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- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x^2+x+1
-
x^2+1/x
-
(x-1)/(x+2)
-
x^2/(x^3+1)
-
Expresiones idénticas
- sin(x)/(3x-x^ dos)
- seno de (x) dividir por (3x menos x al cuadrado )
- seno de (x) dividir por (3x menos x en el grado dos)
- sin(x)/(3x-x2)
- sinx/3x-x2
- sin(x)/(3x-x²)
- sin(x)/(3x-x en el grado 2)
- sinx/3x-x^2
- sin(x) dividir por (3x-x^2)
-
Expresiones semejantes
- sin(x)/(3x+x^2)
- sinx/(3x-x^2)
-
Expresiones con funciones
- Seno sin
- sin(x)+x
- sin(x)^(-2)
- sin(x^2+x-2)
- sin(x)^(2)-cos(x)
- sin(x)^2+sin(2*x)+sin(2)^x
Gráfico de la función y = sin(x)/(3x-x^2)
Solución
Ha introducido
[src]
sin(x)
f(x) = --------
2
3*x - x
$$f{\left(x \right)} = \frac{\sin{\left(x \right)}}{- x^{2} + 3 x}$$
f = sin(x)/(-x^2 + 3*x)
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 0$$
$$x_{2} = 3$$
$$x_{1} = 0$$
$$x_{2} = 3$$
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:
$$\frac{\sin{\left(x \right)}}{- x^{2} + 3 x} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \pi$$
Solución numérica
$$x_{1} = 31.4159265358979$$
$$x_{2} = -47.1238898038469$$
$$x_{3} = 204.203522483337$$
$$x_{4} = -34.5575191894877$$
$$x_{5} = -69.1150383789755$$
$$x_{6} = -12.5663706143592$$
$$x_{7} = -65.9734457253857$$
$$x_{8} = 75.398223686155$$
$$x_{9} = -56.5486677646163$$
$$x_{10} = -153.9380400259$$
$$x_{11} = 59.6902604182061$$
$$x_{12} = -50.2654824574367$$
$$x_{13} = 72.2566310325652$$
$$x_{14} = 91.106186954104$$
$$x_{15} = -91.106186954104$$
$$x_{16} = -62.8318530717959$$
$$x_{17} = -6.28318530717959$$
$$x_{18} = 6.28318530717959$$
$$x_{19} = 62.8318530717959$$
$$x_{20} = -25.1327412287183$$
$$x_{21} = 94.2477796076938$$
$$x_{22} = -9.42477796076938$$
$$x_{23} = -37.6991118430775$$
$$x_{24} = 65.9734457253857$$
$$x_{25} = -100.530964914873$$
$$x_{26} = -43.9822971502571$$
$$x_{27} = 25.1327412287183$$
$$x_{28} = 21.9911485751286$$
$$x_{29} = 87.9645943005142$$
$$x_{30} = -40.8407044966673$$
$$x_{31} = -97.3893722612836$$
$$x_{32} = 43.9822971502571$$
$$x_{33} = -53.4070751110265$$
$$x_{34} = 97.3893722612836$$
$$x_{35} = 100.530964914873$$
$$x_{36} = -94.2477796076938$$
$$x_{37} = -31.4159265358979$$
$$x_{38} = 18.8495559215388$$
$$x_{39} = 78.5398163397448$$
$$x_{40} = -18.8495559215388$$
$$x_{41} = 53.4070751110265$$
$$x_{42} = 47.1238898038469$$
$$x_{43} = 12.5663706143592$$
$$x_{44} = 81.6814089933346$$
$$x_{45} = 34.5575191894877$$
$$x_{46} = -75.398223686155$$
$$x_{47} = -15.707963267949$$
$$x_{48} = 50.2654824574367$$
$$x_{49} = -81.6814089933346$$
$$x_{50} = -3.14159265358979$$
$$x_{51} = -59.6902604182061$$
$$x_{52} = -28.2743338823081$$
$$x_{53} = -87.9645943005142$$
$$x_{54} = 9.42477796076938$$
$$x_{55} = -21.9911485751286$$
$$x_{56} = 56.5486677646163$$
$$x_{57} = 15.707963267949$$
$$x_{58} = 84.8230016469244$$
$$x_{59} = -78.5398163397448$$
$$x_{60} = -119.380520836412$$
$$x_{61} = 37.6991118430775$$
$$x_{62} = -72.2566310325652$$
$$x_{63} = -84.8230016469244$$
$$x_{64} = 69.1150383789755$$
$$x_{65} = 28.2743338823081$$
$$x_{66} = 40.8407044966673$$
o sea hay que resolver la ecuación:
$$\frac{\sin{\left(x \right)}}{- x^{2} + 3 x} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \pi$$
Solución numérica
$$x_{1} = 31.4159265358979$$
$$x_{2} = -47.1238898038469$$
$$x_{3} = 204.203522483337$$
$$x_{4} = -34.5575191894877$$
$$x_{5} = -69.1150383789755$$
$$x_{6} = -12.5663706143592$$
$$x_{7} = -65.9734457253857$$
$$x_{8} = 75.398223686155$$
$$x_{9} = -56.5486677646163$$
$$x_{10} = -153.9380400259$$
$$x_{11} = 59.6902604182061$$
$$x_{12} = -50.2654824574367$$
$$x_{13} = 72.2566310325652$$
$$x_{14} = 91.106186954104$$
$$x_{15} = -91.106186954104$$
$$x_{16} = -62.8318530717959$$
$$x_{17} = -6.28318530717959$$
$$x_{18} = 6.28318530717959$$
$$x_{19} = 62.8318530717959$$
$$x_{20} = -25.1327412287183$$
$$x_{21} = 94.2477796076938$$
$$x_{22} = -9.42477796076938$$
$$x_{23} = -37.6991118430775$$
$$x_{24} = 65.9734457253857$$
$$x_{25} = -100.530964914873$$
$$x_{26} = -43.9822971502571$$
$$x_{27} = 25.1327412287183$$
$$x_{28} = 21.9911485751286$$
$$x_{29} = 87.9645943005142$$
$$x_{30} = -40.8407044966673$$
$$x_{31} = -97.3893722612836$$
$$x_{32} = 43.9822971502571$$
$$x_{33} = -53.4070751110265$$
$$x_{34} = 97.3893722612836$$
$$x_{35} = 100.530964914873$$
$$x_{36} = -94.2477796076938$$
$$x_{37} = -31.4159265358979$$
$$x_{38} = 18.8495559215388$$
$$x_{39} = 78.5398163397448$$
$$x_{40} = -18.8495559215388$$
$$x_{41} = 53.4070751110265$$
$$x_{42} = 47.1238898038469$$
$$x_{43} = 12.5663706143592$$
$$x_{44} = 81.6814089933346$$
$$x_{45} = 34.5575191894877$$
$$x_{46} = -75.398223686155$$
$$x_{47} = -15.707963267949$$
$$x_{48} = 50.2654824574367$$
$$x_{49} = -81.6814089933346$$
$$x_{50} = -3.14159265358979$$
$$x_{51} = -59.6902604182061$$
$$x_{52} = -28.2743338823081$$
$$x_{53} = -87.9645943005142$$
$$x_{54} = 9.42477796076938$$
$$x_{55} = -21.9911485751286$$
$$x_{56} = 56.5486677646163$$
$$x_{57} = 15.707963267949$$
$$x_{58} = 84.8230016469244$$
$$x_{59} = -78.5398163397448$$
$$x_{60} = -119.380520836412$$
$$x_{61} = 37.6991118430775$$
$$x_{62} = -72.2566310325652$$
$$x_{63} = -84.8230016469244$$
$$x_{64} = 69.1150383789755$$
$$x_{65} = 28.2743338823081$$
$$x_{66} = 40.8407044966673$$
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
$$\frac{\left(2 x - 3\right) \sin{\left(x \right)}}{\left(- x^{2} + 3 x\right)^{2}} + \frac{\cos{\left(x \right)}}{- x^{2} + 3 x} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 54.9404345834067$$
$$x_{2} = -64.3722812013129$$
$$x_{3} = -67.5152573767058$$
$$x_{4} = -89.5134134128682$$
$$x_{5} = -48.6547950578543$$
$$x_{6} = -92.6557396818714$$
$$x_{7} = 61.2275627648845$$
$$x_{8} = -54.9424269071032$$
$$x_{9} = -86.3710346431039$$
$$x_{10} = 76.9425054896692$$
$$x_{11} = -70.6581129158171$$
$$x_{12} = 70.6569094215149$$
$$x_{13} = -61.2291662236455$$
$$x_{14} = -73.8008629455869$$
$$x_{15} = 7.51314825625302$$
$$x_{16} = -95.7980185411251$$
$$x_{17} = 39.2168473186151$$
$$x_{18} = -51.7987423415829$$
$$x_{19} = 356.565133372137$$
$$x_{20} = 58.0841083614119$$
$$x_{21} = 26.6238163561621$$
$$x_{22} = 20.3137724727901$$
$$x_{23} = -17.1713626883542$$
$$x_{24} = -26.6323627973555$$
$$x_{25} = -10.8324280431539$$
$$x_{26} = -39.2207660419998$$
$$x_{27} = 42.3625293703483$$
$$x_{28} = 80.0851587913042$$
$$x_{29} = -29.7811340621316$$
$$x_{30} = 89.5126639359415$$
$$x_{31} = -45.5105321791656$$
$$x_{32} = -98.9402544438318$$
$$x_{33} = 92.6550402198056$$
$$x_{34} = 67.5139390487724$$
$$x_{35} = 95.7973642476584$$
$$x_{36} = 36.0704182419511$$
$$x_{37} = -83.2285975252814$$
$$x_{38} = 48.6522528603963$$
$$x_{39} = 86.3702295857644$$
$$x_{40} = 32.92301620939$$
$$x_{41} = 23.4707421747761$$
$$x_{42} = -36.075054175664$$
$$x_{43} = 29.7743136017553$$
$$x_{44} = -416.256239094639$$
$$x_{45} = -7.63260318871025$$
$$x_{46} = -76.9435201641952$$
$$x_{47} = 64.3708307683261$$
$$x_{48} = 13.9759189719399$$
$$x_{49} = 45.5076253057118$$
$$x_{50} = -80.0860953099024$$
$$x_{51} = 152.353985311833$$
$$x_{52} = -42.3658855955268$$
$$x_{53} = 271.740363623689$$
$$x_{54} = 10.7777308434471$$
$$x_{55} = 906.347270240411$$
$$x_{56} = -4.36236151301764$$
$$x_{57} = -14.0077089057568$$
$$x_{58} = -32.9285867081765$$
$$x_{59} = 98.9396410793967$$
$$x_{60} = -58.0858904344489$$
$$x_{61} = -20.3285531693707$$
$$x_{62} = 73.7997598999583$$
$$x_{63} = -23.4817690823466$$
$$x_{64} = 17.1504914179865$$
$$x_{65} = 51.7965001818486$$
$$x_{66} = 83.2277304640382$$
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:
$$x_{1} = -67.5152573767058$$
$$x_{2} = -48.6547950578543$$
$$x_{3} = -92.6557396818714$$
$$x_{4} = -54.9424269071032$$
$$x_{5} = -86.3710346431039$$
$$x_{6} = 76.9425054896692$$
$$x_{7} = 70.6569094215149$$
$$x_{8} = -61.2291662236455$$
$$x_{9} = -73.8008629455869$$
$$x_{10} = 7.51314825625302$$
$$x_{11} = 39.2168473186151$$
$$x_{12} = 58.0841083614119$$
$$x_{13} = 26.6238163561621$$
$$x_{14} = 20.3137724727901$$
$$x_{15} = -17.1713626883542$$
$$x_{16} = -10.8324280431539$$
$$x_{17} = -29.7811340621316$$
$$x_{18} = 89.5126639359415$$
$$x_{19} = -98.9402544438318$$
$$x_{20} = 95.7973642476584$$
$$x_{21} = 32.92301620939$$
$$x_{22} = -36.075054175664$$
$$x_{23} = 64.3708307683261$$
$$x_{24} = 13.9759189719399$$
$$x_{25} = 45.5076253057118$$
$$x_{26} = -80.0860953099024$$
$$x_{27} = 152.353985311833$$
$$x_{28} = -42.3658855955268$$
$$x_{29} = 271.740363623689$$
$$x_{30} = 906.347270240411$$
$$x_{31} = -4.36236151301764$$
$$x_{32} = -23.4817690823466$$
$$x_{33} = 51.7965001818486$$
$$x_{34} = 83.2277304640382$$
Puntos máximos de la función:
$$x_{34} = 54.9404345834067$$
$$x_{34} = -64.3722812013129$$
$$x_{34} = -89.5134134128682$$
$$x_{34} = 61.2275627648845$$
$$x_{34} = -70.6581129158171$$
$$x_{34} = -95.7980185411251$$
$$x_{34} = -51.7987423415829$$
$$x_{34} = 356.565133372137$$
$$x_{34} = -26.6323627973555$$
$$x_{34} = -39.2207660419998$$
$$x_{34} = 42.3625293703483$$
$$x_{34} = 80.0851587913042$$
$$x_{34} = -45.5105321791656$$
$$x_{34} = 92.6550402198056$$
$$x_{34} = 67.5139390487724$$
$$x_{34} = 36.0704182419511$$
$$x_{34} = -83.2285975252814$$
$$x_{34} = 48.6522528603963$$
$$x_{34} = 86.3702295857644$$
$$x_{34} = 23.4707421747761$$
$$x_{34} = 29.7743136017553$$
$$x_{34} = -416.256239094639$$
$$x_{34} = -7.63260318871025$$
$$x_{34} = -76.9435201641952$$
$$x_{34} = 10.7777308434471$$
$$x_{34} = -14.0077089057568$$
$$x_{34} = -32.9285867081765$$
$$x_{34} = 98.9396410793967$$
$$x_{34} = -58.0858904344489$$
$$x_{34} = -20.3285531693707$$
$$x_{34} = 73.7997598999583$$
$$x_{34} = 17.1504914179865$$
Decrece en los intervalos
$$\left[906.347270240411, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.9402544438318\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{\left(2 x - 3\right) \sin{\left(x \right)}}{\left(- x^{2} + 3 x\right)^{2}} + \frac{\cos{\left(x \right)}}{- x^{2} + 3 x} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 54.9404345834067$$
$$x_{2} = -64.3722812013129$$
$$x_{3} = -67.5152573767058$$
$$x_{4} = -89.5134134128682$$
$$x_{5} = -48.6547950578543$$
$$x_{6} = -92.6557396818714$$
$$x_{7} = 61.2275627648845$$
$$x_{8} = -54.9424269071032$$
$$x_{9} = -86.3710346431039$$
$$x_{10} = 76.9425054896692$$
$$x_{11} = -70.6581129158171$$
$$x_{12} = 70.6569094215149$$
$$x_{13} = -61.2291662236455$$
$$x_{14} = -73.8008629455869$$
$$x_{15} = 7.51314825625302$$
$$x_{16} = -95.7980185411251$$
$$x_{17} = 39.2168473186151$$
$$x_{18} = -51.7987423415829$$
$$x_{19} = 356.565133372137$$
$$x_{20} = 58.0841083614119$$
$$x_{21} = 26.6238163561621$$
$$x_{22} = 20.3137724727901$$
$$x_{23} = -17.1713626883542$$
$$x_{24} = -26.6323627973555$$
$$x_{25} = -10.8324280431539$$
$$x_{26} = -39.2207660419998$$
$$x_{27} = 42.3625293703483$$
$$x_{28} = 80.0851587913042$$
$$x_{29} = -29.7811340621316$$
$$x_{30} = 89.5126639359415$$
$$x_{31} = -45.5105321791656$$
$$x_{32} = -98.9402544438318$$
$$x_{33} = 92.6550402198056$$
$$x_{34} = 67.5139390487724$$
$$x_{35} = 95.7973642476584$$
$$x_{36} = 36.0704182419511$$
$$x_{37} = -83.2285975252814$$
$$x_{38} = 48.6522528603963$$
$$x_{39} = 86.3702295857644$$
$$x_{40} = 32.92301620939$$
$$x_{41} = 23.4707421747761$$
$$x_{42} = -36.075054175664$$
$$x_{43} = 29.7743136017553$$
$$x_{44} = -416.256239094639$$
$$x_{45} = -7.63260318871025$$
$$x_{46} = -76.9435201641952$$
$$x_{47} = 64.3708307683261$$
$$x_{48} = 13.9759189719399$$
$$x_{49} = 45.5076253057118$$
$$x_{50} = -80.0860953099024$$
$$x_{51} = 152.353985311833$$
$$x_{52} = -42.3658855955268$$
$$x_{53} = 271.740363623689$$
$$x_{54} = 10.7777308434471$$
$$x_{55} = 906.347270240411$$
$$x_{56} = -4.36236151301764$$
$$x_{57} = -14.0077089057568$$
$$x_{58} = -32.9285867081765$$
$$x_{59} = 98.9396410793967$$
$$x_{60} = -58.0858904344489$$
$$x_{61} = -20.3285531693707$$
$$x_{62} = 73.7997598999583$$
$$x_{63} = -23.4817690823466$$
$$x_{64} = 17.1504914179865$$
$$x_{65} = 51.7965001818486$$
$$x_{66} = 83.2277304640382$$
Signos de extremos en los puntos:
(54.940434583406685, 0.000350185309880627)
(-64.37228120131286, 0.000230472736094432)
(-67.51525737670583, -0.000209958046626072)
(-89.51341341286818, 0.000120726409994599)
(-48.6547950578543, -0.000397574088591686)
(-92.65573968187138, -0.000112802476494125)
(61.22756276488447, 0.000280337221201038)
(-54.94242690710323, -0.00031392262474364)
(-86.37103464310387, -0.000129515743627032)
(76.94250548966916, -0.000175706099044944)
(-70.65811291581707, 0.000192065994254098)
(70.65690942151492, -0.000209098781089558)
(-61.229166223645514, -0.000254149051215051)
(-73.80086294558691, -0.000176367763714622)
(7.513148256253024, -0.027795144301561)
(-95.79801854112513, 0.000105633936613657)
(39.21684731861511, -0.000703080455227028)
(-51.798742341582916, 0.000352049850613586)
(356.56513337213727, 7.93203836154108e-6)
(58.08410836141186, -0.000312352403734362)
(26.62381635616214, -0.00158488624449811)
(20.313772472790074, -0.0028271340204374)
(-17.171362688354222, -0.00287045388295451)
(-26.632362797355544, 0.00126393009526021)
(-10.832428043153927, -0.00658521868971709)
(-39.220766041999816, 0.000603160939108171)
(42.36252937034829, 0.000598982518937337)
(80.08515879130417, 0.000161933417980931)
(-29.781134062131613, -0.001022221141544)
(89.51266393594155, -0.00012909923415002)
(-45.51053217916558, 0.000452541668510378)
(-98.9402544438318, -9.91277259663094e-5)
(92.65504021980564, 0.000120351546988818)
(67.51393904877244, 0.000229484608283392)
(95.79736424765842, -0.000112463886908334)
(36.0704182419511, 0.000836913993338804)
(-83.22859752528143, 0.000139301267328271)
(48.65225286039632, 0.000449825137914034)
(86.37022958576445, 0.000138836705242165)
(32.923016209390006, -0.00101300856958335)
(23.470742174776106, 0.00207267334364847)
(-36.07505417566402, -0.000708397681106311)
(29.77431360175529, 0.00125126700509107)
(-416.25623909463866, 5.73000146077569e-6)
(-7.632603188710249, 0.0120214703302475)
(-76.94352016419518, 0.000162518747743003)
(64.3708307683261, -0.000253004940051749)
(13.975918971939869, -0.00643440021730597)
(45.507625305711755, -0.000516416259359543)
(-80.0860953099024, -0.000150239451859636)
(152.3539853118335, -4.39431507568467e-5)
(-42.365885595526784, -0.000519759375521324)
(271.7403636236889, -1.36930770942029e-5)
(10.777730843447111, 0.0116474892142262)
(906.3472702404113, -1.22137653336016e-6)
(-4.36236151301764, -0.0292479070643904)
(-14.007708905756774, 0.00416234057991184)
(-32.92858670817654, 0.000843825108563041)
(98.93964107939665, 0.000105327086813135)
(-58.08589043444888, 0.000281671967604093)
(-20.328553169370714, 0.0020997772888554)
(73.7997598999583, 0.000191314118625257)
(-23.48176908234661, -0.00160296794280833)
(17.150491417986544, 0.00408666845428565)
(51.79650018184863, -0.000395336778324716)
(83.22773046403825, -0.000149719150056251)
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:
$$x_{1} = -67.5152573767058$$
$$x_{2} = -48.6547950578543$$
$$x_{3} = -92.6557396818714$$
$$x_{4} = -54.9424269071032$$
$$x_{5} = -86.3710346431039$$
$$x_{6} = 76.9425054896692$$
$$x_{7} = 70.6569094215149$$
$$x_{8} = -61.2291662236455$$
$$x_{9} = -73.8008629455869$$
$$x_{10} = 7.51314825625302$$
$$x_{11} = 39.2168473186151$$
$$x_{12} = 58.0841083614119$$
$$x_{13} = 26.6238163561621$$
$$x_{14} = 20.3137724727901$$
$$x_{15} = -17.1713626883542$$
$$x_{16} = -10.8324280431539$$
$$x_{17} = -29.7811340621316$$
$$x_{18} = 89.5126639359415$$
$$x_{19} = -98.9402544438318$$
$$x_{20} = 95.7973642476584$$
$$x_{21} = 32.92301620939$$
$$x_{22} = -36.075054175664$$
$$x_{23} = 64.3708307683261$$
$$x_{24} = 13.9759189719399$$
$$x_{25} = 45.5076253057118$$
$$x_{26} = -80.0860953099024$$
$$x_{27} = 152.353985311833$$
$$x_{28} = -42.3658855955268$$
$$x_{29} = 271.740363623689$$
$$x_{30} = 906.347270240411$$
$$x_{31} = -4.36236151301764$$
$$x_{32} = -23.4817690823466$$
$$x_{33} = 51.7965001818486$$
$$x_{34} = 83.2277304640382$$
Puntos máximos de la función:
$$x_{34} = 54.9404345834067$$
$$x_{34} = -64.3722812013129$$
$$x_{34} = -89.5134134128682$$
$$x_{34} = 61.2275627648845$$
$$x_{34} = -70.6581129158171$$
$$x_{34} = -95.7980185411251$$
$$x_{34} = -51.7987423415829$$
$$x_{34} = 356.565133372137$$
$$x_{34} = -26.6323627973555$$
$$x_{34} = -39.2207660419998$$
$$x_{34} = 42.3625293703483$$
$$x_{34} = 80.0851587913042$$
$$x_{34} = -45.5105321791656$$
$$x_{34} = 92.6550402198056$$
$$x_{34} = 67.5139390487724$$
$$x_{34} = 36.0704182419511$$
$$x_{34} = -83.2285975252814$$
$$x_{34} = 48.6522528603963$$
$$x_{34} = 86.3702295857644$$
$$x_{34} = 23.4707421747761$$
$$x_{34} = 29.7743136017553$$
$$x_{34} = -416.256239094639$$
$$x_{34} = -7.63260318871025$$
$$x_{34} = -76.9435201641952$$
$$x_{34} = 10.7777308434471$$
$$x_{34} = -14.0077089057568$$
$$x_{34} = -32.9285867081765$$
$$x_{34} = 98.9396410793967$$
$$x_{34} = -58.0858904344489$$
$$x_{34} = -20.3285531693707$$
$$x_{34} = 73.7997598999583$$
$$x_{34} = 17.1504914179865$$
Decrece en los intervalos
$$\left[906.347270240411, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.9402544438318\right]$$
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
$$\frac{\sin{\left(x \right)} + \frac{2 \left(1 - \frac{\left(2 x - 3\right)^{2}}{x \left(x - 3\right)}\right) \sin{\left(x \right)}}{x \left(x - 3\right)} + \frac{2 \left(2 x - 3\right) \cos{\left(x \right)}}{x \left(x - 3\right)}}{x \left(x - 3\right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -34.445989594438$$
$$x_{2} = 24.961341658606$$
$$x_{3} = 12.1822044513472$$
$$x_{4} = -43.8940550520313$$
$$x_{5} = 40.7385706479277$$
$$x_{6} = -9.0349615015518$$
$$x_{7} = 87.9182897209429$$
$$x_{8} = -69.058314457774$$
$$x_{9} = -56.479618113704$$
$$x_{10} = -21.8186892386804$$
$$x_{11} = 47.0359234010403$$
$$x_{12} = -40.7458657827672$$
$$x_{13} = 56.4758382714314$$
$$x_{14} = -75.3461459121604$$
$$x_{15} = 1.92475853310599$$
$$x_{16} = 4.67660988487663$$
$$x_{17} = 94.2046171642218$$
$$x_{18} = -5.695018890829$$
$$x_{19} = 15.4161599684918$$
$$x_{20} = 100.490544985862$$
$$x_{21} = -0.899826581246947$$
$$x_{22} = -62.7695712544917$$
$$x_{23} = -37.5966057195081$$
$$x_{24} = 62.7665137961935$$
$$x_{25} = 65.9113011452116$$
$$x_{26} = -59.6247693886956$$
$$x_{27} = 28.1234707204472$$
$$x_{28} = 84.7749477855864$$
$$x_{29} = -24.9810600130324$$
$$x_{30} = -78.4897871542936$$
$$x_{31} = -18.6496085931503$$
$$x_{32} = -65.9140728383026$$
$$x_{33} = -12.2714060368015$$
$$x_{34} = -81.6332731701775$$
$$x_{35} = 75.3440264759478$$
$$x_{36} = -100.491735052398$$
$$x_{37} = -94.2059716213425$$
$$x_{38} = 18.6134803695903$$
$$x_{39} = 72.2000209379011$$
$$x_{40} = 34.4357435023143$$
$$x_{41} = 69.0557902026946$$
$$x_{42} = 8.85039113222326$$
$$x_{43} = -84.7766209743315$$
$$x_{44} = 21.7926305309322$$
$$x_{45} = 53.3298156062769$$
$$x_{46} = -97.3488943758975$$
$$x_{47} = -50.1880105304202$$
$$x_{48} = -354.988749047656$$
$$x_{49} = 81.6314683453896$$
$$x_{50} = -47.0413838099558$$
$$x_{51} = -106.77719881113$$
$$x_{52} = 50.183217429047$$
$$x_{53} = 43.8877769824483$$
$$x_{54} = -28.138927235069$$
$$x_{55} = 446.097159768419$$
$$x_{56} = 59.6213794745684$$
$$x_{57} = 31.2811708976638$$
$$x_{58} = -91.0629584096981$$
$$x_{59} = -31.2936202305249$$
$$x_{60} = -53.3340569417798$$
$$x_{61} = 37.5880231720398$$
$$x_{62} = -15.4698492149447$$
$$x_{63} = 97.3476261027773$$
$$x_{64} = 91.0615086670385$$
$$x_{65} = -72.2023295133061$$
$$x_{66} = -87.9198451802223$$
$$x_{67} = 78.4878345042334$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$x_{2} = 3$$
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)} + \frac{2 \left(1 - \frac{\left(2 x - 3\right)^{2}}{x \left(x - 3\right)}\right) \sin{\left(x \right)}}{x \left(x - 3\right)} + \frac{2 \left(2 x - 3\right) \cos{\left(x \right)}}{x \left(x - 3\right)}}{x \left(x - 3\right)}\right) = - \frac{1}{27}$$
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + \frac{2 \left(1 - \frac{\left(2 x - 3\right)^{2}}{x \left(x - 3\right)}\right) \sin{\left(x \right)}}{x \left(x - 3\right)} + \frac{2 \left(2 x - 3\right) \cos{\left(x \right)}}{x \left(x - 3\right)}}{x \left(x - 3\right)}\right) = - \frac{1}{27}$$
- los límites son iguales, es decir omitimos el punto correspondiente
$$\lim_{x \to 3^-}\left(\frac{\sin{\left(x \right)} + \frac{2 \left(1 - \frac{\left(2 x - 3\right)^{2}}{x \left(x - 3\right)}\right) \sin{\left(x \right)}}{x \left(x - 3\right)} + \frac{2 \left(2 x - 3\right) \cos{\left(x \right)}}{x \left(x - 3\right)}}{x \left(x - 3\right)}\right) = \infty$$
$$\lim_{x \to 3^+}\left(\frac{\sin{\left(x \right)} + \frac{2 \left(1 - \frac{\left(2 x - 3\right)^{2}}{x \left(x - 3\right)}\right) \sin{\left(x \right)}}{x \left(x - 3\right)} + \frac{2 \left(2 x - 3\right) \cos{\left(x \right)}}{x \left(x - 3\right)}}{x \left(x - 3\right)}\right) = -\infty$$
- los límites no son iguales, signo
$$x_{2} = 3$$
- 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
$$\left[446.097159768419, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -106.77719881113\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\sin{\left(x \right)} + \frac{2 \left(1 - \frac{\left(2 x - 3\right)^{2}}{x \left(x - 3\right)}\right) \sin{\left(x \right)}}{x \left(x - 3\right)} + \frac{2 \left(2 x - 3\right) \cos{\left(x \right)}}{x \left(x - 3\right)}}{x \left(x - 3\right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -34.445989594438$$
$$x_{2} = 24.961341658606$$
$$x_{3} = 12.1822044513472$$
$$x_{4} = -43.8940550520313$$
$$x_{5} = 40.7385706479277$$
$$x_{6} = -9.0349615015518$$
$$x_{7} = 87.9182897209429$$
$$x_{8} = -69.058314457774$$
$$x_{9} = -56.479618113704$$
$$x_{10} = -21.8186892386804$$
$$x_{11} = 47.0359234010403$$
$$x_{12} = -40.7458657827672$$
$$x_{13} = 56.4758382714314$$
$$x_{14} = -75.3461459121604$$
$$x_{15} = 1.92475853310599$$
$$x_{16} = 4.67660988487663$$
$$x_{17} = 94.2046171642218$$
$$x_{18} = -5.695018890829$$
$$x_{19} = 15.4161599684918$$
$$x_{20} = 100.490544985862$$
$$x_{21} = -0.899826581246947$$
$$x_{22} = -62.7695712544917$$
$$x_{23} = -37.5966057195081$$
$$x_{24} = 62.7665137961935$$
$$x_{25} = 65.9113011452116$$
$$x_{26} = -59.6247693886956$$
$$x_{27} = 28.1234707204472$$
$$x_{28} = 84.7749477855864$$
$$x_{29} = -24.9810600130324$$
$$x_{30} = -78.4897871542936$$
$$x_{31} = -18.6496085931503$$
$$x_{32} = -65.9140728383026$$
$$x_{33} = -12.2714060368015$$
$$x_{34} = -81.6332731701775$$
$$x_{35} = 75.3440264759478$$
$$x_{36} = -100.491735052398$$
$$x_{37} = -94.2059716213425$$
$$x_{38} = 18.6134803695903$$
$$x_{39} = 72.2000209379011$$
$$x_{40} = 34.4357435023143$$
$$x_{41} = 69.0557902026946$$
$$x_{42} = 8.85039113222326$$
$$x_{43} = -84.7766209743315$$
$$x_{44} = 21.7926305309322$$
$$x_{45} = 53.3298156062769$$
$$x_{46} = -97.3488943758975$$
$$x_{47} = -50.1880105304202$$
$$x_{48} = -354.988749047656$$
$$x_{49} = 81.6314683453896$$
$$x_{50} = -47.0413838099558$$
$$x_{51} = -106.77719881113$$
$$x_{52} = 50.183217429047$$
$$x_{53} = 43.8877769824483$$
$$x_{54} = -28.138927235069$$
$$x_{55} = 446.097159768419$$
$$x_{56} = 59.6213794745684$$
$$x_{57} = 31.2811708976638$$
$$x_{58} = -91.0629584096981$$
$$x_{59} = -31.2936202305249$$
$$x_{60} = -53.3340569417798$$
$$x_{61} = 37.5880231720398$$
$$x_{62} = -15.4698492149447$$
$$x_{63} = 97.3476261027773$$
$$x_{64} = 91.0615086670385$$
$$x_{65} = -72.2023295133061$$
$$x_{66} = -87.9198451802223$$
$$x_{67} = 78.4878345042334$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$x_{2} = 3$$
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)} + \frac{2 \left(1 - \frac{\left(2 x - 3\right)^{2}}{x \left(x - 3\right)}\right) \sin{\left(x \right)}}{x \left(x - 3\right)} + \frac{2 \left(2 x - 3\right) \cos{\left(x \right)}}{x \left(x - 3\right)}}{x \left(x - 3\right)}\right) = - \frac{1}{27}$$
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + \frac{2 \left(1 - \frac{\left(2 x - 3\right)^{2}}{x \left(x - 3\right)}\right) \sin{\left(x \right)}}{x \left(x - 3\right)} + \frac{2 \left(2 x - 3\right) \cos{\left(x \right)}}{x \left(x - 3\right)}}{x \left(x - 3\right)}\right) = - \frac{1}{27}$$
- los límites son iguales, es decir omitimos el punto correspondiente
$$\lim_{x \to 3^-}\left(\frac{\sin{\left(x \right)} + \frac{2 \left(1 - \frac{\left(2 x - 3\right)^{2}}{x \left(x - 3\right)}\right) \sin{\left(x \right)}}{x \left(x - 3\right)} + \frac{2 \left(2 x - 3\right) \cos{\left(x \right)}}{x \left(x - 3\right)}}{x \left(x - 3\right)}\right) = \infty$$
$$\lim_{x \to 3^+}\left(\frac{\sin{\left(x \right)} + \frac{2 \left(1 - \frac{\left(2 x - 3\right)^{2}}{x \left(x - 3\right)}\right) \sin{\left(x \right)}}{x \left(x - 3\right)} + \frac{2 \left(2 x - 3\right) \cos{\left(x \right)}}{x \left(x - 3\right)}}{x \left(x - 3\right)}\right) = -\infty$$
- los límites no son iguales, signo
$$x_{2} = 3$$
- 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
$$\left[446.097159768419, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -106.77719881113\right]$$
Asíntotas verticales
Hay:
$$x_{1} = 0$$
$$x_{2} = 3$$
$$x_{1} = 0$$
$$x_{2} = 3$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{- x^{2} + 3 x}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{- x^{2} + 3 x}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{- x^{2} + 3 x}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{- x^{2} + 3 x}\right) = 0$$
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 sin(x)/(3*x - x^2), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left(- x^{2} + 3 x\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left(- x^{2} + 3 x\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left(- x^{2} + 3 x\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left(- x^{2} + 3 x\right)}\right) = 0$$
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:
$$\frac{\sin{\left(x \right)}}{- x^{2} + 3 x} = - \frac{\sin{\left(x \right)}}{- x^{2} - 3 x}$$
- No
$$\frac{\sin{\left(x \right)}}{- x^{2} + 3 x} = \frac{\sin{\left(x \right)}}{- x^{2} - 3 x}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\sin{\left(x \right)}}{- x^{2} + 3 x} = - \frac{\sin{\left(x \right)}}{- x^{2} - 3 x}$$
- No
$$\frac{\sin{\left(x \right)}}{- x^{2} + 3 x} = \frac{\sin{\left(x \right)}}{- x^{2} - 3 x}$$
- No
es decir, función
no es
par ni impar