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- Gráfico de la función implícita
- Derivadas paso a paso
- Integrales paso a paso
- Gráfico de la función paramétrica
- Gráfico en coordenadas polares
- Superficie especificada paramétricamente
- Superficie dada por la ecuación
- Curva en el espacio especificada paramétricamente
- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x^4-2x^2+10
-
x^3+3*x^2-9*x
-
x^3-6*x+cos(x)+sin(x)
-
-x^3+6x^2
-
Expresiones idénticas
- sin(x+ dos)/(x^ dos - seis *x- dieciséis)
- seno de (x más 2) dividir por (x al cuadrado menos 6 multiplicar por x menos 16)
- seno de (x más dos) dividir por (x en el grado dos menos seis multiplicar por x menos dieciséis)
- sin(x+2)/(x2-6*x-16)
- sinx+2/x2-6*x-16
- sin(x+2)/(x²-6*x-16)
- sin(x+2)/(x en el grado 2-6*x-16)
- sin(x+2)/(x^2-6x-16)
- sin(x+2)/(x2-6x-16)
- sinx+2/x2-6x-16
- sinx+2/x^2-6x-16
- sin(x+2) dividir por (x^2-6*x-16)
-
Expresiones semejantes
- sin(x-2)/(x^2-6*x-16)
- sin(x+2)/(x^2-6*x+16)
- sin(x+2)/(x^2+6*x-16)
-
Expresiones con funciones
- Seno sin
- sin(-x-(pi/3))+1
- sin(x-pi/4)*(-2)
- sin(pi*x)/asin(x)
- sin(x)*e^((1/2)*cot(x)^(2))
- sin(697*t)+sin(1209*t)
Gráfico de la función y = sin(x+2)/(x^2-6*x-16)
Solución
Ha introducido
[src]
sin(x + 2)
f(x) = -------------
2
x - 6*x - 16
$$f{\left(x \right)} = \frac{\sin{\left(x + 2 \right)}}{\left(x^{2} - 6 x\right) - 16}$$
f = sin(x + 2)/(x^2 - 6*x - 16)
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = -2$$
$$x_{2} = 8$$
$$x_{1} = -2$$
$$x_{2} = 8$$
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:
$$\frac{\sin{\left(x + 2 \right)}}{\left(x^{2} - 6 x\right) - 16} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = -2 + \pi$$
Solución numérica
$$x_{1} = 67.1150383789755$$
$$x_{2} = 57.6902604182061$$
$$x_{3} = 63.9734457253857$$
$$x_{4} = 16.8495559215388$$
$$x_{5} = -99.3893722612836$$
$$x_{6} = -52.2654824574367$$
$$x_{7} = 29.4159265358979$$
$$x_{8} = 82.8230016469244$$
$$x_{9} = -115.097335529233$$
$$x_{10} = -83.6814089933346$$
$$x_{11} = 48.2654824574367$$
$$x_{12} = -36.5575191894877$$
$$x_{13} = -61.6902604182061$$
$$x_{14} = -27.1327412287183$$
$$x_{15} = 38.8407044966673$$
$$x_{16} = -33.4159265358979$$
$$x_{17} = 10.5663706143592$$
$$x_{18} = -8.28318530717959$$
$$x_{19} = 23.1327412287183$$
$$x_{20} = 4.28318530717959$$
$$x_{21} = -86.8230016469244$$
$$x_{22} = 70.2566310325652$$
$$x_{23} = -23.9911485751286$$
$$x_{24} = -11.4247779607694$$
$$x_{25} = -89.9645943005142$$
$$x_{26} = 45.1238898038469$$
$$x_{27} = -96.2477796076938$$
$$x_{28} = -14.5663706143592$$
$$x_{29} = -93.106186954104$$
$$x_{30} = 19.9911485751286$$
$$x_{31} = 95.3893722612836$$
$$x_{32} = 26.2743338823081$$
$$x_{33} = 117.380520836412$$
$$x_{34} = -71.1150383789755$$
$$x_{35} = -39.6991118430775$$
$$x_{36} = -5.14159265358979$$
$$x_{37} = 60.8318530717959$$
$$x_{38} = 89.106186954104$$
$$x_{39} = 79.6814089933346$$
$$x_{40} = 54.5486677646163$$
$$x_{41} = 13.707963267949$$
$$x_{42} = -58.5486677646163$$
$$x_{43} = 51.4070751110265$$
$$x_{44} = -77.398223686155$$
$$x_{45} = -74.2566310325652$$
$$x_{46} = -133.946891450771$$
$$x_{47} = 98.5309649148734$$
$$x_{48} = -49.1238898038469$$
$$x_{49} = 41.9822971502571$$
$$x_{50} = -80.5398163397448$$
$$x_{51} = 92.2477796076938$$
$$x_{52} = -30.2743338823081$$
$$x_{53} = -64.8318530717959$$
$$x_{54} = 73.398223686155$$
$$x_{55} = -17.707963267949$$
$$x_{56} = 1.14159265358979$$
$$x_{57} = -67.9734457253857$$
$$x_{58} = -42.8407044966673$$
$$x_{59} = -55.4070751110265$$
$$x_{60} = 32.5575191894877$$
$$x_{61} = -45.9822971502571$$
$$x_{62} = 85.9645943005142$$
$$x_{63} = -20.8495559215388$$
$$x_{64} = 35.6991118430775$$
$$x_{65} = 76.5398163397448$$
o sea hay que resolver la ecuación:
$$\frac{\sin{\left(x + 2 \right)}}{\left(x^{2} - 6 x\right) - 16} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = -2 + \pi$$
Solución numérica
$$x_{1} = 67.1150383789755$$
$$x_{2} = 57.6902604182061$$
$$x_{3} = 63.9734457253857$$
$$x_{4} = 16.8495559215388$$
$$x_{5} = -99.3893722612836$$
$$x_{6} = -52.2654824574367$$
$$x_{7} = 29.4159265358979$$
$$x_{8} = 82.8230016469244$$
$$x_{9} = -115.097335529233$$
$$x_{10} = -83.6814089933346$$
$$x_{11} = 48.2654824574367$$
$$x_{12} = -36.5575191894877$$
$$x_{13} = -61.6902604182061$$
$$x_{14} = -27.1327412287183$$
$$x_{15} = 38.8407044966673$$
$$x_{16} = -33.4159265358979$$
$$x_{17} = 10.5663706143592$$
$$x_{18} = -8.28318530717959$$
$$x_{19} = 23.1327412287183$$
$$x_{20} = 4.28318530717959$$
$$x_{21} = -86.8230016469244$$
$$x_{22} = 70.2566310325652$$
$$x_{23} = -23.9911485751286$$
$$x_{24} = -11.4247779607694$$
$$x_{25} = -89.9645943005142$$
$$x_{26} = 45.1238898038469$$
$$x_{27} = -96.2477796076938$$
$$x_{28} = -14.5663706143592$$
$$x_{29} = -93.106186954104$$
$$x_{30} = 19.9911485751286$$
$$x_{31} = 95.3893722612836$$
$$x_{32} = 26.2743338823081$$
$$x_{33} = 117.380520836412$$
$$x_{34} = -71.1150383789755$$
$$x_{35} = -39.6991118430775$$
$$x_{36} = -5.14159265358979$$
$$x_{37} = 60.8318530717959$$
$$x_{38} = 89.106186954104$$
$$x_{39} = 79.6814089933346$$
$$x_{40} = 54.5486677646163$$
$$x_{41} = 13.707963267949$$
$$x_{42} = -58.5486677646163$$
$$x_{43} = 51.4070751110265$$
$$x_{44} = -77.398223686155$$
$$x_{45} = -74.2566310325652$$
$$x_{46} = -133.946891450771$$
$$x_{47} = 98.5309649148734$$
$$x_{48} = -49.1238898038469$$
$$x_{49} = 41.9822971502571$$
$$x_{50} = -80.5398163397448$$
$$x_{51} = 92.2477796076938$$
$$x_{52} = -30.2743338823081$$
$$x_{53} = -64.8318530717959$$
$$x_{54} = 73.398223686155$$
$$x_{55} = -17.707963267949$$
$$x_{56} = 1.14159265358979$$
$$x_{57} = -67.9734457253857$$
$$x_{58} = -42.8407044966673$$
$$x_{59} = -55.4070751110265$$
$$x_{60} = 32.5575191894877$$
$$x_{61} = -45.9822971502571$$
$$x_{62} = 85.9645943005142$$
$$x_{63} = -20.8495559215388$$
$$x_{64} = 35.6991118430775$$
$$x_{65} = 76.5398163397448$$
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
$$\frac{\left(6 - 2 x\right) \sin{\left(x + 2 \right)}}{\left(\left(x^{2} - 6 x\right) - 16\right)^{2}} + \frac{\cos{\left(x + 2 \right)}}{\left(x^{2} - 6 x\right) - 16} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -31.7864910876843$$
$$x_{2} = 27.7611230930105$$
$$x_{3} = 49.793070679715$$
$$x_{4} = -1.69243756259044$$
$$x_{5} = 6.28795598412434$$
$$x_{6} = 2.68727827277551$$
$$x_{7} = 285.448644769003$$
$$x_{8} = 96.9388208914378$$
$$x_{9} = 81.2265393083297$$
$$x_{10} = 87.511646585983$$
$$x_{11} = 84.3691304713976$$
$$x_{12} = 90.6540958331421$$
$$x_{13} = 30.9128306854896$$
$$x_{14} = 100.081108055878$$
$$x_{15} = -69.5165374834153$$
$$x_{16} = 93.796485243138$$
$$x_{17} = -60.0875725534914$$
$$x_{18} = 15.0816010396296$$
$$x_{19} = -56.9442860032871$$
$$x_{20} = 59.2252173233687$$
$$x_{21} = -34.9331176531621$$
$$x_{22} = -100.940884643295$$
$$x_{23} = -78.9445269495188$$
$$x_{24} = 37.210240969449$$
$$x_{25} = 46.6482893777004$$
$$x_{26} = -12.8564212163698$$
$$x_{27} = -91.5141735765649$$
$$x_{28} = -44.3688291565024$$
$$x_{29} = -16.0247168420022$$
$$x_{30} = -66.3736775903828$$
$$x_{31} = -94.656452380678$$
$$x_{32} = -50.6571036178491$$
$$x_{33} = 24.6060431469901$$
$$x_{34} = 62.3687339914756$$
$$x_{35} = 52.9374378033397$$
$$x_{36} = -41.2241304241125$$
$$x_{37} = 56.0814671580514$$
$$x_{38} = -19.184063957018$$
$$x_{39} = -38.0789367955173$$
$$x_{40} = -107.225173810083$$
$$x_{41} = 74.9410918127375$$
$$x_{42} = 71.7982107905797$$
$$x_{43} = -9.66912755028377$$
$$x_{44} = 21.4454479761881$$
$$x_{45} = 65.5120533445129$$
$$x_{46} = -97.7986880876061$$
$$x_{47} = -53.8008079315373$$
$$x_{48} = -75.801950199115$$
$$x_{49} = 68.6552044181148$$
$$x_{50} = -72.6592906932686$$
$$x_{51} = -82.0870302458297$$
$$x_{52} = -85.2294680416321$$
$$x_{53} = -47.51312932817$$
$$x_{54} = -6.42525079485816$$
$$x_{55} = -63.2306955081109$$
$$x_{56} = 34.0623121686175$$
$$x_{57} = -157.496363552136$$
$$x_{58} = -25.4896392312871$$
$$x_{59} = -28.6387954551956$$
$$x_{60} = 78.0838635183795$$
$$x_{61} = 40.3570407278116$$
$$x_{62} = 11.8143136479515$$
$$x_{63} = 18.2747416504721$$
$$x_{64} = -22.3384065861891$$
$$x_{65} = 43.5029922227953$$
$$x_{66} = -88.371847178991$$
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:
$$x_{1} = 27.7611230930105$$
$$x_{2} = -1.69243756259044$$
$$x_{3} = 6.28795598412434$$
$$x_{4} = 285.448644769003$$
$$x_{5} = 96.9388208914378$$
$$x_{6} = 84.3691304713976$$
$$x_{7} = 90.6540958331421$$
$$x_{8} = -60.0875725534914$$
$$x_{9} = 15.0816010396296$$
$$x_{10} = 59.2252173233687$$
$$x_{11} = -34.9331176531621$$
$$x_{12} = -78.9445269495188$$
$$x_{13} = 46.6482893777004$$
$$x_{14} = -91.5141735765649$$
$$x_{15} = -16.0247168420022$$
$$x_{16} = -66.3736775903828$$
$$x_{17} = 52.9374378033397$$
$$x_{18} = -41.2241304241125$$
$$x_{19} = 71.7982107905797$$
$$x_{20} = -9.66912755028377$$
$$x_{21} = 21.4454479761881$$
$$x_{22} = 65.5120533445129$$
$$x_{23} = -97.7986880876061$$
$$x_{24} = -53.8008079315373$$
$$x_{25} = -72.6592906932686$$
$$x_{26} = -85.2294680416321$$
$$x_{27} = -47.51312932817$$
$$x_{28} = 34.0623121686175$$
$$x_{29} = -28.6387954551956$$
$$x_{30} = 78.0838635183795$$
$$x_{31} = 40.3570407278116$$
$$x_{32} = -22.3384065861891$$
Puntos máximos de la función:
$$x_{32} = -31.7864910876843$$
$$x_{32} = 49.793070679715$$
$$x_{32} = 2.68727827277551$$
$$x_{32} = 81.2265393083297$$
$$x_{32} = 87.511646585983$$
$$x_{32} = 30.9128306854896$$
$$x_{32} = 100.081108055878$$
$$x_{32} = -69.5165374834153$$
$$x_{32} = 93.796485243138$$
$$x_{32} = -56.9442860032871$$
$$x_{32} = -100.940884643295$$
$$x_{32} = 37.210240969449$$
$$x_{32} = -12.8564212163698$$
$$x_{32} = -44.3688291565024$$
$$x_{32} = -94.656452380678$$
$$x_{32} = -50.6571036178491$$
$$x_{32} = 24.6060431469901$$
$$x_{32} = 62.3687339914756$$
$$x_{32} = 56.0814671580514$$
$$x_{32} = -19.184063957018$$
$$x_{32} = -38.0789367955173$$
$$x_{32} = -107.225173810083$$
$$x_{32} = 74.9410918127375$$
$$x_{32} = -75.801950199115$$
$$x_{32} = 68.6552044181148$$
$$x_{32} = -82.0870302458297$$
$$x_{32} = -6.42525079485816$$
$$x_{32} = -63.2306955081109$$
$$x_{32} = -157.496363552136$$
$$x_{32} = -25.4896392312871$$
$$x_{32} = 11.8143136479515$$
$$x_{32} = 18.2747416504721$$
$$x_{32} = 43.5029922227953$$
$$x_{32} = -88.371847178991$$
Decrece en los intervalos
$$\left[285.448644769003, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -97.7986880876061\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{\left(6 - 2 x\right) \sin{\left(x + 2 \right)}}{\left(\left(x^{2} - 6 x\right) - 16\right)^{2}} + \frac{\cos{\left(x + 2 \right)}}{\left(x^{2} - 6 x\right) - 16} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -31.7864910876843$$
$$x_{2} = 27.7611230930105$$
$$x_{3} = 49.793070679715$$
$$x_{4} = -1.69243756259044$$
$$x_{5} = 6.28795598412434$$
$$x_{6} = 2.68727827277551$$
$$x_{7} = 285.448644769003$$
$$x_{8} = 96.9388208914378$$
$$x_{9} = 81.2265393083297$$
$$x_{10} = 87.511646585983$$
$$x_{11} = 84.3691304713976$$
$$x_{12} = 90.6540958331421$$
$$x_{13} = 30.9128306854896$$
$$x_{14} = 100.081108055878$$
$$x_{15} = -69.5165374834153$$
$$x_{16} = 93.796485243138$$
$$x_{17} = -60.0875725534914$$
$$x_{18} = 15.0816010396296$$
$$x_{19} = -56.9442860032871$$
$$x_{20} = 59.2252173233687$$
$$x_{21} = -34.9331176531621$$
$$x_{22} = -100.940884643295$$
$$x_{23} = -78.9445269495188$$
$$x_{24} = 37.210240969449$$
$$x_{25} = 46.6482893777004$$
$$x_{26} = -12.8564212163698$$
$$x_{27} = -91.5141735765649$$
$$x_{28} = -44.3688291565024$$
$$x_{29} = -16.0247168420022$$
$$x_{30} = -66.3736775903828$$
$$x_{31} = -94.656452380678$$
$$x_{32} = -50.6571036178491$$
$$x_{33} = 24.6060431469901$$
$$x_{34} = 62.3687339914756$$
$$x_{35} = 52.9374378033397$$
$$x_{36} = -41.2241304241125$$
$$x_{37} = 56.0814671580514$$
$$x_{38} = -19.184063957018$$
$$x_{39} = -38.0789367955173$$
$$x_{40} = -107.225173810083$$
$$x_{41} = 74.9410918127375$$
$$x_{42} = 71.7982107905797$$
$$x_{43} = -9.66912755028377$$
$$x_{44} = 21.4454479761881$$
$$x_{45} = 65.5120533445129$$
$$x_{46} = -97.7986880876061$$
$$x_{47} = -53.8008079315373$$
$$x_{48} = -75.801950199115$$
$$x_{49} = 68.6552044181148$$
$$x_{50} = -72.6592906932686$$
$$x_{51} = -82.0870302458297$$
$$x_{52} = -85.2294680416321$$
$$x_{53} = -47.51312932817$$
$$x_{54} = -6.42525079485816$$
$$x_{55} = -63.2306955081109$$
$$x_{56} = 34.0623121686175$$
$$x_{57} = -157.496363552136$$
$$x_{58} = -25.4896392312871$$
$$x_{59} = -28.6387954551956$$
$$x_{60} = 78.0838635183795$$
$$x_{61} = 40.3570407278116$$
$$x_{62} = 11.8143136479515$$
$$x_{63} = 18.2747416504721$$
$$x_{64} = -22.3384065861891$$
$$x_{65} = 43.5029922227953$$
$$x_{66} = -88.371847178991$$
Signos de extremos en los puntos:
(-31.786491087684265, 0.000842360350316363)
(27.76112309301045, -0.00169435654290797)
(49.79307067971496, 0.000461549762093301)
(-1.6924375625904449, -0.101554290996157)
(6.2879559841243395, -0.0639424692398677)
(2.6872782727755116, 0.0401444264826029)
(285.44864476900347, -1.25385187160432e-5)
(96.93882089143781, -0.000113616899975756)
(81.22653930832969, 0.000164031249690372)
(87.51164658598304, 0.000140464724572136)
(84.36913047139763, -0.000151562396070581)
(90.65409583314212, -0.000130544097129053)
(30.912830685489627, 0.00132241967583976)
(100.08110805587832, 0.000106363215003366)
(-69.5165374834153, 0.000190997983269388)
(93.79648524313795, 0.000121639591173961)
(-60.087572553491405, -0.00025271325964302)
(15.08160103962958, -0.00810669574032498)
(-56.94428600328711, 0.000280086086360883)
(59.225217323368724, -0.000318644830754798)
(-34.933117653162086, -0.000706236976294377)
(-100.9408846432946, 9.27582523654856e-5)
(-78.94452694951883, -0.000149434107906056)
(37.21024096944899, 0.000871548997581298)
(46.64828937770043, -0.000531293543595867)
(-12.856421216369824, 0.00437376190864768)
(-91.51417357656494, -0.000112234271967402)
(-44.36882915650244, 0.000450282590510704)
(-16.024716842002224, -0.00294914402213017)
(-66.37367759038277, -0.000208780595133749)
(-94.65645238067796, 0.000105110606590734)
(-50.657103617849124, 0.000350127603588176)
(24.6060431469901, 0.00225261125465836)
(62.36873399147561, 0.000285578816624428)
(52.93743780333968, -0.000404732599546242)
(-41.224130424112545, -0.000517384495073603)
(56.08146715805138, 0.000357825366688638)
(-19.184063957017955, 0.00213112833060858)
(-38.07893679551729, 0.000600778169031096)
(-107.22517381008319, 8.24633933265952e-5)
(74.94109181273754, 0.000194079413149621)
(71.79821079057972, -0.00021230502973741)
(-9.669127550283767, -0.00725397642461281)
(21.44544797618815, -0.00315073934894156)
(65.51205334451288, -0.000257415551971328)
(-97.79868808760614, -9.86448150145581e-5)
(-53.80080793153732, -0.000312173508844489)
(-75.80195019911503, 0.000161635620227774)
(68.65520441811485, 0.000233229783697196)
(-72.65929069326857, -0.000175397464638208)
(-82.08703024582971, 0.000138565389373513)
(-85.22946804163212, -0.000128841983967783)
(-47.513129328169974, -0.000395476505330644)
(-6.4252507948581625, 0.0150239366489606)
(-63.23069550811087, 0.000229172986561244)
(34.0623121686175, -0.00106166331812605)
(-157.4963635521364, 3.88559493946351e-5)
(-25.48963923128712, 0.00126787641729337)
(-28.638795455195606, -0.00102242959451557)
(78.08386351837954, -0.000178107233096625)
(40.357040727811636, -0.000728553145949357)
(11.8143136479515, 0.0179976420225405)
(18.274741650472144, 0.00474955964060571)
(-22.338406586189127, -0.00161521563224802)
(43.50299222279528, 0.000618229592200167)
(-88.37184717899098, 0.00012010828642991)
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:
$$x_{1} = 27.7611230930105$$
$$x_{2} = -1.69243756259044$$
$$x_{3} = 6.28795598412434$$
$$x_{4} = 285.448644769003$$
$$x_{5} = 96.9388208914378$$
$$x_{6} = 84.3691304713976$$
$$x_{7} = 90.6540958331421$$
$$x_{8} = -60.0875725534914$$
$$x_{9} = 15.0816010396296$$
$$x_{10} = 59.2252173233687$$
$$x_{11} = -34.9331176531621$$
$$x_{12} = -78.9445269495188$$
$$x_{13} = 46.6482893777004$$
$$x_{14} = -91.5141735765649$$
$$x_{15} = -16.0247168420022$$
$$x_{16} = -66.3736775903828$$
$$x_{17} = 52.9374378033397$$
$$x_{18} = -41.2241304241125$$
$$x_{19} = 71.7982107905797$$
$$x_{20} = -9.66912755028377$$
$$x_{21} = 21.4454479761881$$
$$x_{22} = 65.5120533445129$$
$$x_{23} = -97.7986880876061$$
$$x_{24} = -53.8008079315373$$
$$x_{25} = -72.6592906932686$$
$$x_{26} = -85.2294680416321$$
$$x_{27} = -47.51312932817$$
$$x_{28} = 34.0623121686175$$
$$x_{29} = -28.6387954551956$$
$$x_{30} = 78.0838635183795$$
$$x_{31} = 40.3570407278116$$
$$x_{32} = -22.3384065861891$$
Puntos máximos de la función:
$$x_{32} = -31.7864910876843$$
$$x_{32} = 49.793070679715$$
$$x_{32} = 2.68727827277551$$
$$x_{32} = 81.2265393083297$$
$$x_{32} = 87.511646585983$$
$$x_{32} = 30.9128306854896$$
$$x_{32} = 100.081108055878$$
$$x_{32} = -69.5165374834153$$
$$x_{32} = 93.796485243138$$
$$x_{32} = -56.9442860032871$$
$$x_{32} = -100.940884643295$$
$$x_{32} = 37.210240969449$$
$$x_{32} = -12.8564212163698$$
$$x_{32} = -44.3688291565024$$
$$x_{32} = -94.656452380678$$
$$x_{32} = -50.6571036178491$$
$$x_{32} = 24.6060431469901$$
$$x_{32} = 62.3687339914756$$
$$x_{32} = 56.0814671580514$$
$$x_{32} = -19.184063957018$$
$$x_{32} = -38.0789367955173$$
$$x_{32} = -107.225173810083$$
$$x_{32} = 74.9410918127375$$
$$x_{32} = -75.801950199115$$
$$x_{32} = 68.6552044181148$$
$$x_{32} = -82.0870302458297$$
$$x_{32} = -6.42525079485816$$
$$x_{32} = -63.2306955081109$$
$$x_{32} = -157.496363552136$$
$$x_{32} = -25.4896392312871$$
$$x_{32} = 11.8143136479515$$
$$x_{32} = 18.2747416504721$$
$$x_{32} = 43.5029922227953$$
$$x_{32} = -88.371847178991$$
Decrece en los intervalos
$$\left[285.448644769003, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -97.7986880876061\right]$$
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
$$\frac{- \frac{4 \left(x - 3\right) \cos{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} - \frac{2 \left(\frac{4 \left(x - 3\right)^{2}}{- x^{2} + 6 x + 16} + 1\right) \sin{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} + \sin{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -99.3501947416362$$
$$x_{2} = 73.3410612361785$$
$$x_{3} = -74.204596496154$$
$$x_{4} = -237.602816735387$$
$$x_{5} = -14.3135189034045$$
$$x_{6} = 114.202883064683$$
$$x_{7} = -23.8365720061754$$
$$x_{8} = -71.0607746548393$$
$$x_{9} = -93.0644320739133$$
$$x_{10} = 63.9073143415748$$
$$x_{11} = -80.4917302940608$$
$$x_{12} = -52.1923924048936$$
$$x_{13} = 70.1967638878763$$
$$x_{14} = -33.3035522728338$$
$$x_{15} = 76.4851234890106$$
$$x_{16} = 32.4173777338565$$
$$x_{17} = -7.80595314880128$$
$$x_{18} = -83.6350798427868$$
$$x_{19} = 79.6289801656441$$
$$x_{20} = -96.2073545700785$$
$$x_{21} = -89.9214187975649$$
$$x_{22} = -77.3482416096638$$
$$x_{23} = 13.1818075170047$$
$$x_{24} = 0.54342163985994$$
$$x_{25} = 22.9179395485667$$
$$x_{26} = 19.7276522516312$$
$$x_{27} = -3.74365156883252$$
$$x_{28} = 82.7726560210589$$
$$x_{29} = 45.0273083414577$$
$$x_{30} = 38.7264408089641$$
$$x_{31} = -17.5000759003003$$
$$x_{32} = -118.205868712902$$
$$x_{33} = -67.9167520955876$$
$$x_{34} = 16.5040939830698$$
$$x_{35} = -64.7725002524981$$
$$x_{36} = -11.0984130635893$$
$$x_{37} = -49.0462993588161$$
$$x_{38} = -20.6724375781105$$
$$x_{39} = -86.7783050821665$$
$$x_{40} = 29.2576796489324$$
$$x_{41} = 51.3233786100203$$
$$x_{42} = 41.8776301910141$$
$$x_{43} = 85.9161719734533$$
$$x_{44} = 836.800441020391$$
$$x_{45} = -102.492960023049$$
$$x_{46} = -55.3379877768322$$
$$x_{47} = -61.6279848308926$$
$$x_{48} = -30.1507815570728$$
$$x_{49} = 92.202792811376$$
$$x_{50} = 26.0923176787185$$
$$x_{51} = 148.768973500815$$
$$x_{52} = 57.6163861982496$$
$$x_{53} = 95.3459260715474$$
$$x_{54} = 89.0595458224663$$
$$x_{55} = 35.5732608236774$$
$$x_{56} = -58.483164284657$$
$$x_{57} = 60.7620662546327$$
$$x_{58} = 4.59986410582854$$
$$x_{59} = -42.752190502143$$
$$x_{60} = 67.0521959485079$$
$$x_{61} = -26.9954616531241$$
$$x_{62} = 1210.65145201341$$
$$x_{63} = 54.4701919197543$$
$$x_{64} = 48.1758104994968$$
$$x_{65} = -39.6038752862033$$
$$x_{66} = -121.348299634062$$
$$x_{67} = -36.4544342495191$$
$$x_{68} = 98.4889569770082$$
$$x_{69} = -45.8996084953731$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = -2$$
$$x_{2} = 8$$
$$\lim_{x \to -2^-}\left(\frac{- \frac{4 \left(x - 3\right) \cos{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} - \frac{2 \left(\frac{4 \left(x - 3\right)^{2}}{- x^{2} + 6 x + 16} + 1\right) \sin{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} + \sin{\left(x + 2 \right)}}{- x^{2} + 6 x + 16}\right) = -\infty$$
$$\lim_{x \to -2^+}\left(\frac{- \frac{4 \left(x - 3\right) \cos{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} - \frac{2 \left(\frac{4 \left(x - 3\right)^{2}}{- x^{2} + 6 x + 16} + 1\right) \sin{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} + \sin{\left(x + 2 \right)}}{- x^{2} + 6 x + 16}\right) = \infty$$
- los límites no son iguales, signo
$$x_{1} = -2$$
- es el punto de flexión
$$\lim_{x \to 8^-}\left(\frac{- \frac{4 \left(x - 3\right) \cos{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} - \frac{2 \left(\frac{4 \left(x - 3\right)^{2}}{- x^{2} + 6 x + 16} + 1\right) \sin{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} + \sin{\left(x + 2 \right)}}{- x^{2} + 6 x + 16}\right) = \infty$$
$$\lim_{x \to 8^+}\left(\frac{- \frac{4 \left(x - 3\right) \cos{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} - \frac{2 \left(\frac{4 \left(x - 3\right)^{2}}{- x^{2} + 6 x + 16} + 1\right) \sin{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} + \sin{\left(x + 2 \right)}}{- x^{2} + 6 x + 16}\right) = -\infty$$
- los límites no son iguales, signo
$$x_{2} = 8$$
- 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
$$\left[836.800441020391, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -237.602816735387\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{- \frac{4 \left(x - 3\right) \cos{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} - \frac{2 \left(\frac{4 \left(x - 3\right)^{2}}{- x^{2} + 6 x + 16} + 1\right) \sin{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} + \sin{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -99.3501947416362$$
$$x_{2} = 73.3410612361785$$
$$x_{3} = -74.204596496154$$
$$x_{4} = -237.602816735387$$
$$x_{5} = -14.3135189034045$$
$$x_{6} = 114.202883064683$$
$$x_{7} = -23.8365720061754$$
$$x_{8} = -71.0607746548393$$
$$x_{9} = -93.0644320739133$$
$$x_{10} = 63.9073143415748$$
$$x_{11} = -80.4917302940608$$
$$x_{12} = -52.1923924048936$$
$$x_{13} = 70.1967638878763$$
$$x_{14} = -33.3035522728338$$
$$x_{15} = 76.4851234890106$$
$$x_{16} = 32.4173777338565$$
$$x_{17} = -7.80595314880128$$
$$x_{18} = -83.6350798427868$$
$$x_{19} = 79.6289801656441$$
$$x_{20} = -96.2073545700785$$
$$x_{21} = -89.9214187975649$$
$$x_{22} = -77.3482416096638$$
$$x_{23} = 13.1818075170047$$
$$x_{24} = 0.54342163985994$$
$$x_{25} = 22.9179395485667$$
$$x_{26} = 19.7276522516312$$
$$x_{27} = -3.74365156883252$$
$$x_{28} = 82.7726560210589$$
$$x_{29} = 45.0273083414577$$
$$x_{30} = 38.7264408089641$$
$$x_{31} = -17.5000759003003$$
$$x_{32} = -118.205868712902$$
$$x_{33} = -67.9167520955876$$
$$x_{34} = 16.5040939830698$$
$$x_{35} = -64.7725002524981$$
$$x_{36} = -11.0984130635893$$
$$x_{37} = -49.0462993588161$$
$$x_{38} = -20.6724375781105$$
$$x_{39} = -86.7783050821665$$
$$x_{40} = 29.2576796489324$$
$$x_{41} = 51.3233786100203$$
$$x_{42} = 41.8776301910141$$
$$x_{43} = 85.9161719734533$$
$$x_{44} = 836.800441020391$$
$$x_{45} = -102.492960023049$$
$$x_{46} = -55.3379877768322$$
$$x_{47} = -61.6279848308926$$
$$x_{48} = -30.1507815570728$$
$$x_{49} = 92.202792811376$$
$$x_{50} = 26.0923176787185$$
$$x_{51} = 148.768973500815$$
$$x_{52} = 57.6163861982496$$
$$x_{53} = 95.3459260715474$$
$$x_{54} = 89.0595458224663$$
$$x_{55} = 35.5732608236774$$
$$x_{56} = -58.483164284657$$
$$x_{57} = 60.7620662546327$$
$$x_{58} = 4.59986410582854$$
$$x_{59} = -42.752190502143$$
$$x_{60} = 67.0521959485079$$
$$x_{61} = -26.9954616531241$$
$$x_{62} = 1210.65145201341$$
$$x_{63} = 54.4701919197543$$
$$x_{64} = 48.1758104994968$$
$$x_{65} = -39.6038752862033$$
$$x_{66} = -121.348299634062$$
$$x_{67} = -36.4544342495191$$
$$x_{68} = 98.4889569770082$$
$$x_{69} = -45.8996084953731$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = -2$$
$$x_{2} = 8$$
$$\lim_{x \to -2^-}\left(\frac{- \frac{4 \left(x - 3\right) \cos{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} - \frac{2 \left(\frac{4 \left(x - 3\right)^{2}}{- x^{2} + 6 x + 16} + 1\right) \sin{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} + \sin{\left(x + 2 \right)}}{- x^{2} + 6 x + 16}\right) = -\infty$$
$$\lim_{x \to -2^+}\left(\frac{- \frac{4 \left(x - 3\right) \cos{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} - \frac{2 \left(\frac{4 \left(x - 3\right)^{2}}{- x^{2} + 6 x + 16} + 1\right) \sin{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} + \sin{\left(x + 2 \right)}}{- x^{2} + 6 x + 16}\right) = \infty$$
- los límites no son iguales, signo
$$x_{1} = -2$$
- es el punto de flexión
$$\lim_{x \to 8^-}\left(\frac{- \frac{4 \left(x - 3\right) \cos{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} - \frac{2 \left(\frac{4 \left(x - 3\right)^{2}}{- x^{2} + 6 x + 16} + 1\right) \sin{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} + \sin{\left(x + 2 \right)}}{- x^{2} + 6 x + 16}\right) = \infty$$
$$\lim_{x \to 8^+}\left(\frac{- \frac{4 \left(x - 3\right) \cos{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} - \frac{2 \left(\frac{4 \left(x - 3\right)^{2}}{- x^{2} + 6 x + 16} + 1\right) \sin{\left(x + 2 \right)}}{- x^{2} + 6 x + 16} + \sin{\left(x + 2 \right)}}{- x^{2} + 6 x + 16}\right) = -\infty$$
- los límites no son iguales, signo
$$x_{2} = 8$$
- 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
$$\left[836.800441020391, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -237.602816735387\right]$$
Asíntotas verticales
Hay:
$$x_{1} = -2$$
$$x_{2} = 8$$
$$x_{1} = -2$$
$$x_{2} = 8$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x + 2 \right)}}{\left(x^{2} - 6 x\right) - 16}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x + 2 \right)}}{\left(x^{2} - 6 x\right) - 16}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x + 2 \right)}}{\left(x^{2} - 6 x\right) - 16}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x + 2 \right)}}{\left(x^{2} - 6 x\right) - 16}\right) = 0$$
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 + 2)/(x^2 - 6*x - 16), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x + 2 \right)}}{x \left(\left(x^{2} - 6 x\right) - 16\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x + 2 \right)}}{x \left(\left(x^{2} - 6 x\right) - 16\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x + 2 \right)}}{x \left(\left(x^{2} - 6 x\right) - 16\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x + 2 \right)}}{x \left(\left(x^{2} - 6 x\right) - 16\right)}\right) = 0$$
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:
$$\frac{\sin{\left(x + 2 \right)}}{\left(x^{2} - 6 x\right) - 16} = - \frac{\sin{\left(x - 2 \right)}}{x^{2} + 6 x - 16}$$
- No
$$\frac{\sin{\left(x + 2 \right)}}{\left(x^{2} - 6 x\right) - 16} = \frac{\sin{\left(x - 2 \right)}}{x^{2} + 6 x - 16}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\sin{\left(x + 2 \right)}}{\left(x^{2} - 6 x\right) - 16} = - \frac{\sin{\left(x - 2 \right)}}{x^{2} + 6 x - 16}$$
- No
$$\frac{\sin{\left(x + 2 \right)}}{\left(x^{2} - 6 x\right) - 16} = \frac{\sin{\left(x - 2 \right)}}{x^{2} + 6 x - 16}$$
- No
es decir, función
no es
par ni impar