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- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x^3/(4(2-x)^2)
-
x^3+9x-1
-
(x^3+3x^2-2x-2)/(2-3x^2)
-
x^3+6*x^2+9*x+2
-
Expresiones idénticas
- sin(pi*x)^(uno / tres)/(- cinco +x)
- seno de ( número pi multiplicar por x) en el grado (1 dividir por 3) dividir por ( menos 5 más x)
- seno de ( número pi multiplicar por x) en el grado (uno dividir por tres) dividir por ( menos cinco más x)
- sin(pi*x)(1/3)/(-5+x)
- sinpi*x1/3/-5+x
- sin(pix)^(1/3)/(-5+x)
- sin(pix)(1/3)/(-5+x)
- sinpix1/3/-5+x
- sinpix^1/3/-5+x
- sin(pi*x)^(1 dividir por 3) dividir por (-5+x)
-
Expresiones semejantes
- sin(pi*x)^(1/3)/(5+x)
- sin(pi*x)^(1/3)/(-5-x)
-
Expresiones con funciones
- Seno sin
- sin(3,14*x)^2
- sin(tan((1)/((x-1)(x-1)(x-1))))
- sin(x)*(-x)
- sin(2*(|x|-sqrt(2)/2)-1/2)
- sin(*x)
Gráfico de la función y = sin(pi*x)^(1/3)/(-5+x)
Solución
Ha introducido
[src]
3 ___________
\/ sin(pi*x)
f(x) = -------------
-5 + x
$$f{\left(x \right)} = \frac{\sqrt[3]{\sin{\left(\pi x \right)}}}{x - 5}$$
f = sin(pi*x)^(1/3)/(x - 5)
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 5$$
$$x_{1} = 5$$
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:
$$\frac{\sqrt[3]{\sin{\left(\pi x \right)}}}{x - 5} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = 1$$
Solución numérica
$$x_{1} = 0$$
$$x_{2} = 1$$
o sea hay que resolver la ecuación:
$$\frac{\sqrt[3]{\sin{\left(\pi x \right)}}}{x - 5} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = 1$$
Solución numérica
$$x_{1} = 0$$
$$x_{2} = 1$$
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
$$\frac{\pi \cos{\left(\pi x \right)}}{3 \left(x - 5\right) \sin^{\frac{2}{3}}{\left(\pi x \right)}} - \frac{\sqrt[3]{\sin{\left(\pi x \right)}}}{\left(x - 5\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -89.4967834544883$$
$$x_{2} = 22.4826306576559$$
$$x_{3} = -17.4864905098219$$
$$x_{4} = -77.4963155933236$$
$$x_{5} = -43.4937327103111$$
$$x_{6} = -5.47105113443199$$
$$x_{7} = 6.29811538749029$$
$$x_{8} = -57.4951365831911$$
$$x_{9} = 98.496749052932$$
$$x_{10} = 10.4447350189979$$
$$x_{11} = 38.4909264612995$$
$$x_{12} = -85.496641286731$$
$$x_{13} = 92.4965261308476$$
$$x_{14} = -15.4851725112626$$
$$x_{15} = 76.4957487615285$$
$$x_{16} = 1.58683596346056$$
$$x_{17} = -35.4924947271892$$
$$x_{18} = 80.4959739927052$$
$$x_{19} = -19.4875933250884$$
$$x_{20} = -67.4958073993004$$
$$x_{21} = 90.4964448707506$$
$$x_{22} = 32.4889467803235$$
$$x_{23} = -75.4962240552696$$
$$x_{24} = 74.4956264237311$$
$$x_{25} = 26.4858621616449$$
$$x_{26} = 12.4594717635981$$
$$x_{27} = -39.4931693584383$$
$$x_{28} = -65.4956884602738$$
$$x_{29} = 86.4962703858798$$
$$x_{30} = 66.4950575032433$$
$$x_{31} = 68.4952131724318$$
$$x_{32} = 58.4943184383132$$
$$x_{33} = -83.4965653836064$$
$$x_{34} = -23.4893346125469$$
$$x_{35} = 18.4774841939451$$
$$x_{36} = -93.4969140756257$$
$$x_{37} = 2.62152720496008$$
$$x_{38} = 54.4938593222224$$
$$x_{39} = 34.4896961510675$$
$$x_{40} = 24.4844121275861$$
$$x_{41} = 42.4918943053844$$
$$x_{42} = 96.4966779939798$$
$$x_{43} = 100.496817135593$$
$$x_{44} = 94.4966037592085$$
$$x_{45} = -25.4900339821497$$
$$x_{46} = 46.4926755771317$$
$$x_{47} = 28.4870653815954$$
$$x_{48} = 72.4954968362866$$
$$x_{49} = -7.47568293434139$$
$$x_{50} = 16.4735684148199$$
$$x_{51} = -37.4928479164893$$
$$x_{52} = 50.493319482426$$
$$x_{53} = 82.4960778896675$$
$$x_{54} = -97.4970345019427$$
$$x_{55} = -27.490647275511$$
$$x_{56} = 62.4947136773841$$
$$x_{57} = 44.4923047202881$$
$$x_{58} = -49.4944226871506$$
$$x_{59} = 60.4945231792732$$
$$x_{60} = -45.4939809198007$$
$$x_{61} = 78.4958644414866$$
$$x_{62} = -71.4960266202515$$
$$x_{63} = 88.4963597179543$$
$$x_{64} = 20.4803894546078$$
$$x_{65} = -11.4815779712047$$
$$x_{66} = -41.4934631494683$$
$$x_{67} = -81.4964859705107$$
$$x_{68} = 30.488079861269$$
$$x_{69} = 48.4930123321987$$
$$x_{70} = 64.4948913688995$$
$$x_{71} = -31.4916722315682$$
$$x_{72} = -33.4921048428995$$
$$x_{73} = -59.4952873868124$$
$$x_{74} = 70.495359335105$$
$$x_{75} = -21.4885296777534$$
$$x_{76} = -87.4967139075584$$
$$x_{77} = 40.4914376465524$$
$$x_{78} = -9.47903700525443$$
$$x_{79} = -61.4954291195393$$
$$x_{80} = -55.4949758090825$$
$$x_{81} = -47.4942102180917$$
$$x_{82} = -99.4970912578863$$
$$x_{83} = -53.4948040418727$$
$$x_{84} = -73.4961278528563$$
$$x_{85} = -91.4968501186439$$
$$x_{86} = 14.4680039094037$$
$$x_{87} = 52.4936007673721$$
$$x_{88} = -69.495919952339$$
$$x_{89} = -51.4946201141526$$
$$x_{90} = 8.41316403653945$$
$$x_{91} = -1.45323686184935$$
$$x_{92} = -63.4955625759026$$
$$x_{93} = 36.4903503636438$$
$$x_{94} = -95.496975487056$$
$$x_{95} = -3.46423970898458$$
$$x_{96} = -29.4911894624069$$
$$x_{97} = 84.4961765591096$$
$$x_{98} = -13.4835695403804$$
$$x_{99} = 56.4940977951431$$
$$x_{100} = -79.4964027982152$$
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:
$$x_{1} = -89.4967834544883$$
$$x_{2} = -17.4864905098219$$
$$x_{3} = -77.4963155933236$$
$$x_{4} = -43.4937327103111$$
$$x_{5} = -5.47105113443199$$
$$x_{6} = -57.4951365831911$$
$$x_{7} = -85.496641286731$$
$$x_{8} = -15.4851725112626$$
$$x_{9} = 1.58683596346056$$
$$x_{10} = -35.4924947271892$$
$$x_{11} = -19.4875933250884$$
$$x_{12} = -67.4958073993004$$
$$x_{13} = -75.4962240552696$$
$$x_{14} = -39.4931693584383$$
$$x_{15} = -65.4956884602738$$
$$x_{16} = -83.4965653836064$$
$$x_{17} = -23.4893346125469$$
$$x_{18} = -93.4969140756257$$
$$x_{19} = 2.62152720496008$$
$$x_{20} = -25.4900339821497$$
$$x_{21} = -7.47568293434139$$
$$x_{22} = -37.4928479164893$$
$$x_{23} = -97.4970345019427$$
$$x_{24} = -27.490647275511$$
$$x_{25} = -49.4944226871506$$
$$x_{26} = -45.4939809198007$$
$$x_{27} = -71.4960266202515$$
$$x_{28} = -11.4815779712047$$
$$x_{29} = -41.4934631494683$$
$$x_{30} = -81.4964859705107$$
$$x_{31} = -31.4916722315682$$
$$x_{32} = -33.4921048428995$$
$$x_{33} = -59.4952873868124$$
$$x_{34} = -21.4885296777534$$
$$x_{35} = -87.4967139075584$$
$$x_{36} = -9.47903700525443$$
$$x_{37} = -61.4954291195393$$
$$x_{38} = -55.4949758090825$$
$$x_{39} = -47.4942102180917$$
$$x_{40} = -99.4970912578863$$
$$x_{41} = -53.4948040418727$$
$$x_{42} = -73.4961278528563$$
$$x_{43} = -91.4968501186439$$
$$x_{44} = -69.495919952339$$
$$x_{45} = -51.4946201141526$$
$$x_{46} = -1.45323686184935$$
$$x_{47} = -63.4955625759026$$
$$x_{48} = -95.496975487056$$
$$x_{49} = -3.46423970898458$$
$$x_{50} = -29.4911894624069$$
$$x_{51} = -13.4835695403804$$
$$x_{52} = -79.4964027982152$$
Puntos máximos de la función:
$$x_{52} = 22.4826306576559$$
$$x_{52} = 6.29811538749029$$
$$x_{52} = 98.496749052932$$
$$x_{52} = 10.4447350189979$$
$$x_{52} = 38.4909264612995$$
$$x_{52} = 92.4965261308476$$
$$x_{52} = 76.4957487615285$$
$$x_{52} = 80.4959739927052$$
$$x_{52} = 90.4964448707506$$
$$x_{52} = 32.4889467803235$$
$$x_{52} = 74.4956264237311$$
$$x_{52} = 26.4858621616449$$
$$x_{52} = 12.4594717635981$$
$$x_{52} = 86.4962703858798$$
$$x_{52} = 66.4950575032433$$
$$x_{52} = 68.4952131724318$$
$$x_{52} = 58.4943184383132$$
$$x_{52} = 18.4774841939451$$
$$x_{52} = 54.4938593222224$$
$$x_{52} = 34.4896961510675$$
$$x_{52} = 24.4844121275861$$
$$x_{52} = 42.4918943053844$$
$$x_{52} = 96.4966779939798$$
$$x_{52} = 100.496817135593$$
$$x_{52} = 94.4966037592085$$
$$x_{52} = 46.4926755771317$$
$$x_{52} = 28.4870653815954$$
$$x_{52} = 72.4954968362866$$
$$x_{52} = 16.4735684148199$$
$$x_{52} = 50.493319482426$$
$$x_{52} = 82.4960778896675$$
$$x_{52} = 62.4947136773841$$
$$x_{52} = 44.4923047202881$$
$$x_{52} = 60.4945231792732$$
$$x_{52} = 78.4958644414866$$
$$x_{52} = 88.4963597179543$$
$$x_{52} = 20.4803894546078$$
$$x_{52} = 30.488079861269$$
$$x_{52} = 48.4930123321987$$
$$x_{52} = 64.4948913688995$$
$$x_{52} = 70.495359335105$$
$$x_{52} = 40.4914376465524$$
$$x_{52} = 14.4680039094037$$
$$x_{52} = 52.4936007673721$$
$$x_{52} = 8.41316403653945$$
$$x_{52} = 36.4903503636438$$
$$x_{52} = 84.4961765591096$$
$$x_{52} = 56.4940977951431$$
Decrece en los intervalos
$$\left[2.62152720496008, 6.29811538749029\right]$$
Crece en los intervalos
$$\left(-\infty, -99.4970912578863\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{\pi \cos{\left(\pi x \right)}}{3 \left(x - 5\right) \sin^{\frac{2}{3}}{\left(\pi x \right)}} - \frac{\sqrt[3]{\sin{\left(\pi x \right)}}}{\left(x - 5\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -89.4967834544883$$
$$x_{2} = 22.4826306576559$$
$$x_{3} = -17.4864905098219$$
$$x_{4} = -77.4963155933236$$
$$x_{5} = -43.4937327103111$$
$$x_{6} = -5.47105113443199$$
$$x_{7} = 6.29811538749029$$
$$x_{8} = -57.4951365831911$$
$$x_{9} = 98.496749052932$$
$$x_{10} = 10.4447350189979$$
$$x_{11} = 38.4909264612995$$
$$x_{12} = -85.496641286731$$
$$x_{13} = 92.4965261308476$$
$$x_{14} = -15.4851725112626$$
$$x_{15} = 76.4957487615285$$
$$x_{16} = 1.58683596346056$$
$$x_{17} = -35.4924947271892$$
$$x_{18} = 80.4959739927052$$
$$x_{19} = -19.4875933250884$$
$$x_{20} = -67.4958073993004$$
$$x_{21} = 90.4964448707506$$
$$x_{22} = 32.4889467803235$$
$$x_{23} = -75.4962240552696$$
$$x_{24} = 74.4956264237311$$
$$x_{25} = 26.4858621616449$$
$$x_{26} = 12.4594717635981$$
$$x_{27} = -39.4931693584383$$
$$x_{28} = -65.4956884602738$$
$$x_{29} = 86.4962703858798$$
$$x_{30} = 66.4950575032433$$
$$x_{31} = 68.4952131724318$$
$$x_{32} = 58.4943184383132$$
$$x_{33} = -83.4965653836064$$
$$x_{34} = -23.4893346125469$$
$$x_{35} = 18.4774841939451$$
$$x_{36} = -93.4969140756257$$
$$x_{37} = 2.62152720496008$$
$$x_{38} = 54.4938593222224$$
$$x_{39} = 34.4896961510675$$
$$x_{40} = 24.4844121275861$$
$$x_{41} = 42.4918943053844$$
$$x_{42} = 96.4966779939798$$
$$x_{43} = 100.496817135593$$
$$x_{44} = 94.4966037592085$$
$$x_{45} = -25.4900339821497$$
$$x_{46} = 46.4926755771317$$
$$x_{47} = 28.4870653815954$$
$$x_{48} = 72.4954968362866$$
$$x_{49} = -7.47568293434139$$
$$x_{50} = 16.4735684148199$$
$$x_{51} = -37.4928479164893$$
$$x_{52} = 50.493319482426$$
$$x_{53} = 82.4960778896675$$
$$x_{54} = -97.4970345019427$$
$$x_{55} = -27.490647275511$$
$$x_{56} = 62.4947136773841$$
$$x_{57} = 44.4923047202881$$
$$x_{58} = -49.4944226871506$$
$$x_{59} = 60.4945231792732$$
$$x_{60} = -45.4939809198007$$
$$x_{61} = 78.4958644414866$$
$$x_{62} = -71.4960266202515$$
$$x_{63} = 88.4963597179543$$
$$x_{64} = 20.4803894546078$$
$$x_{65} = -11.4815779712047$$
$$x_{66} = -41.4934631494683$$
$$x_{67} = -81.4964859705107$$
$$x_{68} = 30.488079861269$$
$$x_{69} = 48.4930123321987$$
$$x_{70} = 64.4948913688995$$
$$x_{71} = -31.4916722315682$$
$$x_{72} = -33.4921048428995$$
$$x_{73} = -59.4952873868124$$
$$x_{74} = 70.495359335105$$
$$x_{75} = -21.4885296777534$$
$$x_{76} = -87.4967139075584$$
$$x_{77} = 40.4914376465524$$
$$x_{78} = -9.47903700525443$$
$$x_{79} = -61.4954291195393$$
$$x_{80} = -55.4949758090825$$
$$x_{81} = -47.4942102180917$$
$$x_{82} = -99.4970912578863$$
$$x_{83} = -53.4948040418727$$
$$x_{84} = -73.4961278528563$$
$$x_{85} = -91.4968501186439$$
$$x_{86} = 14.4680039094037$$
$$x_{87} = 52.4936007673721$$
$$x_{88} = -69.495919952339$$
$$x_{89} = -51.4946201141526$$
$$x_{90} = 8.41316403653945$$
$$x_{91} = -1.45323686184935$$
$$x_{92} = -63.4955625759026$$
$$x_{93} = 36.4903503636438$$
$$x_{94} = -95.496975487056$$
$$x_{95} = -3.46423970898458$$
$$x_{96} = -29.4911894624069$$
$$x_{97} = 84.4961765591096$$
$$x_{98} = -13.4835695403804$$
$$x_{99} = 56.4940977951431$$
$$x_{100} = -79.4964027982152$$
Signos de extremos en los puntos:
3 ___________________________ (-89.4967834544883, -0.0105823707796533*\/ -sin(1.49678345448829*pi) )
3 _________________________ (22.4826306576559, 0.0571996297114522*\/ sin(0.4826306576559*pi) )
3 ___________________________ (-17.486490509821905, -0.0444711458892711*\/ -sin(1.48649050982191*pi) )
3 ___________________________ (-77.49631559332357, -0.0121217534723566*\/ -sin(1.49631559332357*pi) )
3 ___________________________ (-43.49373271031109, -0.0206212214261529*\/ -sin(1.49373271031109*pi) )
3 ___________________________ (-5.471051134431993, -0.0955013959116002*\/ -sin(1.47105113443199*pi) )
3 ___________________________ (6.298115387490293, 0.770347543551846*\/ sin(0.298115387490293*pi) )
3 ___________________________ (-57.495136583191055, -0.0160012451315926*\/ -sin(1.49513658319106*pi) )
3 ___________________________ (98.49674905293205, 0.0106955590448804*\/ sin(0.496749052932046*pi) )
3 ___________________________ (10.444735018997855, 0.183663667104236*\/ sin(0.444735018997855*pi) )
3 ___________________________ (38.490926461299495, 0.0298588335905115*\/ sin(0.490926461299495*pi) )
3 ___________________________ (-85.49664128673102, -0.0110501338589085*\/ -sin(1.49664128673102*pi) )
3 ___________________________ (92.49652613084764, 0.0114290251764343*\/ sin(0.496526130847641*pi) )
3 __________________________ (-15.485172511262602, -0.0488157958860345*\/ -sin(1.4851725112626*pi) )
3 ___________________________ (76.4957487615285, 0.0139868456142122*\/ sin(0.495748761528503*pi) )
3 __________________________ (1.5868359634605556, -0.292983281581123*\/ sin(1.58683596346056*pi) )
3 ___________________________ (-35.492494727189154, -0.0246959345611407*\/ -sin(1.49249472718915*pi) )
3 ___________________________ (80.49597399270525, 0.013245739436339*\/ sin(0.495973992705245*pi) )
3 ___________________________ (-19.487593325088355, -0.0408370061820435*\/ -sin(1.48759332508835*pi) )
3 ___________________________ (-67.49580739930036, -0.0137939011354421*\/ -sin(1.49580739930036*pi) )
3 ___________________________ (90.49644487075062, 0.0116963927741294*\/ sin(0.496444870750622*pi) )
3 ___________________________ (32.48894678032347, 0.0363782580682865*\/ sin(0.488946780323467*pi) )
3 ___________________________ (-75.49622405526965, -0.01242294296082*\/ -sin(1.49622405526965*pi) )
3 ___________________________ (74.49562642373114, 0.0143893947210831*\/ sin(0.495626423731139*pi) )
3 ___________________________ (26.485862161644878, 0.0465422328634842*\/ sin(0.485862161644878*pi) )
3 ___________________________ (12.459471763598073, 0.134057749890543*\/ sin(0.459471763598073*pi) )
3 ___________________________ (-39.49316935843832, -0.0224753600253551*\/ -sin(1.49316935843832*pi) )
3 ___________________________ (-65.49568846027377, -0.0141852646855634*\/ -sin(1.49568846027377*pi) )
3 ___________________________ (86.49627038587975, 0.01227050017461*\/ sin(0.496270385879754*pi) )
3 ___________________________ (66.49505750324334, 0.0162614694676439*\/ sin(0.495057503243345*pi) )
3 ___________________________ (68.49521317243179, 0.0157492187211709*\/ sin(0.495213172431789*pi) )
3 ___________________________ (58.49431843831324, 0.0186935739942765*\/ sin(0.494318438313243*pi) )
3 ___________________________ (-83.49656538360638, -0.0112998735675819*\/ -sin(1.49656538360638*pi) )
3 ___________________________ (-23.48933461254693, -0.0351008548848169*\/ -sin(1.48933461254693*pi) )
3 ___________________________ (18.47748419394514, 0.0741978239862642*\/ sin(0.477484193945141*pi) )
3 ___________________________ (-93.49691407562571, -0.0101526023366803*\/ -sin(1.49691407562571*pi) )
3 ___________________________ (2.621527204960085, -0.420437854948523*\/ sin(0.621527204960085*pi) )
3 ___________________________ (54.4938593222224, 0.0202045266563201*\/ sin(0.493859322222399*pi) )
3 __________________________ (34.48969615106747, 0.0339101493239293*\/ sin(0.48969615106747*pi) )
3 ___________________________ (24.484412127586086, 0.0513230778250783*\/ sin(0.484412127586086*pi) )
3 ___________________________ (42.49189430538436, 0.0266724319623505*\/ sin(0.491894305384363*pi) )
3 ___________________________ (96.49667799397982, 0.010929358550764*\/ sin(0.496677993979816*pi) )
3 ___________________________ (100.49681713559309, 0.010471553188837*\/ sin(0.496817135593091*pi) )
3 ___________________________ (94.4966037592085, 0.0111736083604972*\/ sin(0.496603759208497*pi) )
3 ___________________________ (-25.49003398214965, -0.0327976020159718*\/ -sin(1.49003398214965*pi) )
3 ___________________________ (46.49267557713173, 0.0241006391149946*\/ sin(0.492675577131727*pi) )
3 __________________________ (28.48706538159545, 0.0425766260600443*\/ sin(0.48706538159545*pi) )
3 ___________________________ (72.49549683628658, 0.0148158032294443*\/ sin(0.495496836286577*pi) )
3 ___________________________ (-7.4756829343413855, -0.0801559325660108*\/ -sin(1.47568293434139*pi) )
3 ___________________________ (16.473568414819926, 0.0871568429145672*\/ sin(0.473568414819926*pi) )
3 ___________________________ (-37.49284791648929, -0.0235333720621712*\/ -sin(1.49284791648929*pi) )
3 __________________________ (50.49331948242601, 0.0219812493653337*\/ sin(0.49331948242601*pi) )
3 ___________________________ (82.49607788966746, 0.0129038788443425*\/ sin(0.496077889667461*pi) )
3 ___________________________ (-97.49703450194268, -0.0097563798295164*\/ -sin(1.49703450194268*pi) )
3 ___________________________ (-27.490647275511012, -0.0307780879685252*\/ -sin(1.49064727551101*pi) )
3 ___________________________ (62.49471367738412, 0.0173929033825827*\/ sin(0.494713677384119*pi) )
3 ___________________________ (44.49230472028811, 0.0253213887384566*\/ sin(0.492304720288111*pi) )
3 ___________________________ (-49.49442268715064, -0.0183505017704462*\/ -sin(1.49442268715064*pi) )
3 ___________________________ (60.494523179273244, 0.0180197962377212*\/ sin(0.494523179273244*pi) )
3 ___________________________ (-45.49398091980073, -0.0198043406715801*\/ -sin(1.49398091980073*pi) )
3 ___________________________ (78.4958644414866, 0.0136062077451466*\/ sin(0.495864441486603*pi) )
3 ___________________________ (-71.49602662025147, -0.0130725744091819*\/ -sin(1.49602662025147*pi) )
3 ___________________________ (88.49635971795435, 0.0119765700370404*\/ sin(0.496359717954348*pi) )
3 ___________________________ (20.480389454607764, 0.0645978580146347*\/ sin(0.480389454607764*pi) )
3 ___________________________ (-11.481577971204716, -0.0606738020926831*\/ -sin(1.48157797120472*pi) )
3 ___________________________ (-41.49346314946827, -0.0215083999396899*\/ -sin(1.49346314946827*pi) )
3 ___________________________ (-81.49648597051068, -0.0115611633094659*\/ -sin(1.49648597051068*pi) )
3 ___________________________ (30.488079861268975, 0.039234026472099*\/ sin(0.488079861268975*pi) )
3 __________________________ (48.49301233219866, 0.0229921991229769*\/ sin(0.49301233219866*pi) )
3 ___________________________ (64.49489136889952, 0.0168081658272048*\/ sin(0.494891368899516*pi) )
3 ___________________________ (-31.49167223156816, -0.0274035126056767*\/ -sin(1.49167223156816*pi) )
3 ___________________________ (-33.49210484289947, -0.0259793535344812*\/ -sin(1.49210484289947*pi) )
3 ___________________________ (-59.495287386812365, -0.0155050088233962*\/ -sin(1.49528738681236*pi) )
3 ___________________________ (70.49535933510501, 0.0152682573261951*\/ sin(0.495359335105007*pi) )
3 ___________________________ (-21.488529677753363, -0.0377521898031154*\/ -sin(1.48852967775336*pi) )
3 ___________________________ (-87.49671390755837, -0.0108111948820085*\/ -sin(1.49671390755837*pi) )
3 ___________________________ (40.49143764655239, 0.0281758098941686*\/ sin(0.491437646552392*pi) )
3 ___________________________ (-9.479037005254433, -0.069065366684062*\/ -sin(1.47903700525443*pi) )
3 ___________________________ (-61.49542911953925, -0.0150386276657046*\/ -sin(1.49542911953925*pi) )
3 ___________________________ (-55.49497580908252, -0.0165302983698336*\/ -sin(1.49497580908252*pi) )
3 ___________________________ (-47.494210218091666, -0.0190497198804481*\/ -sin(1.49421021809167*pi) )
3 ___________________________ (-99.49709125788631, -0.00956964436007238*\/ -sin(1.49709125788631*pi) )
3 ___________________________ (-53.49480404187269, -0.0170955355160121*\/ -sin(1.49480404187269*pi) )
3 ___________________________ (-73.49612785285632, -0.0127394818999803*\/ -sin(1.49612785285632*pi) )
3 ___________________________ (-91.49685011864392, -0.0103630325629333*\/ -sin(1.49685011864392*pi) )
3 ___________________________ (14.468003909403738, 0.105618883300923*\/ sin(0.468003909403738*pi) )
3 ___________________________ (52.493600767372136, 0.0210554681860845*\/ sin(0.493600767372136*pi) )
3 ___________________________ (-69.495919952339, -0.013423553942817*\/ -sin(1.49591995233899*pi) )
3 ___________________________ (-51.49462011415264, -0.0177008005006389*\/ -sin(1.49462011415264*pi) )
3 ___________________________ (8.413164036539445, 0.292983281581123*\/ sin(0.413164036539445*pi) )
3 ___________________________ (-1.4532368618493523, -0.154960994212356*\/ -sin(1.45323686184935*pi) )
3 __________________________ (-63.4955625759026, -0.0145994858994239*\/ -sin(1.4955625759026*pi) )
3 ___________________________ (36.49035036364376, 0.0317557597312261*\/ sin(0.490350363643763*pi) )
3 ___________________________ (-95.496975487056, -0.00995054821454602*\/ -sin(1.49697548705601*pi) )
3 ___________________________ (-3.464239708984585, -0.118144102055442*\/ -sin(1.46423970898458*pi) )
3 __________________________ (-29.4911894624069, -0.02899291139524*\/ -sin(1.4911894624069*pi) )
3 _________________________ (84.4961765591096, 0.0125792213321913*\/ sin(0.4961765591096*pi) )
3 ___________________________ (-13.48356954038039, -0.054102103915336*\/ -sin(1.48356954038039*pi) )
3 ___________________________ (56.494097795143134, 0.0194197013408849*\/ sin(0.494097795143134*pi) )
3 ___________________________ (-79.49640279821519, -0.0118348233402088*\/ -sin(1.49640279821519*pi) )
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:
$$x_{1} = -89.4967834544883$$
$$x_{2} = -17.4864905098219$$
$$x_{3} = -77.4963155933236$$
$$x_{4} = -43.4937327103111$$
$$x_{5} = -5.47105113443199$$
$$x_{6} = -57.4951365831911$$
$$x_{7} = -85.496641286731$$
$$x_{8} = -15.4851725112626$$
$$x_{9} = 1.58683596346056$$
$$x_{10} = -35.4924947271892$$
$$x_{11} = -19.4875933250884$$
$$x_{12} = -67.4958073993004$$
$$x_{13} = -75.4962240552696$$
$$x_{14} = -39.4931693584383$$
$$x_{15} = -65.4956884602738$$
$$x_{16} = -83.4965653836064$$
$$x_{17} = -23.4893346125469$$
$$x_{18} = -93.4969140756257$$
$$x_{19} = 2.62152720496008$$
$$x_{20} = -25.4900339821497$$
$$x_{21} = -7.47568293434139$$
$$x_{22} = -37.4928479164893$$
$$x_{23} = -97.4970345019427$$
$$x_{24} = -27.490647275511$$
$$x_{25} = -49.4944226871506$$
$$x_{26} = -45.4939809198007$$
$$x_{27} = -71.4960266202515$$
$$x_{28} = -11.4815779712047$$
$$x_{29} = -41.4934631494683$$
$$x_{30} = -81.4964859705107$$
$$x_{31} = -31.4916722315682$$
$$x_{32} = -33.4921048428995$$
$$x_{33} = -59.4952873868124$$
$$x_{34} = -21.4885296777534$$
$$x_{35} = -87.4967139075584$$
$$x_{36} = -9.47903700525443$$
$$x_{37} = -61.4954291195393$$
$$x_{38} = -55.4949758090825$$
$$x_{39} = -47.4942102180917$$
$$x_{40} = -99.4970912578863$$
$$x_{41} = -53.4948040418727$$
$$x_{42} = -73.4961278528563$$
$$x_{43} = -91.4968501186439$$
$$x_{44} = -69.495919952339$$
$$x_{45} = -51.4946201141526$$
$$x_{46} = -1.45323686184935$$
$$x_{47} = -63.4955625759026$$
$$x_{48} = -95.496975487056$$
$$x_{49} = -3.46423970898458$$
$$x_{50} = -29.4911894624069$$
$$x_{51} = -13.4835695403804$$
$$x_{52} = -79.4964027982152$$
Puntos máximos de la función:
$$x_{52} = 22.4826306576559$$
$$x_{52} = 6.29811538749029$$
$$x_{52} = 98.496749052932$$
$$x_{52} = 10.4447350189979$$
$$x_{52} = 38.4909264612995$$
$$x_{52} = 92.4965261308476$$
$$x_{52} = 76.4957487615285$$
$$x_{52} = 80.4959739927052$$
$$x_{52} = 90.4964448707506$$
$$x_{52} = 32.4889467803235$$
$$x_{52} = 74.4956264237311$$
$$x_{52} = 26.4858621616449$$
$$x_{52} = 12.4594717635981$$
$$x_{52} = 86.4962703858798$$
$$x_{52} = 66.4950575032433$$
$$x_{52} = 68.4952131724318$$
$$x_{52} = 58.4943184383132$$
$$x_{52} = 18.4774841939451$$
$$x_{52} = 54.4938593222224$$
$$x_{52} = 34.4896961510675$$
$$x_{52} = 24.4844121275861$$
$$x_{52} = 42.4918943053844$$
$$x_{52} = 96.4966779939798$$
$$x_{52} = 100.496817135593$$
$$x_{52} = 94.4966037592085$$
$$x_{52} = 46.4926755771317$$
$$x_{52} = 28.4870653815954$$
$$x_{52} = 72.4954968362866$$
$$x_{52} = 16.4735684148199$$
$$x_{52} = 50.493319482426$$
$$x_{52} = 82.4960778896675$$
$$x_{52} = 62.4947136773841$$
$$x_{52} = 44.4923047202881$$
$$x_{52} = 60.4945231792732$$
$$x_{52} = 78.4958644414866$$
$$x_{52} = 88.4963597179543$$
$$x_{52} = 20.4803894546078$$
$$x_{52} = 30.488079861269$$
$$x_{52} = 48.4930123321987$$
$$x_{52} = 64.4948913688995$$
$$x_{52} = 70.495359335105$$
$$x_{52} = 40.4914376465524$$
$$x_{52} = 14.4680039094037$$
$$x_{52} = 52.4936007673721$$
$$x_{52} = 8.41316403653945$$
$$x_{52} = 36.4903503636438$$
$$x_{52} = 84.4961765591096$$
$$x_{52} = 56.4940977951431$$
Decrece en los intervalos
$$\left[2.62152720496008, 6.29811538749029\right]$$
Crece en los intervalos
$$\left(-\infty, -99.4970912578863\right]$$
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
$$\frac{- \frac{\pi^{2} \left(3 \sqrt[3]{\sin{\left(\pi x \right)}} + \frac{2 \cos^{2}{\left(\pi x \right)}}{\sin^{\frac{5}{3}}{\left(\pi x \right)}}\right)}{9} - \frac{2 \pi \cos{\left(\pi x \right)}}{3 \left(x - 5\right) \sin^{\frac{2}{3}}{\left(\pi x \right)}} + \frac{2 \sqrt[3]{\sin{\left(\pi x \right)}}}{\left(x - 5\right)^{2}}}{x - 5} = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{- \frac{\pi^{2} \left(3 \sqrt[3]{\sin{\left(\pi x \right)}} + \frac{2 \cos^{2}{\left(\pi x \right)}}{\sin^{\frac{5}{3}}{\left(\pi x \right)}}\right)}{9} - \frac{2 \pi \cos{\left(\pi x \right)}}{3 \left(x - 5\right) \sin^{\frac{2}{3}}{\left(\pi x \right)}} + \frac{2 \sqrt[3]{\sin{\left(\pi x \right)}}}{\left(x - 5\right)^{2}}}{x - 5} = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas verticales
Hay:
$$x_{1} = 5$$
$$x_{1} = 5$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt[3]{\sin{\left(\pi x \right)}}}{x - 5}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sqrt[3]{\sin{\left(\pi x \right)}}}{x - 5}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\sqrt[3]{\sin{\left(\pi x \right)}}}{x - 5}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sqrt[3]{\sin{\left(\pi x \right)}}}{x - 5}\right) = 0$$
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(pi*x)^(1/3)/(-5 + x), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt[3]{\sin{\left(\pi x \right)}}}{x \left(x - 5\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sqrt[3]{\sin{\left(\pi x \right)}}}{x \left(x - 5\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\sqrt[3]{\sin{\left(\pi x \right)}}}{x \left(x - 5\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sqrt[3]{\sin{\left(\pi x \right)}}}{x \left(x - 5\right)}\right) = 0$$
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:
$$\frac{\sqrt[3]{\sin{\left(\pi x \right)}}}{x - 5} = \frac{\sqrt[3]{- \sin{\left(\pi x \right)}}}{- x - 5}$$
- No
$$\frac{\sqrt[3]{\sin{\left(\pi x \right)}}}{x - 5} = - \frac{\sqrt[3]{- \sin{\left(\pi x \right)}}}{- x - 5}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\sqrt[3]{\sin{\left(\pi x \right)}}}{x - 5} = \frac{\sqrt[3]{- \sin{\left(\pi x \right)}}}{- x - 5}$$
- No
$$\frac{\sqrt[3]{\sin{\left(\pi x \right)}}}{x - 5} = - \frac{\sqrt[3]{- \sin{\left(\pi x \right)}}}{- x - 5}$$
- No
es decir, función
no es
par ni impar