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- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x/(x^3+2)
-
x*e^(-x)^2
-
2*x^3-15*x^2+36*x-32
-
x*(x-4)
-
Expresiones idénticas
- (-cos(x)+(dos *sin(x)/(x+ uno))+(dos *cos(x)/(x+ uno)^ dos))/(x+ uno)
- ( menos coseno de (x) más (2 multiplicar por seno de (x) dividir por (x más 1)) más (2 multiplicar por coseno de (x) dividir por (x más 1) al cuadrado )) dividir por (x más 1)
- ( menos coseno de (x) más (dos multiplicar por seno de (x) dividir por (x más uno)) más (dos multiplicar por coseno de (x) dividir por (x más uno) en el grado dos)) dividir por (x más uno)
- (-cos(x)+(2*sin(x)/(x+1))+(2*cos(x)/(x+1)2))/(x+1)
- -cosx+2*sinx/x+1+2*cosx/x+12/x+1
- (-cos(x)+(2*sin(x)/(x+1))+(2*cos(x)/(x+1)²))/(x+1)
- (-cos(x)+(2*sin(x)/(x+1))+(2*cos(x)/(x+1) en el grado 2))/(x+1)
- (-cos(x)+(2sin(x)/(x+1))+(2cos(x)/(x+1)^2))/(x+1)
- (-cos(x)+(2sin(x)/(x+1))+(2cos(x)/(x+1)2))/(x+1)
- -cosx+2sinx/x+1+2cosx/x+12/x+1
- -cosx+2sinx/x+1+2cosx/x+1^2/x+1
- (-cos(x)+(2*sin(x) dividir por (x+1))+(2*cos(x) dividir por (x+1)^2)) dividir por (x+1)
-
Expresiones semejantes
- (-cos(x)+(2*sin(x)/(x+1))-(2*cos(x)/(x+1)^2))/(x+1)
- (-cos(x)+(2*sin(x)/(x+1))+(2*cos(x)/(x+1)^2))/(x-1)
- (cos(x)+(2*sin(x)/(x+1))+(2*cos(x)/(x+1)^2))/(x+1)
- (-cos(x)+(2*sin(x)/(x+1))+(2*cos(x)/(x-1)^2))/(x+1)
- (-cos(x)+(2*sin(x)/(x-1))+(2*cos(x)/(x+1)^2))/(x+1)
- (-cos(x)-(2*sin(x)/(x+1))+(2*cos(x)/(x+1)^2))/(x+1)
- (-cosx+(2*sinx/(x+1))+(2*cosx/(x+1)^2))/(x+1)
-
Expresiones con funciones
- Coseno cos
- cos(x),x
- cos(x)-sqrt(3)*sin(x)
- cos(3x)+3cos(x)
- cos(x)*e^x
- cos(x)+sin(2*x)/2
- Seno sin
- sin(x)/1+cos(x)
- sin(-x+3)
- sin(4*x)^(2)
- sinh(tanh(x))
- sint^2
- Coseno cos
- cos(x),x
- cos(x)-sqrt(3)*sin(x)
- cos(3x)+3cos(x)
- cos(x)*e^x
- cos(x)+sin(2*x)/2
Gráfico de la función y = (-cos(x)+(2*sin(x)/(x+1))+(2*cos(x)/(x+1)^2))/(x+1)
Solución
Ha introducido
[src]
2*sin(x) 2*cos(x)
-cos(x) + -------- + --------
x + 1 2
(x + 1)
f(x) = -----------------------------
x + 1
$$f{\left(x \right)} = \frac{\left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1}\right) + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x + 1}$$
f = (-cos(x) + (2*sin(x))/(x + 1) + (2*cos(x))/(x + 1)^2)/(x + 1)
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = -1$$
$$x_{1} = -1$$
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:
$$\frac{\left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1}\right) + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x + 1} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 86.3709050594723$$
$$x_{2} = -51.7968961320869$$
$$x_{3} = -48.6527034788051$$
$$x_{4} = 14.0034717913284$$
$$x_{5} = -95.7974767616183$$
$$x_{6} = 95.797912862081$$
$$x_{7} = -17.1546413657741$$
$$x_{8} = 10.825651157762$$
$$x_{9} = 80.0859449790141$$
$$x_{10} = -76.9426813176863$$
$$x_{11} = 26.6310922236611$$
$$x_{12} = -58.0844210975337$$
$$x_{13} = -36.0712578833702$$
$$x_{14} = -64.3710840254309$$
$$x_{15} = 61.2289118119026$$
$$x_{16} = 48.654396838104$$
$$x_{17} = -13.9825085391948$$
$$x_{18} = 92.6556268279389$$
$$x_{19} = -98.9397464504172$$
$$x_{20} = 98.9401552763972$$
$$x_{21} = 32.9277399444348$$
$$x_{22} = 42.3653647291314$$
$$x_{23} = 17.1684571899007$$
$$x_{24} = -26.6254109350763$$
$$x_{25} = 54.9421125829153$$
$$x_{26} = -10.7898786754269$$
$$x_{27} = 83.228458145445$$
$$x_{28} = 64.3720505127272$$
$$x_{29} = -86.3703684986956$$
$$x_{30} = -32.9240332040206$$
$$x_{31} = -61.2278434114583$$
$$x_{32} = 23.4801553706306$$
$$x_{33} = -7.54372449628009$$
$$x_{34} = -67.5141687409854$$
$$x_{35} = -4.00507341668955$$
$$x_{36} = 136.64474976163$$
$$x_{37} = -89.5127931011103$$
$$x_{38} = -29.7755709323142$$
$$x_{39} = 70.657920700132$$
$$x_{40} = 73.80068645168$$
$$x_{41} = 39.22016138731$$
$$x_{42} = 36.0743437126941$$
$$x_{43} = 58.0856084395179$$
$$x_{44} = -80.0853208283276$$
$$x_{45} = -83.2278802717944$$
$$x_{46} = -39.2175523279643$$
$$x_{47} = 4.32863617605124$$
$$x_{48} = 29.7801075137773$$
$$x_{49} = 51.7983897861238$$
$$x_{50} = 67.5150472396589$$
$$x_{51} = 7.61991323310644$$
$$x_{52} = -45.5081427660817$$
$$x_{53} = 89.5132926274963$$
$$x_{54} = -73.7999513585394$$
$$x_{55} = -70.6571186927646$$
$$x_{56} = -23.4728313498836$$
$$x_{57} = -54.9407852505616$$
$$x_{58} = -92.6551606286758$$
$$x_{59} = -20.3166301288662$$
$$x_{60} = 76.9433575383977$$
$$x_{61} = -42.3631297553676$$
$$x_{62} = 102.082358062481$$
$$x_{63} = 45.5100787997204$$
$$x_{64} = 20.3264348242219$$
o sea hay que resolver la ecuación:
$$\frac{\left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1}\right) + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x + 1} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 86.3709050594723$$
$$x_{2} = -51.7968961320869$$
$$x_{3} = -48.6527034788051$$
$$x_{4} = 14.0034717913284$$
$$x_{5} = -95.7974767616183$$
$$x_{6} = 95.797912862081$$
$$x_{7} = -17.1546413657741$$
$$x_{8} = 10.825651157762$$
$$x_{9} = 80.0859449790141$$
$$x_{10} = -76.9426813176863$$
$$x_{11} = 26.6310922236611$$
$$x_{12} = -58.0844210975337$$
$$x_{13} = -36.0712578833702$$
$$x_{14} = -64.3710840254309$$
$$x_{15} = 61.2289118119026$$
$$x_{16} = 48.654396838104$$
$$x_{17} = -13.9825085391948$$
$$x_{18} = 92.6556268279389$$
$$x_{19} = -98.9397464504172$$
$$x_{20} = 98.9401552763972$$
$$x_{21} = 32.9277399444348$$
$$x_{22} = 42.3653647291314$$
$$x_{23} = 17.1684571899007$$
$$x_{24} = -26.6254109350763$$
$$x_{25} = 54.9421125829153$$
$$x_{26} = -10.7898786754269$$
$$x_{27} = 83.228458145445$$
$$x_{28} = 64.3720505127272$$
$$x_{29} = -86.3703684986956$$
$$x_{30} = -32.9240332040206$$
$$x_{31} = -61.2278434114583$$
$$x_{32} = 23.4801553706306$$
$$x_{33} = -7.54372449628009$$
$$x_{34} = -67.5141687409854$$
$$x_{35} = -4.00507341668955$$
$$x_{36} = 136.64474976163$$
$$x_{37} = -89.5127931011103$$
$$x_{38} = -29.7755709323142$$
$$x_{39} = 70.657920700132$$
$$x_{40} = 73.80068645168$$
$$x_{41} = 39.22016138731$$
$$x_{42} = 36.0743437126941$$
$$x_{43} = 58.0856084395179$$
$$x_{44} = -80.0853208283276$$
$$x_{45} = -83.2278802717944$$
$$x_{46} = -39.2175523279643$$
$$x_{47} = 4.32863617605124$$
$$x_{48} = 29.7801075137773$$
$$x_{49} = 51.7983897861238$$
$$x_{50} = 67.5150472396589$$
$$x_{51} = 7.61991323310644$$
$$x_{52} = -45.5081427660817$$
$$x_{53} = 89.5132926274963$$
$$x_{54} = -73.7999513585394$$
$$x_{55} = -70.6571186927646$$
$$x_{56} = -23.4728313498836$$
$$x_{57} = -54.9407852505616$$
$$x_{58} = -92.6551606286758$$
$$x_{59} = -20.3166301288662$$
$$x_{60} = 76.9433575383977$$
$$x_{61} = -42.3631297553676$$
$$x_{62} = 102.082358062481$$
$$x_{63} = 45.5100787997204$$
$$x_{64} = 20.3264348242219$$
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
$$\frac{\frac{2 \left(- 2 x - 2\right) \cos{\left(x \right)}}{\left(x + 1\right)^{4}} + \sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x + 1} - \frac{4 \sin{\left(x \right)}}{\left(x + 1\right)^{2}}}{x + 1} - \frac{\left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1}\right) + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{\left(x + 1\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 62.7848083567476$$
$$x_{2} = -53.3497473009545$$
$$x_{3} = -37.6171216613073$$
$$x_{4} = 40.7688393516248$$
$$x_{5} = 47.0614426330123$$
$$x_{6} = -59.6390851053019$$
$$x_{7} = -91.0728764828512$$
$$x_{8} = -81.6442028274391$$
$$x_{9} = 100.501405803988$$
$$x_{10} = -65.9272291780463$$
$$x_{11} = 9.12548663246352$$
$$x_{12} = -25.0075622115342$$
$$x_{13} = -34.4678004641163$$
$$x_{14} = -18.6793158745042$$
$$x_{15} = 91.0736004794182$$
$$x_{16} = -47.0587248004656$$
$$x_{17} = 15.5257534535503$$
$$x_{18} = 84.788026977731$$
$$x_{19} = -56.4945908641657$$
$$x_{20} = -9.04577247969202$$
$$x_{21} = 37.6213824422946$$
$$x_{22} = -119.355171591525$$
$$x_{23} = -31.3168635525413$$
$$x_{24} = 78.5020755017039$$
$$x_{25} = 336.141515515324$$
$$x_{26} = -69.0709572003435$$
$$x_{27} = -21.8469080070317$$
$$x_{28} = 34.4728801053236$$
$$x_{29} = -62.7832835135593$$
$$x_{30} = 25.0172617617707$$
$$x_{31} = -929.908195860891$$
$$x_{32} = 94.2162689101071$$
$$x_{33} = -12.2987046709709$$
$$x_{34} = -28.1637420156702$$
$$x_{35} = 53.3518604974691$$
$$x_{36} = -75.3578709574114$$
$$x_{37} = -40.7652138363894$$
$$x_{38} = -87.9300792369319$$
$$x_{39} = 56.4964748494022$$
$$x_{40} = -5.59257434785924$$
$$x_{41} = 157.060651863799$$
$$x_{42} = 69.0722166780424$$
$$x_{43} = 72.2156485357275$$
$$x_{44} = -97.3582350860498$$
$$x_{45} = -78.5011007629945$$
$$x_{46} = 21.8596598669415$$
$$x_{47} = 18.696850464782$$
$$x_{48} = 43.9154718440336$$
$$x_{49} = -103.643327369698$$
$$x_{50} = 1.94603737394059$$
$$x_{51} = 81.6451038844529$$
$$x_{52} = 75.3589288068687$$
$$x_{53} = -50.2044871816413$$
$$x_{54} = 12.3401926554614$$
$$x_{55} = 50.206874194731$$
$$x_{56} = -15.5000638473804$$
$$x_{57} = -100.500811360292$$
$$x_{58} = 31.3230242457502$$
$$x_{59} = -94.2155924451548$$
$$x_{60} = -596.897569745606$$
$$x_{61} = -84.7871915483544$$
$$x_{62} = -43.9123491536953$$
$$x_{63} = 1.02874435561827$$
$$x_{64} = 28.1713718816254$$
$$x_{65} = 97.3588685530428$$
$$x_{66} = 65.9286118303933$$
$$x_{67} = 87.9308559596614$$
$$x_{68} = -72.2144964647543$$
$$x_{69} = 5.83393874788382$$
$$x_{70} = 59.6407752865768$$
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:
$$x_{1} = 62.7848083567476$$
$$x_{2} = -53.3497473009545$$
$$x_{3} = -59.6390851053019$$
$$x_{4} = -91.0728764828512$$
$$x_{5} = 100.501405803988$$
$$x_{6} = -65.9272291780463$$
$$x_{7} = -34.4678004641163$$
$$x_{8} = -47.0587248004656$$
$$x_{9} = -9.04577247969202$$
$$x_{10} = 37.6213824422946$$
$$x_{11} = -21.8469080070317$$
$$x_{12} = 25.0172617617707$$
$$x_{13} = 94.2162689101071$$
$$x_{14} = -28.1637420156702$$
$$x_{15} = -40.7652138363894$$
$$x_{16} = 56.4964748494022$$
$$x_{17} = 157.060651863799$$
$$x_{18} = 69.0722166780424$$
$$x_{19} = -97.3582350860498$$
$$x_{20} = -78.5011007629945$$
$$x_{21} = 18.696850464782$$
$$x_{22} = 43.9154718440336$$
$$x_{23} = -103.643327369698$$
$$x_{24} = 81.6451038844529$$
$$x_{25} = 75.3589288068687$$
$$x_{26} = 12.3401926554614$$
$$x_{27} = 50.206874194731$$
$$x_{28} = -15.5000638473804$$
$$x_{29} = 31.3230242457502$$
$$x_{30} = -84.7871915483544$$
$$x_{31} = 1.02874435561827$$
$$x_{32} = 87.9308559596614$$
$$x_{33} = -72.2144964647543$$
$$x_{34} = 5.83393874788382$$
Puntos máximos de la función:
$$x_{34} = -37.6171216613073$$
$$x_{34} = 40.7688393516248$$
$$x_{34} = 47.0614426330123$$
$$x_{34} = -81.6442028274391$$
$$x_{34} = 9.12548663246352$$
$$x_{34} = -25.0075622115342$$
$$x_{34} = -18.6793158745042$$
$$x_{34} = 91.0736004794182$$
$$x_{34} = 15.5257534535503$$
$$x_{34} = 84.788026977731$$
$$x_{34} = -56.4945908641657$$
$$x_{34} = -119.355171591525$$
$$x_{34} = -31.3168635525413$$
$$x_{34} = 78.5020755017039$$
$$x_{34} = 336.141515515324$$
$$x_{34} = -69.0709572003435$$
$$x_{34} = 34.4728801053236$$
$$x_{34} = -62.7832835135593$$
$$x_{34} = -929.908195860891$$
$$x_{34} = -12.2987046709709$$
$$x_{34} = 53.3518604974691$$
$$x_{34} = -75.3578709574114$$
$$x_{34} = -87.9300792369319$$
$$x_{34} = -5.59257434785924$$
$$x_{34} = 72.2156485357275$$
$$x_{34} = 21.8596598669415$$
$$x_{34} = 1.94603737394059$$
$$x_{34} = -50.2044871816413$$
$$x_{34} = -100.500811360292$$
$$x_{34} = -94.2155924451548$$
$$x_{34} = -596.897569745606$$
$$x_{34} = -43.9123491536953$$
$$x_{34} = 28.1713718816254$$
$$x_{34} = 97.3588685530428$$
$$x_{34} = 65.9286118303933$$
$$x_{34} = 59.6407752865768$$
Decrece en los intervalos
$$\left[157.060651863799, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -103.643327369698\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{\frac{2 \left(- 2 x - 2\right) \cos{\left(x \right)}}{\left(x + 1\right)^{4}} + \sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x + 1} - \frac{4 \sin{\left(x \right)}}{\left(x + 1\right)^{2}}}{x + 1} - \frac{\left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1}\right) + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{\left(x + 1\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 62.7848083567476$$
$$x_{2} = -53.3497473009545$$
$$x_{3} = -37.6171216613073$$
$$x_{4} = 40.7688393516248$$
$$x_{5} = 47.0614426330123$$
$$x_{6} = -59.6390851053019$$
$$x_{7} = -91.0728764828512$$
$$x_{8} = -81.6442028274391$$
$$x_{9} = 100.501405803988$$
$$x_{10} = -65.9272291780463$$
$$x_{11} = 9.12548663246352$$
$$x_{12} = -25.0075622115342$$
$$x_{13} = -34.4678004641163$$
$$x_{14} = -18.6793158745042$$
$$x_{15} = 91.0736004794182$$
$$x_{16} = -47.0587248004656$$
$$x_{17} = 15.5257534535503$$
$$x_{18} = 84.788026977731$$
$$x_{19} = -56.4945908641657$$
$$x_{20} = -9.04577247969202$$
$$x_{21} = 37.6213824422946$$
$$x_{22} = -119.355171591525$$
$$x_{23} = -31.3168635525413$$
$$x_{24} = 78.5020755017039$$
$$x_{25} = 336.141515515324$$
$$x_{26} = -69.0709572003435$$
$$x_{27} = -21.8469080070317$$
$$x_{28} = 34.4728801053236$$
$$x_{29} = -62.7832835135593$$
$$x_{30} = 25.0172617617707$$
$$x_{31} = -929.908195860891$$
$$x_{32} = 94.2162689101071$$
$$x_{33} = -12.2987046709709$$
$$x_{34} = -28.1637420156702$$
$$x_{35} = 53.3518604974691$$
$$x_{36} = -75.3578709574114$$
$$x_{37} = -40.7652138363894$$
$$x_{38} = -87.9300792369319$$
$$x_{39} = 56.4964748494022$$
$$x_{40} = -5.59257434785924$$
$$x_{41} = 157.060651863799$$
$$x_{42} = 69.0722166780424$$
$$x_{43} = 72.2156485357275$$
$$x_{44} = -97.3582350860498$$
$$x_{45} = -78.5011007629945$$
$$x_{46} = 21.8596598669415$$
$$x_{47} = 18.696850464782$$
$$x_{48} = 43.9154718440336$$
$$x_{49} = -103.643327369698$$
$$x_{50} = 1.94603737394059$$
$$x_{51} = 81.6451038844529$$
$$x_{52} = 75.3589288068687$$
$$x_{53} = -50.2044871816413$$
$$x_{54} = 12.3401926554614$$
$$x_{55} = 50.206874194731$$
$$x_{56} = -15.5000638473804$$
$$x_{57} = -100.500811360292$$
$$x_{58} = 31.3230242457502$$
$$x_{59} = -94.2155924451548$$
$$x_{60} = -596.897569745606$$
$$x_{61} = -84.7871915483544$$
$$x_{62} = -43.9123491536953$$
$$x_{63} = 1.02874435561827$$
$$x_{64} = 28.1713718816254$$
$$x_{65} = 97.3588685530428$$
$$x_{66} = 65.9286118303933$$
$$x_{67} = 87.9308559596614$$
$$x_{68} = -72.2144964647543$$
$$x_{69} = 5.83393874788382$$
$$x_{70} = 59.6407752865768$$
Signos de extremos en los puntos:
(62.78480835674761, -0.0156757879001206)
(-53.34974730095452, -0.019098804711862)
(-37.61712166130734, 0.0272994503266694)
(40.76883935162482, 0.0239344339680559)
(47.06144263301231, 0.0208021972460696)
(-59.63908510530185, -0.0170509928499818)
(-91.07287648285123, -0.0111014371491547)
(-81.64420282743907, 0.0123991941382204)
(100.50140580398805, -0.0098516021906735)
(-65.92722917804628, -0.0154000320808205)
(9.125486632463517, 0.0982810375274557)
(-25.00756221153416, 0.0416174506063621)
(-34.467800464116344, -0.029866136470405)
(-18.679315874504162, 0.056472976963668)
(91.07360047941823, 0.0108602359489118)
(-47.05872480046564, -0.0217062976280438)
(15.5257534535503, 0.0604010821630529)
(84.78802697773095, 0.0116558464333725)
(-56.49459086416567, 0.018016849341145)
(-9.045772479692022, -0.123332237193204)
(37.62138244229462, -0.0258837175784044)
(-119.35517159152471, 0.00844884337907207)
(-31.31686355254135, 0.0329670124977501)
(78.50207550170391, 0.0125772930624508)
(336.14151551532404, 0.00296610045068425)
(-69.07095720034349, 0.0146889679865376)
(-21.846908007031654, -0.0479136419208277)
(34.47288010532358, 0.0281793548228721)
(-62.78328351355932, 0.0161834881804617)
(25.01726176177074, -0.0384076583836474)
(-929.9081958608914, 0.00107653202436486)
(94.21626891010708, -0.0105018277413189)
(-12.298704670970862, 0.088160405448785)
(-28.163742015670245, -0.036788857009578)
(53.35186049746913, 0.0183955209318672)
(-75.3578709574114, 0.0134472594299687)
(-40.765213836389364, -0.0251396597254002)
(-87.93007923693195, 0.0115027370281061)
(56.49647484940217, -0.0173897406615397)
(-5.592574347859239, 0.212336087777758)
(157.06065186379865, -0.00632655866804922)
(69.0722166780424, -0.0142695383737192)
(72.21564853572748, 0.0136570085576436)
(-97.35823508604976, -0.0103773813687567)
(-78.50110076299448, -0.0129019685684665)
(21.85965986694152, 0.0437033815775418)
(18.696850464782038, -0.0507042198994302)
(43.915471844033625, -0.022258527079826)
(-103.64332736969784, -0.00974201218027633)
(1.9460373739405887, 0.310139768644169)
(81.64510388445292, -0.0120990446721193)
(75.35892880686873, -0.0130949225371441)
(-50.2044871816413, 0.020319153817462)
(12.34019265546136, -0.0747514914894685)
(50.20687419473105, -0.0195249053120444)
(-15.50006384738036, -0.0688016699422005)
(-100.50081136029176, 0.0100496617785023)
(31.32302424575023, -0.03092290346948)
(-94.2155924451548, 0.0107272017015871)
(-596.8975697456058, 0.00167813839608656)
(-84.78719154835436, -0.0119341484896391)
(-43.91234915369529, 0.0232969903870678)
(1.0287443556182734, 0.28555062804704)
(28.171371881625443, 0.0342600591376061)
(97.3588685530428, 0.0101663259883443)
(65.92861183039332, 0.0149396255210373)
(87.93085595966139, -0.011243980209497)
(-72.21449646475432, -0.0140407006160139)
(5.833938747883824, -0.144762325739142)
(59.640775286576826, 0.0164883125831885)
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:
$$x_{1} = 62.7848083567476$$
$$x_{2} = -53.3497473009545$$
$$x_{3} = -59.6390851053019$$
$$x_{4} = -91.0728764828512$$
$$x_{5} = 100.501405803988$$
$$x_{6} = -65.9272291780463$$
$$x_{7} = -34.4678004641163$$
$$x_{8} = -47.0587248004656$$
$$x_{9} = -9.04577247969202$$
$$x_{10} = 37.6213824422946$$
$$x_{11} = -21.8469080070317$$
$$x_{12} = 25.0172617617707$$
$$x_{13} = 94.2162689101071$$
$$x_{14} = -28.1637420156702$$
$$x_{15} = -40.7652138363894$$
$$x_{16} = 56.4964748494022$$
$$x_{17} = 157.060651863799$$
$$x_{18} = 69.0722166780424$$
$$x_{19} = -97.3582350860498$$
$$x_{20} = -78.5011007629945$$
$$x_{21} = 18.696850464782$$
$$x_{22} = 43.9154718440336$$
$$x_{23} = -103.643327369698$$
$$x_{24} = 81.6451038844529$$
$$x_{25} = 75.3589288068687$$
$$x_{26} = 12.3401926554614$$
$$x_{27} = 50.206874194731$$
$$x_{28} = -15.5000638473804$$
$$x_{29} = 31.3230242457502$$
$$x_{30} = -84.7871915483544$$
$$x_{31} = 1.02874435561827$$
$$x_{32} = 87.9308559596614$$
$$x_{33} = -72.2144964647543$$
$$x_{34} = 5.83393874788382$$
Puntos máximos de la función:
$$x_{34} = -37.6171216613073$$
$$x_{34} = 40.7688393516248$$
$$x_{34} = 47.0614426330123$$
$$x_{34} = -81.6442028274391$$
$$x_{34} = 9.12548663246352$$
$$x_{34} = -25.0075622115342$$
$$x_{34} = -18.6793158745042$$
$$x_{34} = 91.0736004794182$$
$$x_{34} = 15.5257534535503$$
$$x_{34} = 84.788026977731$$
$$x_{34} = -56.4945908641657$$
$$x_{34} = -119.355171591525$$
$$x_{34} = -31.3168635525413$$
$$x_{34} = 78.5020755017039$$
$$x_{34} = 336.141515515324$$
$$x_{34} = -69.0709572003435$$
$$x_{34} = 34.4728801053236$$
$$x_{34} = -62.7832835135593$$
$$x_{34} = -929.908195860891$$
$$x_{34} = -12.2987046709709$$
$$x_{34} = 53.3518604974691$$
$$x_{34} = -75.3578709574114$$
$$x_{34} = -87.9300792369319$$
$$x_{34} = -5.59257434785924$$
$$x_{34} = 72.2156485357275$$
$$x_{34} = 21.8596598669415$$
$$x_{34} = 1.94603737394059$$
$$x_{34} = -50.2044871816413$$
$$x_{34} = -100.500811360292$$
$$x_{34} = -94.2155924451548$$
$$x_{34} = -596.897569745606$$
$$x_{34} = -43.9123491536953$$
$$x_{34} = 28.1713718816254$$
$$x_{34} = 97.3588685530428$$
$$x_{34} = 65.9286118303933$$
$$x_{34} = 59.6407752865768$$
Decrece en los intervalos
$$\left[157.060651863799, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -103.643327369698\right]$$
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
$$\frac{\cos{\left(x \right)} - \frac{2 \left(\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x + 1} - \frac{4 \sin{\left(x \right)}}{\left(x + 1\right)^{2}} - \frac{4 \cos{\left(x \right)}}{\left(x + 1\right)^{3}}\right)}{x + 1} - \frac{2 \sin{\left(x \right)}}{x + 1} + \frac{2 \left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1} + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}\right)}{\left(x + 1\right)^{2}} - \frac{6 \cos{\left(x \right)}}{\left(x + 1\right)^{2}} + \frac{12 \sin{\left(x \right)}}{\left(x + 1\right)^{3}} + \frac{12 \cos{\left(x \right)}}{\left(x + 1\right)^{4}}}{x + 1} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -7.19519596149036$$
$$x_{2} = -89.4901840227415$$
$$x_{3} = 98.9201339417676$$
$$x_{4} = 70.6299849073373$$
$$x_{5} = 42.3191299769022$$
$$x_{6} = 13.867289840921$$
$$x_{7} = -13.8239463102427$$
$$x_{8} = 1.40731154138569$$
$$x_{9} = 92.6342606247044$$
$$x_{10} = -318.859070062961$$
$$x_{11} = -86.34692614697$$
$$x_{12} = 10.6504896158414$$
$$x_{13} = -39.1650521132014$$
$$x_{14} = 36.0202136918045$$
$$x_{15} = 45.4669842133418$$
$$x_{16} = 20.2316727993409$$
$$x_{17} = 86.3480001337381$$
$$x_{18} = -76.9163243230484$$
$$x_{19} = 83.2046975626699$$
$$x_{20} = -48.6106465468566$$
$$x_{21} = -80.0600127890459$$
$$x_{22} = 17.0567762419777$$
$$x_{23} = -29.7056715201635$$
$$x_{24} = -70.6283789578667$$
$$x_{25} = -26.54680103954$$
$$x_{26} = -36.0140131736861$$
$$x_{27} = -42.3146449333139$$
$$x_{28} = -67.484068177944$$
$$x_{29} = 67.4858274972604$$
$$x_{30} = -61.194593351618$$
$$x_{31} = -95.7763681868513$$
$$x_{32} = 76.9176781393433$$
$$x_{33} = -73.7724545812494$$
$$x_{34} = -20.2117653772165$$
$$x_{35} = -58.049334955595$$
$$x_{36} = 58.0517138854424$$
$$x_{37} = 39.1702908298917$$
$$x_{38} = -10.5745820582157$$
$$x_{39} = -23.3829976938101$$
$$x_{40} = 61.1967335889873$$
$$x_{41} = 54.9063078762122$$
$$x_{42} = 48.614041915914$$
$$x_{43} = 3.85793818596233$$
$$x_{44} = 23.3978106353706$$
$$x_{45} = 89.4911838253213$$
$$x_{46} = -45.4631008294892$$
$$x_{47} = 32.8685502646286$$
$$x_{48} = -64.3394871333198$$
$$x_{49} = -98.9193157877185$$
$$x_{50} = -83.203540811653$$
$$x_{51} = 64.3414229191161$$
$$x_{52} = 95.7772409591446$$
$$x_{53} = 29.7148073593336$$
$$x_{54} = -32.8610950559118$$
$$x_{55} = 80.0612622615078$$
$$x_{56} = -149.198659428593$$
$$x_{57} = 7.37124034627779$$
$$x_{58} = 26.5582622916455$$
$$x_{59} = -92.6333275731548$$
$$x_{60} = -17.0285445585894$$
$$x_{61} = -164.909209954026$$
$$x_{62} = -51.7574521916164$$
$$x_{63} = 51.7604462256561$$
$$x_{64} = -54.9036479029571$$
$$x_{65} = 230.889810252879$$
$$x_{66} = 73.7739263925337$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = -1$$
$$\lim_{x \to -1^-}\left(\frac{\cos{\left(x \right)} - \frac{2 \left(\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x + 1} - \frac{4 \sin{\left(x \right)}}{\left(x + 1\right)^{2}} - \frac{4 \cos{\left(x \right)}}{\left(x + 1\right)^{3}}\right)}{x + 1} - \frac{2 \sin{\left(x \right)}}{x + 1} + \frac{2 \left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1} + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}\right)}{\left(x + 1\right)^{2}} - \frac{6 \cos{\left(x \right)}}{\left(x + 1\right)^{2}} + \frac{12 \sin{\left(x \right)}}{\left(x + 1\right)^{3}} + \frac{12 \cos{\left(x \right)}}{\left(x + 1\right)^{4}}}{x + 1}\right) = -\infty$$
$$\lim_{x \to -1^+}\left(\frac{\cos{\left(x \right)} - \frac{2 \left(\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x + 1} - \frac{4 \sin{\left(x \right)}}{\left(x + 1\right)^{2}} - \frac{4 \cos{\left(x \right)}}{\left(x + 1\right)^{3}}\right)}{x + 1} - \frac{2 \sin{\left(x \right)}}{x + 1} + \frac{2 \left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1} + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}\right)}{\left(x + 1\right)^{2}} - \frac{6 \cos{\left(x \right)}}{\left(x + 1\right)^{2}} + \frac{12 \sin{\left(x \right)}}{\left(x + 1\right)^{3}} + \frac{12 \cos{\left(x \right)}}{\left(x + 1\right)^{4}}}{x + 1}\right) = \infty$$
- los límites no son iguales, signo
$$x_{1} = -1$$
- 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
$$\left[230.889810252879, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -318.859070062961\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\cos{\left(x \right)} - \frac{2 \left(\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x + 1} - \frac{4 \sin{\left(x \right)}}{\left(x + 1\right)^{2}} - \frac{4 \cos{\left(x \right)}}{\left(x + 1\right)^{3}}\right)}{x + 1} - \frac{2 \sin{\left(x \right)}}{x + 1} + \frac{2 \left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1} + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}\right)}{\left(x + 1\right)^{2}} - \frac{6 \cos{\left(x \right)}}{\left(x + 1\right)^{2}} + \frac{12 \sin{\left(x \right)}}{\left(x + 1\right)^{3}} + \frac{12 \cos{\left(x \right)}}{\left(x + 1\right)^{4}}}{x + 1} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -7.19519596149036$$
$$x_{2} = -89.4901840227415$$
$$x_{3} = 98.9201339417676$$
$$x_{4} = 70.6299849073373$$
$$x_{5} = 42.3191299769022$$
$$x_{6} = 13.867289840921$$
$$x_{7} = -13.8239463102427$$
$$x_{8} = 1.40731154138569$$
$$x_{9} = 92.6342606247044$$
$$x_{10} = -318.859070062961$$
$$x_{11} = -86.34692614697$$
$$x_{12} = 10.6504896158414$$
$$x_{13} = -39.1650521132014$$
$$x_{14} = 36.0202136918045$$
$$x_{15} = 45.4669842133418$$
$$x_{16} = 20.2316727993409$$
$$x_{17} = 86.3480001337381$$
$$x_{18} = -76.9163243230484$$
$$x_{19} = 83.2046975626699$$
$$x_{20} = -48.6106465468566$$
$$x_{21} = -80.0600127890459$$
$$x_{22} = 17.0567762419777$$
$$x_{23} = -29.7056715201635$$
$$x_{24} = -70.6283789578667$$
$$x_{25} = -26.54680103954$$
$$x_{26} = -36.0140131736861$$
$$x_{27} = -42.3146449333139$$
$$x_{28} = -67.484068177944$$
$$x_{29} = 67.4858274972604$$
$$x_{30} = -61.194593351618$$
$$x_{31} = -95.7763681868513$$
$$x_{32} = 76.9176781393433$$
$$x_{33} = -73.7724545812494$$
$$x_{34} = -20.2117653772165$$
$$x_{35} = -58.049334955595$$
$$x_{36} = 58.0517138854424$$
$$x_{37} = 39.1702908298917$$
$$x_{38} = -10.5745820582157$$
$$x_{39} = -23.3829976938101$$
$$x_{40} = 61.1967335889873$$
$$x_{41} = 54.9063078762122$$
$$x_{42} = 48.614041915914$$
$$x_{43} = 3.85793818596233$$
$$x_{44} = 23.3978106353706$$
$$x_{45} = 89.4911838253213$$
$$x_{46} = -45.4631008294892$$
$$x_{47} = 32.8685502646286$$
$$x_{48} = -64.3394871333198$$
$$x_{49} = -98.9193157877185$$
$$x_{50} = -83.203540811653$$
$$x_{51} = 64.3414229191161$$
$$x_{52} = 95.7772409591446$$
$$x_{53} = 29.7148073593336$$
$$x_{54} = -32.8610950559118$$
$$x_{55} = 80.0612622615078$$
$$x_{56} = -149.198659428593$$
$$x_{57} = 7.37124034627779$$
$$x_{58} = 26.5582622916455$$
$$x_{59} = -92.6333275731548$$
$$x_{60} = -17.0285445585894$$
$$x_{61} = -164.909209954026$$
$$x_{62} = -51.7574521916164$$
$$x_{63} = 51.7604462256561$$
$$x_{64} = -54.9036479029571$$
$$x_{65} = 230.889810252879$$
$$x_{66} = 73.7739263925337$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = -1$$
$$\lim_{x \to -1^-}\left(\frac{\cos{\left(x \right)} - \frac{2 \left(\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x + 1} - \frac{4 \sin{\left(x \right)}}{\left(x + 1\right)^{2}} - \frac{4 \cos{\left(x \right)}}{\left(x + 1\right)^{3}}\right)}{x + 1} - \frac{2 \sin{\left(x \right)}}{x + 1} + \frac{2 \left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1} + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}\right)}{\left(x + 1\right)^{2}} - \frac{6 \cos{\left(x \right)}}{\left(x + 1\right)^{2}} + \frac{12 \sin{\left(x \right)}}{\left(x + 1\right)^{3}} + \frac{12 \cos{\left(x \right)}}{\left(x + 1\right)^{4}}}{x + 1}\right) = -\infty$$
$$\lim_{x \to -1^+}\left(\frac{\cos{\left(x \right)} - \frac{2 \left(\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x + 1} - \frac{4 \sin{\left(x \right)}}{\left(x + 1\right)^{2}} - \frac{4 \cos{\left(x \right)}}{\left(x + 1\right)^{3}}\right)}{x + 1} - \frac{2 \sin{\left(x \right)}}{x + 1} + \frac{2 \left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1} + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}\right)}{\left(x + 1\right)^{2}} - \frac{6 \cos{\left(x \right)}}{\left(x + 1\right)^{2}} + \frac{12 \sin{\left(x \right)}}{\left(x + 1\right)^{3}} + \frac{12 \cos{\left(x \right)}}{\left(x + 1\right)^{4}}}{x + 1}\right) = \infty$$
- los límites no son iguales, signo
$$x_{1} = -1$$
- 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
$$\left[230.889810252879, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -318.859070062961\right]$$
Asíntotas verticales
Hay:
$$x_{1} = -1$$
$$x_{1} = -1$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1}\right) + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x + 1}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1}\right) + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x + 1}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1}\right) + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x + 1}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1}\right) + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x + 1}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (-cos(x) + (2*sin(x))/(x + 1) + (2*cos(x))/(x + 1)^2)/(x + 1), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1}\right) + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x \left(x + 1\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1}\right) + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x \left(x + 1\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1}\right) + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x \left(x + 1\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1}\right) + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x \left(x + 1\right)}\right) = 0$$
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:
$$\frac{\left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1}\right) + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x + 1} = \frac{- \cos{\left(x \right)} - \frac{2 \sin{\left(x \right)}}{1 - x} + \frac{2 \cos{\left(x \right)}}{\left(1 - x\right)^{2}}}{1 - x}$$
- No
$$\frac{\left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1}\right) + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x + 1} = - \frac{- \cos{\left(x \right)} - \frac{2 \sin{\left(x \right)}}{1 - x} + \frac{2 \cos{\left(x \right)}}{\left(1 - x\right)^{2}}}{1 - x}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1}\right) + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x + 1} = \frac{- \cos{\left(x \right)} - \frac{2 \sin{\left(x \right)}}{1 - x} + \frac{2 \cos{\left(x \right)}}{\left(1 - x\right)^{2}}}{1 - x}$$
- No
$$\frac{\left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1}\right) + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x + 1} = - \frac{- \cos{\left(x \right)} - \frac{2 \sin{\left(x \right)}}{1 - x} + \frac{2 \cos{\left(x \right)}}{\left(1 - x\right)^{2}}}{1 - x}$$
- No
es decir, función
no es
par ni impar