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  • Gráfico de la función y =:
  • (1/3)x^3-4x (1/3)x^3-4x
  • y=-x³+2x y=-x³+2x
  • y=x^3-14x^2+60x-72 y=x^3-14x^2+60x-72
  • y=(x^2-3) y=(x^2-3)

  • Expresiones idénticas

  • veinte *(-exp(x)- dos *x)/(x^ dos +exp(x))^ dos
  • 20 multiplicar por ( menos exponente de (x) menos 2 multiplicar por x) dividir por (x al cuadrado más exponente de (x)) al cuadrado
  • veinte multiplicar por ( menos exponente de (x) menos dos multiplicar por x) dividir por (x en el grado dos más exponente de (x)) en el grado dos
  • 20*(-exp(x)-2*x)/(x2+exp(x))2
  • 20*-expx-2*x/x2+expx2
  • 20*(-exp(x)-2*x)/(x²+exp(x))²
  • 20*(-exp(x)-2*x)/(x en el grado 2+exp(x)) en el grado 2
  • 20(-exp(x)-2x)/(x^2+exp(x))^2
  • 20(-exp(x)-2x)/(x2+exp(x))2
  • 20-expx-2x/x2+expx2
  • 20-expx-2x/x^2+expx^2
  • 20*(-exp(x)-2*x) dividir por (x^2+exp(x))^2

  • Expresiones semejantes

  • 20*(-exp(x)+2*x)/(x^2+exp(x))^2
  • 20*(exp(x)-2*x)/(x^2+exp(x))^2
  • 20*(-exp(x)-2*x)/(x^2-exp(x))^2
Trazado el gráfico de la función/ 20*(-exp(x)-2*x)/(x^2+exp(x))^2

Gráfico de la función y = 20*(-exp(x)-2*x)/(x^2+exp(x))^2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          /   x      \
       20*\- e  - 2*x/
f(x) = ---------------
                   2  
          / 2    x\   
          \x  + e /
f(x)=20(−2x−ex)(x2+ex)2f{\left(x \right)} = \frac{20 \left(- 2 x - e^{x}\right)}{\left(x^{2} + e^{x}\right)^{2}} 
f = (20*(-2*x - exp(x)))/(x^2 + exp(x))^2
Gráfico de la función
02468-8-6-4-2-1010-5050
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:
20(−2x−ex)(x2+ex)2=0\frac{20 \left(- 2 x - e^{x}\right)}{\left(x^{2} + e^{x}\right)^{2}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−W(12)x_{1} = - W\left(\frac{1}{2}\right)
Solución numérica
x1=117.181662037845x_{1} = 117.181662037845
x2=105.181662037845x_{2} = 105.181662037845
x3=−37229.0173858502x_{3} = -37229.0173858502
x4=−33838.605204445x_{4} = -33838.605204445
x5=−34686.2081625031x_{5} = -34686.2081625031
x6=75.1816620378446x_{6} = 75.1816620378446
x7=49.1816620392208x_{7} = 49.1816620392208
x8=−32143.3994905401x_{8} = -32143.3994905401
x9=−35533.8111815333x_{9} = -35533.8111815333
x10=113.181662037845x_{10} = 113.181662037845
x11=119.181662037845x_{11} = 119.181662037845
x12=93.1816620378446x_{12} = 93.1816620378446
x13=65.1816620378446x_{13} = 65.1816620378446
x14=−32991.0023120587x_{14} = -32991.0023120587
x15=89.1816620378446x_{15} = 89.1816620378446
x16=99.1816620378446x_{16} = 99.1816620378446
x17=71.1816620378446x_{17} = 71.1816620378446
x18=−38076.6205637343x_{18} = -38076.6205637343
x19=77.1816620378446x_{19} = 77.1816620378446
x20=103.181662037845x_{20} = 103.181662037845
x21=101.181662037845x_{21} = 101.181662037845
x22=37.1817321074615x_{22} = 37.1817321074615
x23=59.1816620378447x_{23} = 59.1816620378447
x24=109.181662037845x_{24} = 109.181662037845
x25=73.1816620378446x_{25} = 73.1816620378446
x26=67.1816620378446x_{26} = 67.1816620378446
x27=43.1816623753418x_{27} = 43.1816623753418
x28=121.181662037845x_{28} = 121.181662037845
x29=85.1816620378446x_{29} = 85.1816620378446
x30=83.1816620378446x_{30} = 83.1816620378446
x31=35.2096381854689x_{31} = 35.2096381854689
x32=63.1816620378446x_{32} = 63.1816620378446
x33=81.1816620378446x_{33} = 81.1816620378446
x34=115.181662037845x_{34} = 115.181662037845
x35=−41467.0337078697x_{35} = -41467.0337078697
x36=91.1816620378446x_{36} = 91.1816620378446
x37=−38924.2237877047x_{37} = -38924.2237877047
x38=−39771.8270548144x_{38} = -39771.8270548144
x39=55.1816620378497x_{39} = 55.1816620378497
x40=−42314.6370890542x_{40} = -42314.6370890542
x41=−36381.4142572736x_{41} = -36381.4142572736
x42=33.1837407610153x_{42} = 33.1837407610153
x43=53.1816620378778x_{43} = 53.1816620378778
x44=−40619.4303623627x_{44} = -40619.4303623627
x45=61.1816620378446x_{45} = 61.1816620378446
x46=111.181662037845x_{46} = 111.181662037845
x47=35.1820526789567x_{47} = 35.1820526789567
x48=51.1816620380592x_{48} = 51.1816620380592
x49=47.1816620465744x_{49} = 47.1816620465744
x50=87.1816620378446x_{50} = 87.1816620378446
x51=97.1816620378446x_{51} = 97.1816620378446
x52=79.1816620378446x_{52} = 79.1816620378446
x53=31.1920178742905x_{53} = 31.1920178742905
x54=45.1816620925295x_{54} = 45.1816620925295
x55=95.1816620378446x_{55} = 95.1816620378446
x56=69.1816620378446x_{56} = 69.1816620378446
x57=39.1816741753077x_{57} = 39.1816741753077
x58=107.181662037845x_{58} = 107.181662037845
x59=41.1816640839418x_{59} = 41.1816640839418
x60=57.1816620378454x_{60} = 57.1816620378454
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 (20*(-exp(x) - 2*x))/(x^2 + exp(x))^2.
20(−e0−0)(02+e0)2\frac{20 \left(- e^{0} - 0\right)}{\left(0^{2} + e^{0}\right)^{2}}
Resultado:
f(0)=−20f{\left(0 \right)} = -20
Punto:
(0, -20)
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
−20(−2x−ex)(4x+2ex)(x2+ex)3+−20ex−40(x2+ex)2=0- \frac{20 \left(- 2 x - e^{x}\right) \left(4 x + 2 e^{x}\right)}{\left(x^{2} + e^{x}\right)^{3}} + \frac{- 20 e^{x} - 40}{\left(x^{2} + e^{x}\right)^{2}} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga extremos
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
20(4(2x+ex)(ex+2)x2+ex+2(2x+ex)(−3(2x+ex)2x2+ex+ex+2)x2+ex−ex)(x2+ex)2=0\frac{20 \left(\frac{4 \left(2 x + e^{x}\right) \left(e^{x} + 2\right)}{x^{2} + e^{x}} + \frac{2 \left(2 x + e^{x}\right) \left(- \frac{3 \left(2 x + e^{x}\right)^{2}}{x^{2} + e^{x}} + e^{x} + 2\right)}{x^{2} + e^{x}} - e^{x}\right)}{\left(x^{2} + e^{x}\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−2303.2918927703x_{1} = -2303.2918927703
x2=−4428.87861739056x_{2} = -4428.87861739056
x3=−3966.79374691734x_{3} = -3966.79374691734
x4=39.1817050340071x_{4} = 39.1817050340071
x5=109.181662037845x_{5} = 109.181662037845
x6=119.181662037845x_{6} = 119.181662037845
x7=37.1819061174749x_{7} = 37.1819061174749
x8=−4613.71262767709x_{8} = -4613.71262767709
x9=101.181662037845x_{9} = 101.181662037845
x10=67.1816620378446x_{10} = 67.1816620378446
x11=−3412.29229821251x_{11} = -3412.29229821251
x12=51.1816620386448x_{12} = 51.1816620386448
x13=−824.649596542509x_{13} = -824.649596542509
x14=−3319.87544611241x_{14} = -3319.87544611241
x15=−1563.96297388422x_{15} = -1563.96297388422
x16=71.1816620378446x_{16} = 71.1816620378446
x17=105.181662037845x_{17} = 105.181662037845
x18=89.1816620378446x_{18} = 89.1816620378446
x19=−2026.04280825936x_{19} = -2026.04280825936
x20=−3689.54295776183x_{20} = -3689.54295776183
x21=45.1816622379575x_{21} = 45.1816622379575
x22=115.181662037845x_{22} = 115.181662037845
x23=−3504.70916842872x_{23} = -3504.70916842872
x24=−3227.45861368543x_{24} = -3227.45861368543
x25=73.1816620378446x_{25} = 73.1816620378446
x26=−1471.54747044516x_{26} = -1471.54747044516
x27=−3781.95987448783x_{27} = -3781.95987448783
x28=75.1816620378446x_{28} = 75.1816620378446
x29=−2765.37481571999x_{29} = -2765.37481571999
x30=−4244.04464026923x_{30} = -4244.04464026923
x31=53.1816620379688x_{31} = 53.1816620379688
x32=93.1816620378446x_{32} = 93.1816620378446
x33=79.1816620378446x_{33} = 79.1816620378446
x34=−1656.37866630998x_{34} = -1656.37866630998
x35=69.1816620378446x_{35} = 69.1816620378446
x36=63.1816620378446x_{36} = 63.1816620378446
x37=−917.061625378965x_{37} = -917.061625378965
x38=−1841.2105041965x_{38} = -1841.2105041965
x39=113.181662037845x_{39} = 113.181662037845
x40=59.1816620378451x_{40} = 59.1816620378451
x41=87.1816620378446x_{41} = 87.1816620378446
x42=95.1816620378446x_{42} = 95.1816620378446
x43=117.181662037845x_{43} = 117.181662037845
x44=−2118.45909694788x_{44} = -2118.45909694788
x45=83.1816620378446x_{45} = 83.1816620378446
x46=−2395.70838303981x_{46} = -2395.70838303981
x47=−4059.21070084534x_{47} = -4059.21070084534
x48=35.1829930535804x_{48} = 35.1829930535804
x49=−4336.4616244191x_{49} = -4336.4616244191
x50=−2488.12492572786x_{50} = -2488.12492572786
x51=0.434233107519508x_{51} = 0.434233107519508
x52=121.181662037845x_{52} = 121.181662037845
x53=31.214488746646x_{53} = 31.214488746646
x54=85.1816620378446x_{54} = 85.1816620378446
x55=−1286.71719360562x_{55} = -1286.71719360562
x56=103.181662037845x_{56} = 103.181662037845
x57=−3135.04180267225x_{57} = -3135.04180267225
x58=33.1885354420719x_{58} = 33.1885354420719
x59=107.181662037845x_{59} = 107.181662037845
x60=−2857.79151890169x_{60} = -2857.79151890169
x61=−4151.6276655303x_{61} = -4151.6276655303
x62=−1009.4746077854x_{62} = -1009.4746077854
x63=43.1816632621693x_{63} = 43.1816632621693
x64=91.1816620378446x_{64} = 91.1816620378446
x65=99.1816620378446x_{65} = 99.1816620378446
x66=−3042.62501502515x_{66} = -3042.62501502515
x67=−1933.62660631779x_{67} = -1933.62660631779
x68=−2210.8754614974x_{68} = -2210.8754614974
x69=61.1816620378447x_{69} = 61.1816620378447
x70=−2580.54151519927x_{70} = -2580.54151519927
x71=111.181662037845x_{71} = 111.181662037845
x72=−4521.29561864248x_{72} = -4521.29561864248
x73=49.1816620429486x_{73} = 49.1816620429486
x74=77.1816620378446x_{74} = 77.1816620378446
x75=81.1816620378446x_{75} = 81.1816620378446
x76=−1101.8882989557x_{76} = -1101.8882989557
x77=−2672.95814659864x_{77} = -2672.95814659864
x78=−1194.30253333697x_{78} = -1194.30253333697
x79=65.1816620378446x_{79} = 65.1816620378446
x80=−1379.13219402731x_{80} = -1379.13219402731
x81=−3597.12605536413x_{81} = -3597.12605536413
x82=47.1816620700234x_{82} = 47.1816620700234
x83=−1748.79451773475x_{83} = -1748.79451773475
x84=41.1816693821433x_{84} = 41.1816693821433
x85=57.1816620378475x_{85} = 57.1816620378475
x86=−2950.20825294117x_{86} = -2950.20825294117
x87=97.1816620378446x_{87} = 97.1816620378446
x88=−3874.37680451638x_{88} = -3874.37680451638
x89=55.1816620378637x_{89} = 55.1816620378637

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
(−∞,0.434233107519508]\left(-\infty, 0.434233107519508\right]
Convexa en los intervalos
[0.434233107519508,∞)\left[0.434233107519508, \infty\right)
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(20(−2x−ex)(x2+ex)2)=0\lim_{x \to -\infty}\left(\frac{20 \left(- 2 x - e^{x}\right)}{\left(x^{2} + e^{x}\right)^{2}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(20(−2x−ex)(x2+ex)2)=0\lim_{x \to \infty}\left(\frac{20 \left(- 2 x - e^{x}\right)}{\left(x^{2} + e^{x}\right)^{2}}\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 (20*(-exp(x) - 2*x))/(x^2 + exp(x))^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(20(−2x−ex)x(x2+ex)2)=0\lim_{x \to -\infty}\left(\frac{20 \left(- 2 x - e^{x}\right)}{x \left(x^{2} + e^{x}\right)^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(20(−2x−ex)x(x2+ex)2)=0\lim_{x \to \infty}\left(\frac{20 \left(- 2 x - e^{x}\right)}{x \left(x^{2} + e^{x}\right)^{2}}\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:
20(−2x−ex)(x2+ex)2=40x−20e−x(x2+e−x)2\frac{20 \left(- 2 x - e^{x}\right)}{\left(x^{2} + e^{x}\right)^{2}} = \frac{40 x - 20 e^{- x}}{\left(x^{2} + e^{- x}\right)^{2}}
- No
20(−2x−ex)(x2+ex)2=−40x−20e−x(x2+e−x)2\frac{20 \left(- 2 x - e^{x}\right)}{\left(x^{2} + e^{x}\right)^{2}} = - \frac{40 x - 20 e^{- x}}{\left(x^{2} + e^{- x}\right)^{2}}
- No
es decir, función
no es
par ni impar
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