240014x_{20} = 8578.6271240014
x21=5307.84532286983x_{21} = 5307.84532286983
x22=3563.63525073288x_{22} = 3563.63525073288
x23=8796.68646966634x_{23} = 8796.68646966634
x24=−5274.08063017358x_{24} = -5274.08063017358
x25=−2875.91346755289x_{25} = -2875.91346755289
x26=−10071.2844248865x_{26} = -10071.2844248865
x27=9668.92878863151x_{27} = 9668.92878863151
x28=8360.56834969763x_{28} = 8360.56834969763
x29=2255.84284777311x_{29} = 2255.84284777311
x30=−1786.39023902939x_{30} = -1786.39023902939
x31=−4183.91755155239x_{31} = -4183.91755155239
x32=−1568.62798192089x_{32} = -1568.62798192089
x33=−5492.1222907781x_{33} = -5492.1222907781
x34=−4838.00511805066x_{34} = -4838.00511805066
x35=−10725.4723482245x_{35} = -10725.4723482245
x36=2037.95787900293x_{36} = 2037.95787900293
x37=1602.34377731744x_{37} = 1602.34377731744
x38=2473.76111677168x_{38} = 2473.76111677168
x39=−8544.85957816161x_{39} = -8544.85957816161
x40=6616.12555682936x_{40} = 6616.12555682936
x41=10105.0524601947x_{41} = 10105.0524601947
x42=3127.64505064611x_{42} = 3127.64505064611
x43=3999.65863276751x_{43} = 3999.65863276751
x44=9232.80671223416x_{44} = 9232.80671223416
x45=−1350.96857451446x_{45} = -1350.96857451446
x46=3781.64352081394x_{46} = 3781.64352081394
x47=−8980.97863348592x_{47} = -8980.97863348592
x48=−7018.46399104355x_{48} = -7018.46399104355
x49=−8326.80089666195x_{49} = -8326.80089666195
x50=10323.1148306644x_{50} = 10323.1148306644
x51=5743.93152489527x_{51} = 5743.93152489527
x52=−9199.03892662363x_{52} = -9199.03892662363
x53=2691.70445475524x_{53} = 2691.70445475524
x54=−7890.68546137018x_{54} = -7890.68546137018
x55=9014.74634499065x_{55} = 9014.74634499065
x56=1820.11848049166x_{56} = 1820.11848049166
x57=4871.76893866043x_{57} = 4871.76893866043
x58=−6364.30867056898x_{58} = -6364.30867056898
x59=−5928.2119725539x_{59} = -5928.2119725539
x60=7488.33997130571x_{60} = 7488.33997130571
x61=−6800.41102478015x_{61} = -6800.41102478015
x62=5089.8057442349x_{62} = 5089.8057442349
x63=4653.73530019236x_{63} = 4653.73530019236
x64=10977.303824696x_{64} = 10977.303824696
x65=6398.07480973781x_{65} = 6398.07480973781
x66=1384.66529443179x_{66} = 1384.66529443179
x67=−10289.3467426694x_{67} = -10289.3467426694
x68=−4619.97201179297x_{68} = -4619.97201179297
x69=7270.28486085012x_{69} = 7270.28486085012
x70=3345.63518005307x_{70} = 3345.63518005307
x71=−6582.35921146526x_{71} = -6582.35921146526
x72=−10507.4093905257x_{72} = -10507.4093905257
x73=2909.66713500245x_{73} = 2909.66713500245
x74=7706.39594324275x_{74} = 7706.39594324275
x75=−2004.22107401941x_{75} = -2004.22107401941
x76=10759.2405320859x_{76} = 10759.2405320859
x77=−3529.87628681345x_{77} = -3529.87628681345
x78=10541.1775279508x_{78} = 10541.1775279508
x79=−9853.22245920811x_{79} = -9853.22245920811
x80=4435.70530122338x_{80} = 4435.70530122338
x81=−3093.88923759967x_{81} = -3093.88923759967
x82=−3965.89750630056x_{82} = -3965.89750630056
x83=−7672.62881789392x_{83} = -7672.62881789392
x84=6834.1775567283x_{84} = 6834.1775567283
x85=−3311.87763474688x_{85} = -3311.87763474688
x86=−7454.57297490512x_{86} = -7454.57297490512
x87=−10943.5355971552x_{87} = -10943.5355971552
x88=−2657.95348889226x_{88} = -2657.95348889226
x89=−3747.8833805644x_{89} = -3747.8833805644
x90=−2222.09990170263x_{90} = -2222.09990170263
Signos de extremos en los puntos:
(-9635.160869670677, -1.11776958145545e-12)

(7924.452705080006, 2.00912093288887e-12)

(-5710.16615325996, -5.369505499223e-12)

(-8762.918837890646, -1.48585278088761e-12)

(-6146.259538655606, -4.30582444498396e-12)

(7052.23069237461, 2.85051763938332e-12)

(6180.0254491705355, 4.23563785298216e-12)

(4217.6795129765505, 1.33234752979403e-11)

(-5056.041459206507, -7.73451687364544e-12)

(9450.867537158769, 1.18443924224715e-12)

(8142.510192952174, 1.85200128943106e-12)

(5525.887343273277, 5.92475591907681e-12)

(-2440.0136183683203, -6.87929048602917e-11)

(-8108.742840340623, -1.87523321172825e-12)

(-4401.942626687983, -1.17195756557215e-11)

(9886.990438280332, 1.03451944451266e-12)

(-9417.099682549078, -1.19722576549115e-12)

(-7236.518005286297, -2.63825510263823e-12)

(5961.977628572791, 4.71752866551541e-12)

(8578.627124001401, 1.58367879911771e-12)

(5307.845322869832, 6.68523527959022e-12)

(3563.6352507328766, 2.20865978104899e-11)

(8796.686469666336, 1.46880826811012e-12)

(-5274.080630173582, -6.8144428561162e-12)

(-2875.913467552889, -4.20179961154258e-11)

(-10071.28442488646, -9.78763306996241e-13)

(9668.928788631507, 1.10609991563322e-12)

(8360.56834969763, 1.71084631052185e-12)

(2255.842847773106, 8.70504551674902e-11)

(-1786.3902390293924, -1.75262529830896e-10)

(-4183.917551552389, -1.36485838112269e-11)

(-1568.6279819208949, -2.58824384730569e-10)

(-5492.122290778105, -6.03469308883257e-12)

(-4838.005118050664, -8.82795313995201e-12)

(-10725.472348224523, -8.10377010365927e-13)

(2037.9578790029336, 1.18053705543013e-10)

(1602.343777317439, 2.42832673982205e-10)

(2473.7611167716786, 6.60162631030637e-11)

(-8544.859578161615, -1.60252706153691e-12)

(6616.125556829361, 3.45211831990863e-12)

(10105.052460194725, 9.68984370258066e-13)

(3127.6450506461083, 3.26685225521898e-11)

(3999.658632767512, 1.56228640389328e-11)

(9232.806712234158, 1.27035435320526e-12)

(-1350.9685745144648, -4.05097200287e-10)

(3781.6435208139396, 1.84832286538086e-11)

(-8980.97863348592, -1.38023504438876e-12)

(-7018.463991043547, -2.89185538223141e-12)

(-8326.80089666195, -1.73174335887348e-12)

(10323.114830664428, 9.08869732367151e-13)

(5743.931524895269, 5.27537659773035e-12)

(-9199.038926623629, -1.28439459797409e-12)

(2691.7044547552405, 5.12465254472768e-11)

(-7890.685461370175, -2.03502312117089e-12)

(9014.746344990652, 1.36478358583227e-12)

(1820.1184804916636, 1.65701403634991e-10)

(4871.768938660432, 8.64569519810368e-12)

(-6364.308670568983, -3.87827948720368e-12)

(-5928.211972553896, -4.79859116263575e-12)

(7488.339971305707, 2.38096076334153e-12)

(-6800.411024780154, -3.17902683397301e-12)

(5089.805744234899, 7.5816257601581e-12)

(4653.735300192361, 9.9185542641628e-12)

(10977.303824695986, 7.55876424564143e-13)

(6398.074809737811, 3.81720454449933e-12)

(1384.6652944317864, 3.76246832259478e-10)

(-10289.346742669373, -9.17846994344074e-13)

(-4619.972011792972, -1.01375801500692e-11)

(7270.284860850115, 2.60166804735459e-12)

(3345.6351800530674, 2.66907231397105e-11)

(-6582.359211465264, -3.50551332472551e-12)

(-10507.40939052567, -8.61882601343844e-13)

(2909.6671350024462, 4.05728584898987e-11)

(7706.395943242754, 2.18452751894021e-12)

(-2004.2210740194118, -1.24114558285075e-10)

(10759.240532085852, 8.02771127626173e-13)

(-3529.87628681345, -2.27262759870871e-11)

(10541.177527950755, 8.53626521964498e-13)

(-9853.222459208115, -1.04519157327288e-12)

(4435.705301223378, 1.14540254982792e-11)

(-3093.889237599666, -3.37493341432853e-11)

(-3965.8975063005573, -1.60252023237651e-11)

(-7672.628817893916, -2.21349487302131e-12)

(6834.1775567282975, 3.13214171811575e-12)

(-3311.8776347468806, -2.75151041710324e-11)

(-7454.572974905121, -2.41346038082211e-12)

(-10943.535597155234, -7.62894874802996e-13)

(-2657.9534888922576, -5.3223226078453e-11)

(-3747.883380564403, -1.89871493516154e-11)

(-2222.0999017026343, -9.10756411810749e-11)


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(1(x2+1)2−4(1−1x2+1)x2+10(x−atan⁡(x))x3)x3=0\frac{2 \left(\frac{1}{\left(x^{2} + 1\right)^{2}} - \frac{4 \left(1 - \frac{1}{x^{2} + 1}\right)}{x^{2}} + \frac{10 \left(x - \operatorname{atan}{\left(x \right)}\right)}{x^{3}}\right)}{x^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−572.465199827019x_{1} = -572.465199827019
x2=−2512.09779708369x_{2} = -2512.09779708369
x3=2434.00343656067x_{3} = 2434.00343656067
x4=−3898.24169087343x_{4} = -3898.24169087343
x5=2156.79462690364x_{5} = 2156.79462690364
x6=−2789.31742643728x_{6} = -2789.31742643728
x7=−3251.36234604233x_{7} = -3251.36234604233
x8=402.591722762769x_{8} = 402.591722762769
x9=−2974.13385043974x_{9} = -2974.13385043974
x10=−3713.41755254606x_{10} = -3713.41755254606
x11=−2419.69289812792x_{11} = -2419.69289812792
x12=−4452.71924174689x_{12} = -4452.71924174689
x13=4282.20453670567x_{13} = 4282.20453670567
x14=4651.85844778548x_{14} = 4651.85844778548
x15=3635.31751990797x_{15} = 3635.31751990797
x16=−2881.72534784275x_{16} = -2881.72534784275
x17=955.850637591234x_{17} = 955.850637591234
x18=2711.22107442408x_{18} = 2711.22107442408
x19=4097.3785935636x_{19} = 4097.3785935636
x20=−4637.54646101171x_{20} = -4637.54646101171
x21=−2234.88618505185x_{21} = -2234.88618505185
x22=1048.19102428609x_{22} = 1048.19102428609
x23=1510.03615690086x_{23} = 1510.03615690086
x24=2249.19637885223x_{24} = 2249.19637885223
x25=3080.85417255796x_{25} = 3080.85417255796
x26=−941.551515553724x_{26} = -941.551515553724
x27=−1033.88958061081x_{27} = -1033.88958061081
x28=863.527135536658x_{28} = 863.527135536658
x29=1787.20381100282x_{29} = 1787.20381100282
x30=2526.40847983995x_{30} = 2526.40847983995
x31=1971.9955197485x_{31} = 1971.9955197485
x32=2988.44506351476x_{32} = 2988.44506351476
x33=−2050.08452317832x_{33} = -2050.08452317832
x34=−2142.4846400842x_{34} = -2142.4846400842
x35=3358.08418998989x_{35} = 3358.08418998989
x36=−4083.06676315289x_{36} = -4083.06676315289
x37=2896.03647500433x_{37} = 2896.03647500433
x38=4004.96591622832x_{38} = 4004.96591622832
x39=−3158.95239358517x_{39} = -3158.95239358517
x40=586.740679799955x_{40} = 586.740679799955
x41=−1680.50311600375x_{41} = -1680.50311600375
x42=2341.59936263935x_{42} = 2341.59936263935
x43=−756.935146423658x_{43} = -756.935146423658
x44=3542.90609021798x_{44} = 3542.90609021798
x45=−1310.97190874837x_{45} = -1310.97190874837
x46=1879.59860242853x_{46} = 1879.59860242853
x47=4374.61777750788x_{47} = 4374.61777750788
x48=−2696.91014651591x_{48} = -2696.91014651591
x49=−3528.59449597132x_{49} = -3528.59449597132
x50=4189.79147140841x_{50} = 4189.79147140841
x51=−3343.77270186947x_{51} = -3343.77270186947
x52=1602.42209898023x_{52} = 1602.42209898023
x53=1325.27758809026x_{53} = 1325.27758809026
x54=−849.231147792715x_{54} = -849.231147792715
x55=−4267.89264733485x_{55} = -4267.89264733485
x56=3173.26375630385x_{56} = 3173.26375630385
x57=1694.81149812614x_{57} = 1694.81149812614
x58=2618.81438890368x_{58} = 2618.81438890368
x59=−1588.11421823202x_{59} = -1588.11421823202
x60=4467.03118286166x_{60} = 4467.03118286166
x61=−664.674000933506x_{61} = -664.674000933506
x62=678.95924905112x_{62} = 678.95924905112
x63=3820.14122167262x_{63} = 3820.14122167262
x64=2803.62845895159x_{64} = 2803.62845895159
x65=−1865.28943360022x_{65} = -1865.28943360022
x66=−1126.24090185439x_{66} = -1126.24090185439
x67=3450.49497162093x_{67} = 3450.49497162093
x68=−4545.13277806334x_{68} = -4545.13277806334
x69=3912.55345370101x_{69} = 3912.55345370101
x70=−3805.82949640548x_{70} = -3805.82949640548
x71=−4175.47961053501x_{71} = -4175.47961053501
x72=771.226759521683x_{72} = 771.226759521683
x73=−3621.00587861527x_{73} = -3621.00587861527
x74=1417.65435918043x_{74} = 1417.65435918043
x75=4559.44474270646x_{75} = 4559.44474270646
x76=494.599578700493x_{76} = 494.599578700493
x77=1232.90699523937x_{77} = 1232.90699523937
x78=−2604.50357700807x_{78} = -2604.50357700807
x79=−1495.72887435697x_{79} = -1495.72887435697
x80=3727.72923740033x_{80} = 3727.72923740033
x81=−1218.6024082584x_{81} = -1218.6024082584
x82=−3066.54288126103x_{82} = -3066.54288126103
x83=−4360.30586143941x_{83} = -4360.30586143941
x84=3265.6737741485x_{84} = 3265.6737741485
x85=−1403.34779829348x_{85} = -1403.34779829348
x86=−3436.18342828653x_{86} = -3436.18342828653
x87=−480.340190036038x_{87} = -480.340190036038
x88=−1772.89500451929x_{88} = -1772.89500451929
x89=2064.39427416289x_{89} = 2064.39427416289
x90=1140.54411273606x_{90} = 1140.54411273606
x91=−2327.28898619093x_{91} = -2327.28898619093
x92=−1957.68603907431x_{92} = -1957.68603907431
x93=−3990.65411843047x_{93} = -3990.65411843047
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0

lim⁡x→0−(2(1(x2+1)2−4(1−1x2+1)x2+10(x−atan⁡(x))x3)x3)=−∞\lim_{x \to 0^-}\left(\frac{2 \left(\frac{1}{\left(x^{2} + 1\right)^{2}} - \frac{4 \left(1 - \frac{1}{x^{2} + 1}\right)}{x^{2}} + \frac{10 \left(x - \operatorname{atan}{\left(x \right)}\right)}{x^{3}}\right)}{x^{3}}\right) = -\infty
lim⁡x→0+(2(1(x2+1)2−4(1−1x2+1)x2+10(x−atan⁡(x))x3)x3)=∞\lim_{x \to 0^+}\left(\frac{2 \left(\frac{1}{\left(x^{2} + 1\right)^{2}} - \frac{4 \left(1 - \frac{1}{x^{2} + 1}\right)}{x^{2}} + \frac{10 \left(x - \operatorname{atan}{\left(x \right)}\right)}{x^{3}}\right)}{x^{3}}\right) = \infty
- los límites no son iguales, signo
x1=0x_{1} = 0
- es el punto de flexión

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:
No tiene corvaduras en todo el eje numérico
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(x−atan⁡(x)x4)=0\lim_{x \to -\infty}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(x−atan⁡(x)x4)=0\lim_{x \to \infty}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}}\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 (x - atan(x))/x^4, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(x−atan⁡(x)xx4)=0\lim_{x \to -\infty}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x x^{4}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(x−atan⁡(x)xx4)=0\lim_{x \to \infty}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x x^{4}}\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:
x−atan⁡(x)x4=−x+atan⁡(x)x4\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}} = \frac{- x + \operatorname{atan}{\left(x \right)}}{x^{4}}
- No
x−atan⁡(x)x4=−−x+atan⁡(x)x4\frac{x - \operatorname{atan}{\left(x \right)}}{x^{4}} = - \frac{- x + \operatorname{atan}{\left(x \right)}}{x^{4}}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор