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Trazado el gráfico de la función/ (-1-cos(x))/x^3

Gráfico de la función y = (-1-cos(x))/x^3

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       -1 - cos(x)
f(x) = -----------
             3    
            x
f(x)=−cos⁡(x)−1x3f{\left(x \right)} = \frac{- \cos{\left(x \right)} - 1}{x^{3}} 
f = (-cos(x) - 1)/x^3
Gráfico de la función
02468-8-6-4-2-1010-5000050000
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:
−cos⁡(x)−1x3=0\frac{- \cos{\left(x \right)} - 1}{x^{3}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=πx_{1} = \pi
Solución numérica
x1=−78.5398153894579x_{1} = -78.5398153894579
x2=−47.1238986800577x_{2} = -47.1238986800577
x3=3.141591496203x_{3} = 3.141591496203
x4=47.1238901884455x_{4} = 47.1238901884455
x5=−15.7079632962567x_{5} = -15.7079632962567
x6=−97.3893717160488x_{6} = -97.3893717160488
x7=59.6902598572919x_{7} = 59.6902598572919
x8=53.407074602786x_{8} = 53.407074602786
x9=97.3893721745645x_{9} = 97.3893721745645
x10=72.2566310277155x_{10} = 72.2566310277155
x11=21.9911482907622x_{11} = 21.9911482907622
x12=84.8230013568118x_{12} = 84.8230013568118
x13=−40.8407072269748x_{13} = -40.8407072269748
x14=−116.238927904696x_{14} = -116.238927904696
x15=21.9911488739842x_{15} = 21.9911488739842
x16=−34.5575196610773x_{16} = -34.5575196610773
x17=97.3893725741756x_{17} = 97.3893725741756
x18=−15.7079624377685x_{18} = -15.7079624377685
x19=28.2743332641658x_{19} = 28.2743332641658
x20=−2334.20334291192x_{20} = -2334.20334291192
x21=78.5398168147997x_{21} = 78.5398168147997
x22=−9.42477764607965x_{22} = -9.42477764607965
x23=65.9734458772162x_{23} = 65.9734458772162
x24=−9.42477811581359x_{24} = -9.42477811581359
x25=−116.238928747448x_{25} = -116.238928747448
x26=28.2743341948265x_{26} = 28.2743341948265
x27=78.5398161783123x_{27} = 78.5398161783123
x28=47.1238893842218x_{28} = 47.1238893842218
x29=−72.2566308631921x_{29} = -72.2566308631921
x30=−34.5575188707681x_{30} = -34.5575188707681
x31=−72.2566314968061x_{31} = -72.2566314968061
x32=103.672556745932x_{32} = 103.672556745932
x33=−15.7079602650305x_{33} = -15.7079602650305
x34=−34.5575196069889x_{34} = -34.5575196069889
x35=21.9911485851755x_{35} = 21.9911485851755
x36=59.6902593569372x_{36} = 59.6902593569372
x37=−59.690260457629x_{37} = -59.690260457629
x38=9.42477687669227x_{38} = 9.42477687669227
x39=354.999968063691x_{39} = 354.999968063691
x40=−47.1238892817945x_{40} = -47.1238892817945
x41=72.2566314467486x_{41} = 72.2566314467486
x42=−97.3893724507843x_{42} = -97.3893724507843
x43=−59.6902604870089x_{43} = -59.6902604870089
x44=34.5575196158589x_{44} = 34.5575196158589
x45=−59.6902598423497x_{45} = -59.6902598423497
x46=−65.9734452405333x_{46} = -65.9734452405333
x47=91.1061873567288x_{47} = 91.1061873567288
x48=−40.8407048701465x_{48} = -40.8407048701465
x49=−65.9734461164612x_{49} = -65.9734461164612
x50=34.5575190163577x_{50} = 34.5575190163577
x51=−3.14159279790805x_{51} = -3.14159279790805
x52=−39637.4744613621x_{52} = -39637.4744613621
x53=15.7079634378574x_{53} = 15.7079634378574
x54=9.42477820368308x_{54} = 9.42477820368308
x55=72.2566305794936x_{55} = 72.2566305794936
x56=47.1238881370041x_{56} = 47.1238881370041
x57=40.8407041911131x_{57} = 40.8407041911131
x58=9.42477721314894x_{58} = 9.42477721314894
x59=−53.4070745362574x_{59} = -53.4070745362574
x60=−21.9911478628404x_{60} = -21.9911478628404
x61=−78.5398160394446x_{61} = -78.5398160394446
x62=−21.9911485864039x_{62} = -21.9911485864039
x63=65.9734457529054x_{63} = 65.9734457529054
x64=−53.407075290761x_{64} = -53.407075290761
x65=−72.2566308926902x_{65} = -72.2566308926902
x66=−40.8407040630831x_{66} = -40.8407040630831
x67=59.6902606062886x_{67} = 59.6902606062886
x68=−28.2743336994448x_{68} = -28.2743336994448
x69=−84.8230012365264x_{69} = -84.8230012365264
x70=−3.14159047055643x_{70} = -3.14159047055643
x71=−91.1061872578797x_{71} = -91.1061872578797
x72=21.9911478600072x_{72} = 21.9911478600072
x73=−9.4247770072291x_{73} = -9.4247770072291
x74=78.5398163004214x_{74} = 78.5398163004214
x75=53.4070754116207x_{75} = 53.4070754116207
x76=−53.4070737434634x_{76} = -53.4070737434634
x77=40.8407049289347x_{77} = 40.8407049289347
x78=97.3893717754184x_{78} = 97.3893717754184
x79=−28.2743342815076x_{79} = -28.2743342815076
x80=−28.2743334339593x_{80} = -28.2743334339593
x81=−78.5398174722281x_{81} = -78.5398174722281
x82=−78.5398167921365x_{82} = -78.5398167921365
x83=34.5575188844436x_{83} = 34.5575188844436
x84=53.4070872303504x_{84} = 53.4070872303504
x85=15.7079625304308x_{85} = 15.7079625304308
x86=−97.3893711480408x_{86} = -97.3893711480408
x87=−84.8230020409408x_{87} = -84.8230020409408
x88=91.106186553725x_{88} = 91.106186553725
x89=−91.106186458973x_{89} = -91.106186458973
x90=−47.1238900945067x_{90} = -47.1238900945067
x91=84.8230021077614x_{91} = 84.8230021077614
x92=−15.707962145847x_{92} = -15.707962145847
x93=28.2743338651116x_{93} = 28.2743338651116
x94=−53.4070765670325x_{94} = -53.4070765670325
x95=40.8407053715323x_{95} = 40.8407053715323
x96=3.14159275476265x_{96} = 3.14159275476265
x97=65.9734451663027x_{97} = 65.9734451663027
x98=−21.9911488037302x_{98} = -21.9911488037302
x99=84.8230027570787x_{99} = 84.8230027570787
x100=−65.9734457647733x_{100} = -65.9734457647733
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 (-1 - cos(x))/x^3.
−1−cos⁡(0)03\frac{-1 - \cos{\left(0 \right)}}{0^{3}}
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
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
sin⁡(x)x3−3(−cos⁡(x)−1)x4=0\frac{\sin{\left(x \right)}}{x^{3}} - \frac{3 \left(- \cos{\left(x \right)} - 1\right)}{x^{4}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−75.3186041804901x_{1} = -75.3186041804901
x2=81.6079198094314x_{2} = 81.6079198094314
x3=−24.8928651811575x_{3} = -24.8928651811575
x4=−100.471264085445x_{4} = -100.471264085445
x5=40.8407044966673x_{5} = 40.8407044966673
x6=53.4070751110265x_{6} = 53.4070751110265
x7=97.3893722612836x_{7} = 97.3893722612836
x8=34.5575191894877x_{8} = 34.5575191894877
x9=47.1238898038469x_{9} = 47.1238898038469
x10=−62.7362874405728x_{10} = -62.7362874405728
x11=−97.3893722612836x_{11} = -97.3893722612836
x12=−21.9911485751286x_{12} = -21.9911485751286
x13=3.14159265358979x_{13} = 3.14159265358979
x14=12.0795120816332x_{14} = 12.0795120816332
x15=65.9734457253857x_{15} = 65.9734457253857
x16=−69.0281720102481x_{16} = -69.0281720102481
x17=100.471264085445x_{17} = 100.471264085445
x18=62.7362874405728x_{18} = 62.7362874405728
x19=−94.1840961163538x_{19} = -94.1840961163538
x20=37.5396196644034x_{20} = 37.5396196644034
x21=18.5285168299698x_{21} = 18.5285168299698
x22=−53.4070751110265x_{22} = -53.4070751110265
x23=−9.42477796076938x_{23} = -9.42477796076938
x24=5.24388943952117x_{24} = 5.24388943952117
x25=50.1459742160797x_{25} = 50.1459742160797
x26=−34.5575191894877x_{26} = -34.5575191894877
x27=24.8928651811575x_{27} = 24.8928651811575
x28=21.9911485751286x_{28} = 21.9911485751286
x29=−47.1238898038469x_{29} = -47.1238898038469
x30=28.2743338823081x_{30} = 28.2743338823081
x31=−3.14159265358979x_{31} = -3.14159265358979
x32=−103.672557568463x_{32} = -103.672557568463
x33=−12.0795120816332x_{33} = -12.0795120816332
x34=−65.9734457253857x_{34} = -65.9734457253857
x35=72.2566310325652x_{35} = 72.2566310325652
x36=−59.6902604182061x_{36} = -59.6902604182061
x37=31.2243568800933x_{37} = 31.2243568800933
x38=−50.1459742160797x_{38} = -50.1459742160797
x39=43.8456664725751x_{39} = 43.8456664725751
x40=75.3186041804901x_{40} = 75.3186041804901
x41=−91.106186954104x_{41} = -91.106186954104
x42=−40.8407044966673x_{42} = -40.8407044966673
x43=59.6902604182061x_{43} = 59.6902604182061
x44=−56.4424647590253x_{44} = -56.4424647590253
x45=91.106186954104x_{45} = 91.106186954104
x46=78.5398163397448x_{46} = 78.5398163397448
x47=84.8230016469244x_{47} = 84.8230016469244
x48=−37.5396196644034x_{48} = -37.5396196644034
x49=87.8963585754007x_{49} = 87.8963585754007
x50=−87.8963585754007x_{50} = -87.8963585754007
x51=−5.24388943952117x_{51} = -5.24388943952117
x52=9.42477796076938x_{52} = 9.42477796076938
x53=69.0281720102481x_{53} = 69.0281720102481
x54=−84.8230016469244x_{54} = -84.8230016469244
x55=56.4424647590253x_{55} = 56.4424647590253
x56=−18.5285168299698x_{56} = -18.5285168299698
x57=−78.5398163397448x_{57} = -78.5398163397448
x58=15.707963267949x_{58} = 15.707963267949
x59=94.1840961163538x_{59} = 94.1840961163538
x60=−28.2743338823081x_{60} = -28.2743338823081
x61=−15.707963267949x_{61} = -15.707963267949
x62=−43.8456664725751x_{62} = -43.8456664725751
x63=−72.2566310325652x_{63} = -72.2566310325652
x64=−81.6079198094314x_{64} = -81.6079198094314
x65=−31.2243568800933x_{65} = -31.2243568800933
Signos de extremos en los puntos:
(-75.31860418049011, 4.67341930056756e-6)

(81.60791980943141, -3.67490884511663e-6)

(-24.892865181157465, 0.000127803549693016)

(-100.47126408544507, 1.97023197093293e-6)

(40.840704496667314, 0)

(53.40707511102649, 0)

(97.3893722612836, 0)

(34.55751918948773, 0)

(47.1238898038469, 0)

(-62.73628744057283, 8.08130678143302e-6)

(-97.3893722612836, 0)

(-21.991148575128552, 0)

(3.141592653589793, 0)

(12.079512081633247, -0.00106877975605313)

(65.97344572538566, 0)

(-69.02817201024808, 6.06919859876142e-6)

(100.47126408544507, -1.97023197093293e-6)

(62.73628744057283, -8.08130678143302e-6)

(-94.18409611635384, 2.39142557137598e-6)

(37.53961966440337, -3.75660549595209e-5)

(18.52851682996978, -0.000306386325939657)

(-53.40707511102649, 0)

(-9.42477796076938, 0)

(5.243889439521166, -0.0104496783518742)

(50.145974216079686, -1.58041149777668e-5)

(-34.55751918948773, 0)

(24.892865181157465, -0.000127803549693016)

(21.991148575128552, 0)

(-47.1238898038469, 0)

(28.274333882308138, 0)

(-3.141592653589793, 0)

(-103.67255756846318, 0)

(-12.079512081633247, 0.00106877975605313)

(-65.97344572538566, 0)

(72.25663103256524, 0)

(-59.69026041820607, 0)

(31.224356880093286, -6.50966808887725e-5)

(-50.145974216079686, 1.58041149777668e-5)

(43.845666472575125, -2.36168267311103e-5)

(75.31860418049011, -4.67341930056756e-6)

(-91.106186954104, 0)

(-40.840704496667314, 0)

(59.69026041820607, 0)

(-56.44246475902532, 1.10914135327694e-5)

(91.106186954104, 0)

(78.53981633974483, 0)

(84.82300164692441, 0)

(-37.53961966440337, 3.75660549595209e-5)

(87.89635857540073, -2.94179033438626e-6)

(-87.89635857540073, 2.94179033438626e-6)

(-5.243889439521166, 0.0104496783518742)

(9.42477796076938, 0)

(69.02817201024808, -6.06919859876142e-6)

(-84.82300164692441, 0)

(56.44246475902532, -1.10914135327694e-5)

(-18.52851682996978, 0.000306386325939657)

(-78.53981633974483, 0)

(15.707963267948966, 0)

(94.18409611635384, -2.39142557137598e-6)

(-28.274333882308138, 0)

(-15.707963267948966, 0)

(-43.845666472575125, 2.36168267311103e-5)

(-72.25663103256524, 0)

(-81.60791980943141, 3.67490884511663e-6)

(-31.224356880093286, 6.50966808887725e-5)


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:
Puntos mínimos de la función:
x1=81.6079198094314x_{1} = 81.6079198094314
x2=−97.3893722612836x_{2} = -97.3893722612836
x3=−21.9911485751286x_{3} = -21.9911485751286
x4=12.0795120816332x_{4} = 12.0795120816332
x5=100.471264085445x_{5} = 100.471264085445
x6=62.7362874405728x_{6} = 62.7362874405728
x7=37.5396196644034x_{7} = 37.5396196644034
x8=18.5285168299698x_{8} = 18.5285168299698
x9=−53.4070751110265x_{9} = -53.4070751110265
x10=−9.42477796076938x_{10} = -9.42477796076938
x11=5.24388943952117x_{11} = 5.24388943952117
x12=50.1459742160797x_{12} = 50.1459742160797
x13=−34.5575191894877x_{13} = -34.5575191894877
x14=24.8928651811575x_{14} = 24.8928651811575
x15=−47.1238898038469x_{15} = -47.1238898038469
x16=−3.14159265358979x_{16} = -3.14159265358979
x17=−103.672557568463x_{17} = -103.672557568463
x18=−65.9734457253857x_{18} = -65.9734457253857
x19=−59.6902604182061x_{19} = -59.6902604182061
x20=31.2243568800933x_{20} = 31.2243568800933
x21=43.8456664725751x_{21} = 43.8456664725751
x22=75.3186041804901x_{22} = 75.3186041804901
x23=−91.106186954104x_{23} = -91.106186954104
x24=−40.8407044966673x_{24} = -40.8407044966673
x25=87.8963585754007x_{25} = 87.8963585754007
x26=69.0281720102481x_{26} = 69.0281720102481
x27=−84.8230016469244x_{27} = -84.8230016469244
x28=56.4424647590253x_{28} = 56.4424647590253
x29=−78.5398163397448x_{29} = -78.5398163397448
x30=94.1840961163538x_{30} = 94.1840961163538
x31=−28.2743338823081x_{31} = -28.2743338823081
x32=−15.707963267949x_{32} = -15.707963267949
x33=−72.2566310325652x_{33} = -72.2566310325652
Puntos máximos de la función:
x33=−75.3186041804901x_{33} = -75.3186041804901
x33=−24.8928651811575x_{33} = -24.8928651811575
x33=−100.471264085445x_{33} = -100.471264085445
x33=40.8407044966673x_{33} = 40.8407044966673
x33=53.4070751110265x_{33} = 53.4070751110265
x33=97.3893722612836x_{33} = 97.3893722612836
x33=34.5575191894877x_{33} = 34.5575191894877
x33=47.1238898038469x_{33} = 47.1238898038469
x33=−62.7362874405728x_{33} = -62.7362874405728
x33=3.14159265358979x_{33} = 3.14159265358979
x33=65.9734457253857x_{33} = 65.9734457253857
x33=−69.0281720102481x_{33} = -69.0281720102481
x33=−94.1840961163538x_{33} = -94.1840961163538
x33=21.9911485751286x_{33} = 21.9911485751286
x33=28.2743338823081x_{33} = 28.2743338823081
x33=−12.0795120816332x_{33} = -12.0795120816332
x33=72.2566310325652x_{33} = 72.2566310325652
x33=−50.1459742160797x_{33} = -50.1459742160797
x33=59.6902604182061x_{33} = 59.6902604182061
x33=−56.4424647590253x_{33} = -56.4424647590253
x33=91.106186954104x_{33} = 91.106186954104
x33=78.5398163397448x_{33} = 78.5398163397448
x33=84.8230016469244x_{33} = 84.8230016469244
x33=−37.5396196644034x_{33} = -37.5396196644034
x33=−87.8963585754007x_{33} = -87.8963585754007
x33=−5.24388943952117x_{33} = -5.24388943952117
x33=9.42477796076938x_{33} = 9.42477796076938
x33=−18.5285168299698x_{33} = -18.5285168299698
x33=15.707963267949x_{33} = 15.707963267949
x33=−43.8456664725751x_{33} = -43.8456664725751
x33=−81.6079198094314x_{33} = -81.6079198094314
x33=−31.2243568800933x_{33} = -31.2243568800933
Decrece en los intervalos
[100.471264085445,∞)\left[100.471264085445, \infty\right)
Crece en los intervalos
(−∞,−103.672557568463]\left(-\infty, -103.672557568463\right]
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
cos⁡(x)−6sin⁡(x)x−12(cos⁡(x)+1)x2x3=0\frac{\cos{\left(x \right)} - \frac{6 \sin{\left(x \right)}}{x} - \frac{12 \left(\cos{\left(x \right)} + 1\right)}{x^{2}}}{x^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=80.03751936527x_{1} = 80.03751936527
x2=51.7157837904777x_{2} = 51.7157837904777
x3=6.74389560830075x_{3} = 6.74389560830075
x4=10.5294302230251x_{4} = 10.5294302230251
x5=−89.4668286273486x_{5} = -89.4668286273486
x6=35.9707475149935x_{6} = 35.9707475149935
x7=−70.5984423541311x_{7} = -70.5984423541311
x8=67.4579314078806x_{8} = 67.4579314078806
x9=−39.1086764039796x_{9} = -39.1086764039796
x10=−54.8725048633783x_{10} = -54.8725048633783
x11=−83.1783382561883x_{11} = -83.1783382561883
x12=26.4597983844386x_{12} = 26.4597983844386
x13=−80.03751936527x_{13} = -80.03751936527
x14=−42.2762700825632x_{14} = -42.2762700825632
x15=356.554032824969x_{15} = 356.554032824969
x16=−10.5294302230251x_{16} = -10.5294302230251
x17=48.5762421680465x_{17} = 48.5762421680465
x18=−29.6563714086934x_{18} = -29.6563714086934
x19=23.3265629591175x_{19} = 23.3265629591175
x20=−13.6349951156977x_{20} = -13.6349951156977
x21=−58.0124790381597x_{21} = -58.0124790381597
x22=−67.4579314078806x_{22} = -67.4579314078806
x23=45.4151787911182x_{23} = 45.4151787911182
x24=−98.9007277835613x_{24} = -98.9007277835613
x25=−64.3064486934137x_{25} = -64.3064486934137
x26=83.1783382561883x_{26} = 83.1783382561883
x27=−76.888957724397x_{27} = -76.888957724397
x28=13.6349951156977x_{28} = 13.6349951156977
x29=86.3259029330052x_{29} = 86.3259029330052
x30=−23.3265629591175x_{30} = -23.3265629591175
x31=70.5984423541311x_{31} = 70.5984423541311
x32=−73.748273533974x_{32} = -73.748273533974
x33=73.748273533974x_{33} = 73.748273533974
x34=−20.0925011495984x_{34} = -20.0925011495984
x35=−92.613596068641x_{35} = -92.613596068641
x36=−48.5762421680465x_{36} = -48.5762421680465
x37=3.79881290481705x_{37} = 3.79881290481705
x38=54.8725048633783x_{38} = 54.8725048633783
x39=−35.9707475149935x_{39} = -35.9707475149935
x40=−51.7157837904777x_{40} = -51.7157837904777
x41=92.613596068641x_{41} = 92.613596068641
x42=−16.9660549331947x_{42} = -16.9660549331947
x43=61.166165799029x_{43} = 61.166165799029
x44=−45.4151787911182x_{44} = -45.4151787911182
x45=−249.732782672848x_{45} = -249.732782672848
x46=42.2762700825632x_{46} = 42.2762700825632
x47=−3.79881290481705x_{47} = -3.79881290481705
x48=16.9660549331947x_{48} = 16.9660549331947
x49=89.4668286273486x_{49} = 89.4668286273486
x50=39.1086764039796x_{50} = 39.1086764039796
x51=−26.4597983844386x_{51} = -26.4597983844386
x52=−117.759638242453x_{52} = -117.759638242453
x53=−95.7546078767246x_{53} = -95.7546078767246
x54=29.6563714086934x_{54} = 29.6563714086934
x55=−61.166165799029x_{55} = -61.166165799029
x56=95.7546078767246x_{56} = 95.7546078767246
x57=−32.7926627594127x_{57} = -32.7926627594127
x58=−6.74389560830075x_{58} = -6.74389560830075
x59=64.3064486934137x_{59} = 64.3064486934137
x60=58.0124790381597x_{60} = 58.0124790381597
x61=32.7926627594127x_{61} = 32.7926627594127
x62=20.0925011495984x_{62} = 20.0925011495984
x63=−86.3259029330052x_{63} = -86.3259029330052
x64=76.888957724397x_{64} = 76.888957724397
x65=98.9007277835613x_{65} = 98.9007277835613
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−(cos⁡(x)−6sin⁡(x)x−12(cos⁡(x)+1)x2x3)=∞\lim_{x \to 0^-}\left(\frac{\cos{\left(x \right)} - \frac{6 \sin{\left(x \right)}}{x} - \frac{12 \left(\cos{\left(x \right)} + 1\right)}{x^{2}}}{x^{3}}\right) = \infty
lim⁡x→0+(cos⁡(x)−6sin⁡(x)x−12(cos⁡(x)+1)x2x3)=−∞\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)} - \frac{6 \sin{\left(x \right)}}{x} - \frac{12 \left(\cos{\left(x \right)} + 1\right)}{x^{2}}}{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:
Cóncava en los intervalos
[356.554032824969,∞)\left[356.554032824969, \infty\right)
Convexa en los intervalos
(−∞,−249.732782672848]\left(-\infty, -249.732782672848\right]
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→−∞(−cos⁡(x)−1x3)=0\lim_{x \to -\infty}\left(\frac{- \cos{\left(x \right)} - 1}{x^{3}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(−cos⁡(x)−1x3)=0\lim_{x \to \infty}\left(\frac{- \cos{\left(x \right)} - 1}{x^{3}}\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 (-1 - cos(x))/x^3, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(−cos⁡(x)−1xx3)=0\lim_{x \to -\infty}\left(\frac{- \cos{\left(x \right)} - 1}{x x^{3}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(−cos⁡(x)−1xx3)=0\lim_{x \to \infty}\left(\frac{- \cos{\left(x \right)} - 1}{x x^{3}}\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:
−cos⁡(x)−1x3=−−cos⁡(x)−1x3\frac{- \cos{\left(x \right)} - 1}{x^{3}} = - \frac{- \cos{\left(x \right)} - 1}{x^{3}}
- No
−cos⁡(x)−1x3=−cos⁡(x)−1x3\frac{- \cos{\left(x \right)} - 1}{x^{3}} = \frac{- \cos{\left(x \right)} - 1}{x^{3}}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор