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-
¿Cómo usar?
- Gráfico de la función y =:
-
x^3/(4(2-x)^2)
-
x^3-x-1
-
(x^3-3x)/(x^2-1)
-
x^3-6*x^2-15*x-2
-
Expresiones idénticas
- exp(uno /x)/(x^ dos (exp(uno /x)- uno)^ dos)
- exponente de (1 dividir por x) dividir por (x al cuadrado ( exponente de (1 dividir por x) menos 1) al cuadrado )
- exponente de (uno dividir por x) dividir por (x en el grado dos ( exponente de (uno dividir por x) menos uno) en el grado dos)
- exp(1/x)/(x2(exp(1/x)-1)2)
- exp1/x/x2exp1/x-12
- exp(1/x)/(x²(exp(1/x)-1)²)
- exp(1/x)/(x en el grado 2(exp(1/x)-1) en el grado 2)
- exp1/x/x^2exp1/x-1^2
- exp(1 dividir por x) dividir por (x^2(exp(1 dividir por x)-1)^2)
-
Expresiones semejantes
- exp(1/x)/(x^2(exp(1/x)+1)^2)
-
Expresiones con funciones
- Exponente exp
- exp(x^3)*sqrt(x^2)-x-1
- exp(x)-ex
- exp^(-10t)*(2*cos(10*t)+2*sin(10*t))
- expsin(x)
- exp(1/sin(x))
- Exponente exp
- exp(x^3)*sqrt(x^2)-x-1
- exp(x)-ex
- exp^(-10t)*(2*cos(10*t)+2*sin(10*t))
- expsin(x)
- exp(1/sin(x))
Gráfico de la función y = exp(1/x)/(x^2(exp(1/x)-1)^2)
Solución
Ha introducido
[src]
1
-
x
e
f(x) = ------------
2
/ 1 \
| - |
2 | x |
x *\e - 1/
$$f{\left(x \right)} = \frac{e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}}$$
f = exp(1/x)/((x^2*(exp(1/x) - 1)^2))
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\frac{e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}} = 0$$
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
o sea hay que resolver la ecuación:
$$\frac{e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}} = 0$$
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$- \frac{\frac{1}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}} e^{\frac{1}{x}}}{x^{2}} + \frac{\left(- 2 x \left(e^{\frac{1}{x}} - 1\right)^{2} + 2 \left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}\right) e^{\frac{1}{x}}}{x^{4} \left(e^{\frac{1}{x}} - 1\right)^{4}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 36512.8341459926$$
$$x_{2} = 13627.9152059125$$
$$x_{3} = -38924.3992160704$$
$$x_{4} = -17734.5572561786$$
$$x_{5} = 16170.6287045655$$
$$x_{6} = 23798.8964012984$$
$$x_{7} = 41598.4307745321$$
$$x_{8} = -40619.5984694662$$
$$x_{9} = 12780.3528127123$$
$$x_{10} = 26341.6742356719$$
$$x_{11} = 30579.6501812154$$
$$x_{12} = 10237.7072058338$$
$$x_{13} = 28884.4582101405$$
$$x_{14} = 39055.6315231319$$
$$x_{15} = -28753.2265233594$$
$$x_{16} = 28036.8629728761$$
$$x_{17} = -12649.1267832051$$
$$x_{18} = -39771.9987445352$$
$$x_{19} = -29600.822196685$$
$$x_{20} = -41467.1983788078$$
$$x_{21} = 20408.5394735307$$
$$x_{22} = 33970.0390640156$$
$$x_{23} = -30448.4183467446$$
$$x_{24} = 15323.0538797709$$
$$x_{25} = 18713.3691647864$$
$$x_{26} = 35665.2355060803$$
$$x_{27} = 14475.4824993254$$
$$x_{28} = 33122.4413047548$$
$$x_{29} = 22103.7156359971$$
$$x_{30} = -32143.6119262135$$
$$x_{31} = -23667.6653632422$$
$$x_{32} = 25494.0808512544$$
$$x_{33} = -27905.6313702123$$
$$x_{34} = -38076.7998971915$$
$$x_{35} = -8411.44815555337$$
$$x_{36} = -14344.2549142001$$
$$x_{37} = -31296.0149348033$$
$$x_{38} = -18582.1393762791$$
$$x_{39} = 37360.4330398325$$
$$x_{40} = 7695.1720429073$$
$$x_{41} = 32274.8438856766$$
$$x_{42} = -5869.07851287475$$
$$x_{43} = -10106.4851277259$$
$$x_{44} = 8542.66540404055$$
$$x_{45} = -35534.0033480628$$
$$x_{46} = 34817.6371386206$$
$$x_{47} = 6000.27930908944$$
$$x_{48} = 11085.2472232554$$
$$x_{49} = -15191.8257038658$$
$$x_{50} = 42446.0308832849$$
$$x_{51} = -34686.4050248436$$
$$x_{52} = -26210.4428263164$$
$$x_{53} = -20277.3091634513$$
$$x_{54} = -42314.7984614978$$
$$x_{55} = 39903.2310830382$$
$$x_{56} = 17865.7867262802$$
$$x_{57} = -32991.2092898821$$
$$x_{58} = -9258.95879064063$$
$$x_{59} = -21972.4849200131$$
$$x_{60} = -10954.0235214509$$
$$x_{61} = 29732.0539604744$$
$$x_{62} = -11801.5713738084$$
$$x_{63} = -21124.8963853792$$
$$x_{64} = -16886.977356312$$
$$x_{65} = -22820.0746210752$$
$$x_{66} = 5152.91086370651$$
$$x_{67} = 31427.246834305$$
$$x_{68} = -13496.6883257698$$
$$x_{69} = -7563.95850210681$$
$$x_{70} = 17018.2064590623$$
$$x_{71} = 9390.17878234122$$
$$x_{72} = 24646.4882132137$$
$$x_{73} = 22951.3055070382$$
$$x_{74} = -33838.8069978398$$
$$x_{75} = -24515.2570385241$$
$$x_{76} = 40750.8308374762$$
$$x_{77} = -36381.6019467943$$
$$x_{78} = -25362.8495533661$$
$$x_{79} = 6847.70613289343$$
$$x_{80} = 19560.9534923245$$
$$x_{81} = 38208.032170686$$
$$x_{82} = -27058.0367861288$$
$$x_{83} = -16039.4000286848$$
$$x_{84} = -19429.7234260341$$
$$x_{85} = -37229.2008022157$$
$$x_{86} = 21256.1269105821$$
$$x_{87} = -6716.49777568092$$
$$x_{88} = 11932.7963639546$$
$$x_{89} = 27189.2682966688$$
Signos de extremos en los puntos:
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:
$$x_{1} = 13627.9152059125$$
$$x_{2} = 41598.4307745321$$
$$x_{3} = -40619.5984694662$$
$$x_{4} = 26341.6742356719$$
$$x_{5} = 30579.6501812154$$
$$x_{6} = 28884.4582101405$$
$$x_{7} = 28036.8629728761$$
$$x_{8} = 35665.2355060803$$
$$x_{9} = 14475.4824993254$$
$$x_{10} = -23667.6653632422$$
$$x_{11} = -8411.44815555337$$
$$x_{12} = -5869.07851287475$$
$$x_{13} = -15191.8257038658$$
$$x_{14} = 42446.0308832849$$
$$x_{15} = -32991.2092898821$$
$$x_{16} = -11801.5713738084$$
$$x_{17} = 5152.91086370651$$
$$x_{18} = 31427.246834305$$
$$x_{19} = -13496.6883257698$$
$$x_{20} = 19560.9534923245$$
$$x_{21} = -37229.2008022157$$
$$x_{22} = 11932.7963639546$$
Puntos máximos de la función:
$$x_{22} = 36512.8341459926$$
$$x_{22} = 12780.3528127123$$
$$x_{22} = 15323.0538797709$$
$$x_{22} = -27905.6313702123$$
$$x_{22} = -14344.2549142001$$
$$x_{22} = -18582.1393762791$$
$$x_{22} = 11085.2472232554$$
$$x_{22} = -34686.4050248436$$
$$x_{22} = 39903.2310830382$$
$$x_{22} = -21972.4849200131$$
$$x_{22} = -21124.8963853792$$
$$x_{22} = -16886.977356312$$
$$x_{22} = -22820.0746210752$$
$$x_{22} = -7563.95850210681$$
$$x_{22} = 17018.2064590623$$
$$x_{22} = 24646.4882132137$$
$$x_{22} = 22951.3055070382$$
$$x_{22} = 40750.8308374762$$
$$x_{22} = 38208.032170686$$
$$x_{22} = -27058.0367861288$$
$$x_{22} = -6716.49777568092$$
Decrece en los intervalos
$$\left[42446.0308832849, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -40619.5984694662\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$- \frac{\frac{1}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}} e^{\frac{1}{x}}}{x^{2}} + \frac{\left(- 2 x \left(e^{\frac{1}{x}} - 1\right)^{2} + 2 \left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}\right) e^{\frac{1}{x}}}{x^{4} \left(e^{\frac{1}{x}} - 1\right)^{4}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 36512.8341459926$$
$$x_{2} = 13627.9152059125$$
$$x_{3} = -38924.3992160704$$
$$x_{4} = -17734.5572561786$$
$$x_{5} = 16170.6287045655$$
$$x_{6} = 23798.8964012984$$
$$x_{7} = 41598.4307745321$$
$$x_{8} = -40619.5984694662$$
$$x_{9} = 12780.3528127123$$
$$x_{10} = 26341.6742356719$$
$$x_{11} = 30579.6501812154$$
$$x_{12} = 10237.7072058338$$
$$x_{13} = 28884.4582101405$$
$$x_{14} = 39055.6315231319$$
$$x_{15} = -28753.2265233594$$
$$x_{16} = 28036.8629728761$$
$$x_{17} = -12649.1267832051$$
$$x_{18} = -39771.9987445352$$
$$x_{19} = -29600.822196685$$
$$x_{20} = -41467.1983788078$$
$$x_{21} = 20408.5394735307$$
$$x_{22} = 33970.0390640156$$
$$x_{23} = -30448.4183467446$$
$$x_{24} = 15323.0538797709$$
$$x_{25} = 18713.3691647864$$
$$x_{26} = 35665.2355060803$$
$$x_{27} = 14475.4824993254$$
$$x_{28} = 33122.4413047548$$
$$x_{29} = 22103.7156359971$$
$$x_{30} = -32143.6119262135$$
$$x_{31} = -23667.6653632422$$
$$x_{32} = 25494.0808512544$$
$$x_{33} = -27905.6313702123$$
$$x_{34} = -38076.7998971915$$
$$x_{35} = -8411.44815555337$$
$$x_{36} = -14344.2549142001$$
$$x_{37} = -31296.0149348033$$
$$x_{38} = -18582.1393762791$$
$$x_{39} = 37360.4330398325$$
$$x_{40} = 7695.1720429073$$
$$x_{41} = 32274.8438856766$$
$$x_{42} = -5869.07851287475$$
$$x_{43} = -10106.4851277259$$
$$x_{44} = 8542.66540404055$$
$$x_{45} = -35534.0033480628$$
$$x_{46} = 34817.6371386206$$
$$x_{47} = 6000.27930908944$$
$$x_{48} = 11085.2472232554$$
$$x_{49} = -15191.8257038658$$
$$x_{50} = 42446.0308832849$$
$$x_{51} = -34686.4050248436$$
$$x_{52} = -26210.4428263164$$
$$x_{53} = -20277.3091634513$$
$$x_{54} = -42314.7984614978$$
$$x_{55} = 39903.2310830382$$
$$x_{56} = 17865.7867262802$$
$$x_{57} = -32991.2092898821$$
$$x_{58} = -9258.95879064063$$
$$x_{59} = -21972.4849200131$$
$$x_{60} = -10954.0235214509$$
$$x_{61} = 29732.0539604744$$
$$x_{62} = -11801.5713738084$$
$$x_{63} = -21124.8963853792$$
$$x_{64} = -16886.977356312$$
$$x_{65} = -22820.0746210752$$
$$x_{66} = 5152.91086370651$$
$$x_{67} = 31427.246834305$$
$$x_{68} = -13496.6883257698$$
$$x_{69} = -7563.95850210681$$
$$x_{70} = 17018.2064590623$$
$$x_{71} = 9390.17878234122$$
$$x_{72} = 24646.4882132137$$
$$x_{73} = 22951.3055070382$$
$$x_{74} = -33838.8069978398$$
$$x_{75} = -24515.2570385241$$
$$x_{76} = 40750.8308374762$$
$$x_{77} = -36381.6019467943$$
$$x_{78} = -25362.8495533661$$
$$x_{79} = 6847.70613289343$$
$$x_{80} = 19560.9534923245$$
$$x_{81} = 38208.032170686$$
$$x_{82} = -27058.0367861288$$
$$x_{83} = -16039.4000286848$$
$$x_{84} = -19429.7234260341$$
$$x_{85} = -37229.2008022157$$
$$x_{86} = 21256.1269105821$$
$$x_{87} = -6716.49777568092$$
$$x_{88} = 11932.7963639546$$
$$x_{89} = 27189.2682966688$$
Signos de extremos en los puntos:
(36512.834145992565, 0.999999999943895)
(13627.915205912484, 0.999999999548784)
(-38924.399216070444, 0.999999999946964)
(-17734.55725617861, 0.99999999973503)
(16170.62870456547, 0.999999999682074)
(23798.896401298392, 0.999999999848202)
(41598.43077453214, 0.999999999943836)
(-40619.59846946623, 0.999999999949068)
(12780.352812712275, 0.999999999491137)
(26341.67423567186, 0.999999999878544)
(30579.650181215387, 0.999999999904241)
(10237.707205833769, 0.999999999204571)
(28884.458210140507, 0.999999999895273)
(39055.63152313194, 0.999999999939318)
(-28753.226523359415, 0.999999999898373)
(28036.86297287614, 0.999999999889254)
(-12649.126783205136, 0.999999999479564)
(-39771.998744535245, 0.999999999946935)
(-29600.822196685, 0.999999999903124)
(-41467.198378807785, 0.999999999951626)
(20408.53947353069, 0.999999999802226)
(33970.03906401565, 0.9999999999319)
(-30448.418346744576, 0.99999999990841)
(15323.053879770945, 0.99999999964837)
(18713.369164786425, 0.999999999762062)
(35665.23550608033, 0.999999999929527)
(14475.482499325431, 0.999999999600144)
(33122.44130475478, 0.999999999925651)
(22103.715635997123, 0.999999999830137)
(-32143.61192621353, 0.999999999918018)
(-23667.665363242217, 0.99999999984895)
(25494.080851254443, 0.999999999875735)
(-27905.631370212333, 0.99999999989415)
(-38076.79989719149, 0.999999999939754)
(-8411.448155553366, 0.999999998821564)
(-14344.254914200106, 0.999999999596252)
(-31296.014934803297, 0.999999999912904)
(-18582.139376279127, 0.999999999760366)
(37360.43303983254, 0.999999999943504)
(7695.172042907301, 0.999999998594119)
(32274.84388567656, 0.999999999921211)
(-5869.078512874751, 0.999999997580107)
(-10106.485127725944, 0.999999999183488)
(8542.665404040546, 0.999999998857507)
(-35534.003348062775, 0.999999999931898)
(34817.63713862056, 0.999999999929905)
(6000.279309089443, 0.999999997684569)
(11085.247223255352, 0.999999999323776)
(-15191.825703865761, 0.999999999637211)
(42446.03088328492, 0.999999999949988)
(-34686.405024843574, 0.99999999993407)
(-26210.44282631641, 0.9999999998789)
(-20277.309163451333, 0.999999999795962)
(-42314.79846149776, 0.99999999995506)
(39903.23108303823, 0.999999999954422)
(17865.786726280232, 0.999999999739458)
(-32991.209289882085, 0.999999999919875)
(-9258.958790640634, 0.99999999902765)
(-21972.484920013052, 0.999999999828907)
(-10954.023521450948, 0.999999999305978)
(29732.053960474437, 0.999999999902336)
(-11801.571373808376, 0.999999999400516)
(-21124.89638537924, 0.999999999814505)
(-16886.977356312047, 0.999999999708111)
(-22820.07462107522, 0.999999999840901)
(5152.910863706506, 0.999999996860707)
(31427.24683430499, 0.999999999909635)
(-13496.688325769803, 0.999999999542028)
(-7563.958502106809, 0.999999998544044)
(17018.2064590623, 0.999999999715423)
(9390.178782341221, 0.999999999056659)
(24646.48821321372, 0.999999999867149)
(22951.30550703823, 0.999999999842787)
(-33838.80699783976, 0.999999999926482)
(-24515.257038524076, 0.999999999859095)
(40750.83083747623, 0.99999999995989)
(-36381.60194679426, 0.999999999935505)
(-25362.849553366064, 0.999999999871507)
(6847.706132893425, 0.99999999822187)
(19560.953492324457, 0.999999999780259)
(38208.03217068598, 0.999999999951217)
(-27058.03678612882, 0.99999999988755)
(-16039.400028684786, 0.999999999677059)
(-19429.72342603411, 0.99999999977758)
(-37229.200802215695, 0.999999999936696)
(21256.126910582065, 0.999999999813405)
(-6716.49777568092, 0.999999998153448)
(11932.796363954556, 0.999999999412703)
(27189.26829666882, 0.999999999884665)
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:
$$x_{1} = 13627.9152059125$$
$$x_{2} = 41598.4307745321$$
$$x_{3} = -40619.5984694662$$
$$x_{4} = 26341.6742356719$$
$$x_{5} = 30579.6501812154$$
$$x_{6} = 28884.4582101405$$
$$x_{7} = 28036.8629728761$$
$$x_{8} = 35665.2355060803$$
$$x_{9} = 14475.4824993254$$
$$x_{10} = -23667.6653632422$$
$$x_{11} = -8411.44815555337$$
$$x_{12} = -5869.07851287475$$
$$x_{13} = -15191.8257038658$$
$$x_{14} = 42446.0308832849$$
$$x_{15} = -32991.2092898821$$
$$x_{16} = -11801.5713738084$$
$$x_{17} = 5152.91086370651$$
$$x_{18} = 31427.246834305$$
$$x_{19} = -13496.6883257698$$
$$x_{20} = 19560.9534923245$$
$$x_{21} = -37229.2008022157$$
$$x_{22} = 11932.7963639546$$
Puntos máximos de la función:
$$x_{22} = 36512.8341459926$$
$$x_{22} = 12780.3528127123$$
$$x_{22} = 15323.0538797709$$
$$x_{22} = -27905.6313702123$$
$$x_{22} = -14344.2549142001$$
$$x_{22} = -18582.1393762791$$
$$x_{22} = 11085.2472232554$$
$$x_{22} = -34686.4050248436$$
$$x_{22} = 39903.2310830382$$
$$x_{22} = -21972.4849200131$$
$$x_{22} = -21124.8963853792$$
$$x_{22} = -16886.977356312$$
$$x_{22} = -22820.0746210752$$
$$x_{22} = -7563.95850210681$$
$$x_{22} = 17018.2064590623$$
$$x_{22} = 24646.4882132137$$
$$x_{22} = 22951.3055070382$$
$$x_{22} = 40750.8308374762$$
$$x_{22} = 38208.032170686$$
$$x_{22} = -27058.0367861288$$
$$x_{22} = -6716.49777568092$$
Decrece en los intervalos
$$\left[42446.0308832849, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -40619.5984694662\right]$$
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$\frac{\left(- \frac{2 \left(\frac{\left(e^{\frac{1}{x}} - 1\right)^{2} - \frac{2 \left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}}{x} + \frac{\left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}}{x^{2}} + \frac{e^{\frac{2}{x}}}{x^{2}}}{e^{\frac{1}{x}} - 1} - \frac{2 \left(1 - \frac{e^{\frac{1}{x}}}{x \left(e^{\frac{1}{x}} - 1\right)}\right) \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x} - \frac{2 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x} + \frac{2 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right) e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)}\right)}{e^{\frac{1}{x}} - 1} + \frac{2 + \frac{1}{x}}{x} + \frac{4 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x^{2} \left(e^{\frac{1}{x}} - 1\right)}\right) e^{\frac{1}{x}}}{x^{4} \left(e^{\frac{1}{x}} - 1\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 4378.45489343224$$
$$x_{2} = -8051.80396529387$$
$$x_{3} = 6122.96954310366$$
$$x_{4} = 9612.05023180978$$
$$x_{5} = 2415.95737140673$$
$$x_{6} = 6559.10212778111$$
$$x_{7} = 8085.57286482282$$
$$x_{8} = -9796.34976305168$$
$$x_{9} = -5216.93993305863$$
$$x_{10} = 2634.00275086831$$
$$x_{11} = 3070.10409968592$$
$$x_{12} = -4998.8757695151$$
$$x_{13} = -7179.53399063609$$
$$x_{14} = 10484.3248752431$$
$$x_{15} = 10702.3936959841$$
$$x_{16} = 8957.84505151282$$
$$x_{17} = -6961.46692563361$$
$$x_{18} = 5468.77302131714$$
$$x_{19} = -2600.23715088493$$
$$x_{20} = -4126.62552828055$$
$$x_{21} = 7431.37007033236$$
$$x_{22} = -7615.66866449652$$
$$x_{23} = -10450.5558183163$$
$$x_{24} = 2197.91692269672$$
$$x_{25} = -6089.20093298288$$
$$x_{26} = 889.980807079997$$
$$x_{27} = -856.244713551305$$
$$x_{28} = 10048.1874172931$$
$$x_{29} = -1946.12030197999$$
$$x_{30} = 7649.43751843264$$
$$x_{31} = -2382.19247235659$$
$$x_{32} = 7867.50512075532$$
$$x_{33} = 6995.23569448843$$
$$x_{34} = 5032.6440531647$$
$$x_{35} = 5250.70829861068$$
$$x_{36} = -8269.87181979443$$
$$x_{37} = 10266.25611438$$
$$x_{38} = 9175.91335764308$$
$$x_{39} = -7397.60124226819$$
$$x_{40} = -8706.00788587186$$
$$x_{41} = 9830.11878822658$$
$$x_{42} = -1074.14251057279$$
$$x_{43} = 1107.89055578526$$
$$x_{44} = 1543.84605417316$$
$$x_{45} = -8924.07607999643$$
$$x_{46} = -10886.6934980213$$
$$x_{47} = -6743.40006536826$$
$$x_{48} = 8303.64073946823$$
$$x_{49} = -10232.4870673643$$
$$x_{50} = 6777.1688002299$$
$$x_{51} = -2818.28578470162$$
$$x_{52} = 5686.83816638426$$
$$x_{53} = 8521.70873466762$$
$$x_{54} = 10920.4625730159$$
$$x_{55} = -1728.09715000669$$
$$x_{56} = 4814.58034982471$$
$$x_{57} = -4780.81215949822$$
$$x_{58} = -9142.14437126442$$
$$x_{59} = 2852.05193017091$$
$$x_{60} = 4160.39334481353$$
$$x_{61} = -6525.33343036833$$
$$x_{62} = 5904.90368702149$$
$$x_{63} = -2164.15294523236$$
$$x_{64} = -10014.4183808417$$
$$x_{65} = 6341.03570002487$$
$$x_{66} = 1979.88303384799$$
$$x_{67} = 1761.85814045997$$
$$x_{68} = -3908.56510799633$$
$$x_{69} = 1325.85309427006$$
$$x_{70} = -6307.26704400142$$
$$x_{71} = -1292.09855768527$$
$$x_{72} = 3506.21517784186$$
$$x_{73} = -5871.13512799763$$
$$x_{74} = -3036.33752139337$$
$$x_{75} = 3724.27329735441$$
$$x_{76} = -9578.28121868891$$
$$x_{77} = 9393.98175310191$$
$$x_{78} = -3690.50584726659$$
$$x_{79} = 8739.77684139745$$
$$x_{80} = -3472.44796459652$$
$$x_{81} = -10668.6246297469$$
$$x_{82} = -7833.73624307318$$
$$x_{83} = 4596.51726574094$$
$$x_{84} = 3288.15866454085$$
$$x_{85} = -8487.93979637861$$
$$x_{86} = -5435.00458349307$$
$$x_{87} = -4344.68693348994$$
$$x_{88} = -3254.39173707057$$
$$x_{89} = 7213.30279044085$$
$$x_{90} = -1510.08760379866$$
$$x_{91} = -9360.2127528858$$
$$x_{92} = -5653.06966446584$$
$$x_{93} = -4562.74918238466$$
$$x_{94} = 3942.33275655982$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{\left(- \frac{2 \left(\frac{\left(e^{\frac{1}{x}} - 1\right)^{2} - \frac{2 \left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}}{x} + \frac{\left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}}{x^{2}} + \frac{e^{\frac{2}{x}}}{x^{2}}}{e^{\frac{1}{x}} - 1} - \frac{2 \left(1 - \frac{e^{\frac{1}{x}}}{x \left(e^{\frac{1}{x}} - 1\right)}\right) \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x} - \frac{2 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x} + \frac{2 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right) e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)}\right)}{e^{\frac{1}{x}} - 1} + \frac{2 + \frac{1}{x}}{x} + \frac{4 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x^{2} \left(e^{\frac{1}{x}} - 1\right)}\right) e^{\frac{1}{x}}}{x^{4} \left(e^{\frac{1}{x}} - 1\right)^{2}}\right) = 0$$
$$\lim_{x \to 0^+}\left(\frac{\left(- \frac{2 \left(\frac{\left(e^{\frac{1}{x}} - 1\right)^{2} - \frac{2 \left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}}{x} + \frac{\left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}}{x^{2}} + \frac{e^{\frac{2}{x}}}{x^{2}}}{e^{\frac{1}{x}} - 1} - \frac{2 \left(1 - \frac{e^{\frac{1}{x}}}{x \left(e^{\frac{1}{x}} - 1\right)}\right) \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x} - \frac{2 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x} + \frac{2 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right) e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)}\right)}{e^{\frac{1}{x}} - 1} + \frac{2 + \frac{1}{x}}{x} + \frac{4 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x^{2} \left(e^{\frac{1}{x}} - 1\right)}\right) e^{\frac{1}{x}}}{x^{4} \left(e^{\frac{1}{x}} - 1\right)^{2}}\right) = -\infty$$
- los límites no son iguales, signo
$$x_{1} = 0$$
- 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
$$\left[10048.1874172931, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -10450.5558183163\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$\frac{\left(- \frac{2 \left(\frac{\left(e^{\frac{1}{x}} - 1\right)^{2} - \frac{2 \left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}}{x} + \frac{\left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}}{x^{2}} + \frac{e^{\frac{2}{x}}}{x^{2}}}{e^{\frac{1}{x}} - 1} - \frac{2 \left(1 - \frac{e^{\frac{1}{x}}}{x \left(e^{\frac{1}{x}} - 1\right)}\right) \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x} - \frac{2 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x} + \frac{2 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right) e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)}\right)}{e^{\frac{1}{x}} - 1} + \frac{2 + \frac{1}{x}}{x} + \frac{4 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x^{2} \left(e^{\frac{1}{x}} - 1\right)}\right) e^{\frac{1}{x}}}{x^{4} \left(e^{\frac{1}{x}} - 1\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 4378.45489343224$$
$$x_{2} = -8051.80396529387$$
$$x_{3} = 6122.96954310366$$
$$x_{4} = 9612.05023180978$$
$$x_{5} = 2415.95737140673$$
$$x_{6} = 6559.10212778111$$
$$x_{7} = 8085.57286482282$$
$$x_{8} = -9796.34976305168$$
$$x_{9} = -5216.93993305863$$
$$x_{10} = 2634.00275086831$$
$$x_{11} = 3070.10409968592$$
$$x_{12} = -4998.8757695151$$
$$x_{13} = -7179.53399063609$$
$$x_{14} = 10484.3248752431$$
$$x_{15} = 10702.3936959841$$
$$x_{16} = 8957.84505151282$$
$$x_{17} = -6961.46692563361$$
$$x_{18} = 5468.77302131714$$
$$x_{19} = -2600.23715088493$$
$$x_{20} = -4126.62552828055$$
$$x_{21} = 7431.37007033236$$
$$x_{22} = -7615.66866449652$$
$$x_{23} = -10450.5558183163$$
$$x_{24} = 2197.91692269672$$
$$x_{25} = -6089.20093298288$$
$$x_{26} = 889.980807079997$$
$$x_{27} = -856.244713551305$$
$$x_{28} = 10048.1874172931$$
$$x_{29} = -1946.12030197999$$
$$x_{30} = 7649.43751843264$$
$$x_{31} = -2382.19247235659$$
$$x_{32} = 7867.50512075532$$
$$x_{33} = 6995.23569448843$$
$$x_{34} = 5032.6440531647$$
$$x_{35} = 5250.70829861068$$
$$x_{36} = -8269.87181979443$$
$$x_{37} = 10266.25611438$$
$$x_{38} = 9175.91335764308$$
$$x_{39} = -7397.60124226819$$
$$x_{40} = -8706.00788587186$$
$$x_{41} = 9830.11878822658$$
$$x_{42} = -1074.14251057279$$
$$x_{43} = 1107.89055578526$$
$$x_{44} = 1543.84605417316$$
$$x_{45} = -8924.07607999643$$
$$x_{46} = -10886.6934980213$$
$$x_{47} = -6743.40006536826$$
$$x_{48} = 8303.64073946823$$
$$x_{49} = -10232.4870673643$$
$$x_{50} = 6777.1688002299$$
$$x_{51} = -2818.28578470162$$
$$x_{52} = 5686.83816638426$$
$$x_{53} = 8521.70873466762$$
$$x_{54} = 10920.4625730159$$
$$x_{55} = -1728.09715000669$$
$$x_{56} = 4814.58034982471$$
$$x_{57} = -4780.81215949822$$
$$x_{58} = -9142.14437126442$$
$$x_{59} = 2852.05193017091$$
$$x_{60} = 4160.39334481353$$
$$x_{61} = -6525.33343036833$$
$$x_{62} = 5904.90368702149$$
$$x_{63} = -2164.15294523236$$
$$x_{64} = -10014.4183808417$$
$$x_{65} = 6341.03570002487$$
$$x_{66} = 1979.88303384799$$
$$x_{67} = 1761.85814045997$$
$$x_{68} = -3908.56510799633$$
$$x_{69} = 1325.85309427006$$
$$x_{70} = -6307.26704400142$$
$$x_{71} = -1292.09855768527$$
$$x_{72} = 3506.21517784186$$
$$x_{73} = -5871.13512799763$$
$$x_{74} = -3036.33752139337$$
$$x_{75} = 3724.27329735441$$
$$x_{76} = -9578.28121868891$$
$$x_{77} = 9393.98175310191$$
$$x_{78} = -3690.50584726659$$
$$x_{79} = 8739.77684139745$$
$$x_{80} = -3472.44796459652$$
$$x_{81} = -10668.6246297469$$
$$x_{82} = -7833.73624307318$$
$$x_{83} = 4596.51726574094$$
$$x_{84} = 3288.15866454085$$
$$x_{85} = -8487.93979637861$$
$$x_{86} = -5435.00458349307$$
$$x_{87} = -4344.68693348994$$
$$x_{88} = -3254.39173707057$$
$$x_{89} = 7213.30279044085$$
$$x_{90} = -1510.08760379866$$
$$x_{91} = -9360.2127528858$$
$$x_{92} = -5653.06966446584$$
$$x_{93} = -4562.74918238466$$
$$x_{94} = 3942.33275655982$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{\left(- \frac{2 \left(\frac{\left(e^{\frac{1}{x}} - 1\right)^{2} - \frac{2 \left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}}{x} + \frac{\left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}}{x^{2}} + \frac{e^{\frac{2}{x}}}{x^{2}}}{e^{\frac{1}{x}} - 1} - \frac{2 \left(1 - \frac{e^{\frac{1}{x}}}{x \left(e^{\frac{1}{x}} - 1\right)}\right) \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x} - \frac{2 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x} + \frac{2 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right) e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)}\right)}{e^{\frac{1}{x}} - 1} + \frac{2 + \frac{1}{x}}{x} + \frac{4 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x^{2} \left(e^{\frac{1}{x}} - 1\right)}\right) e^{\frac{1}{x}}}{x^{4} \left(e^{\frac{1}{x}} - 1\right)^{2}}\right) = 0$$
$$\lim_{x \to 0^+}\left(\frac{\left(- \frac{2 \left(\frac{\left(e^{\frac{1}{x}} - 1\right)^{2} - \frac{2 \left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}}{x} + \frac{\left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}}{x^{2}} + \frac{e^{\frac{2}{x}}}{x^{2}}}{e^{\frac{1}{x}} - 1} - \frac{2 \left(1 - \frac{e^{\frac{1}{x}}}{x \left(e^{\frac{1}{x}} - 1\right)}\right) \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x} - \frac{2 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x} + \frac{2 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right) e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)}\right)}{e^{\frac{1}{x}} - 1} + \frac{2 + \frac{1}{x}}{x} + \frac{4 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x^{2} \left(e^{\frac{1}{x}} - 1\right)}\right) e^{\frac{1}{x}}}{x^{4} \left(e^{\frac{1}{x}} - 1\right)^{2}}\right) = -\infty$$
- los límites no son iguales, signo
$$x_{1} = 0$$
- 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
$$\left[10048.1874172931, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -10450.5558183163\right]$$
Asíntotas verticales
Hay:
$$x_{1} = 0$$
$$x_{1} = 0$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}}\right) = 1$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 1$$
$$\lim_{x \to \infty}\left(\frac{e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}}\right) = 1$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 1$$
$$\lim_{x \to -\infty}\left(\frac{e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}}\right) = 1$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 1$$
$$\lim_{x \to \infty}\left(\frac{e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}}\right) = 1$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 1$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función exp(1/x)/((x^2*(exp(1/x) - 1)^2)), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{1}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}} e^{\frac{1}{x}}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\frac{1}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}} e^{\frac{1}{x}}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\frac{1}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}} e^{\frac{1}{x}}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\frac{1}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}} e^{\frac{1}{x}}}{x}\right) = 0$$
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:
$$\frac{e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}} = \frac{e^{- \frac{1}{x}}}{x^{2} \left(-1 + e^{- \frac{1}{x}}\right)^{2}}$$
- No
$$\frac{e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}} = - \frac{e^{- \frac{1}{x}}}{x^{2} \left(-1 + e^{- \frac{1}{x}}\right)^{2}}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}} = \frac{e^{- \frac{1}{x}}}{x^{2} \left(-1 + e^{- \frac{1}{x}}\right)^{2}}$$
- No
$$\frac{e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}} = - \frac{e^{- \frac{1}{x}}}{x^{2} \left(-1 + e^{- \frac{1}{x}}\right)^{2}}$$
- No
es decir, función
no es
par ni impar