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- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x^3/3-4*x
-
x^3-6*x^2+9*x+1
-
y=x+2
-
(x^3-x^2)/(4-x^2)
-
Expresiones idénticas
- sin(x)^ tres *((-atan(x))/ tres +x/ tres)
- seno de (x) al cubo multiplicar por (( menos arco tangente de gente de (x)) dividir por 3 más x dividir por 3)
- seno de (x) en el grado tres multiplicar por (( menos arco tangente de gente de (x)) dividir por tres más x dividir por tres)
- sin(x)3*((-atan(x))/3+x/3)
- sinx3*-atanx/3+x/3
- sin(x)³*((-atan(x))/3+x/3)
- sin(x) en el grado 3*((-atan(x))/3+x/3)
- sin(x)^3((-atan(x))/3+x/3)
- sin(x)3((-atan(x))/3+x/3)
- sinx3-atanx/3+x/3
- sinx^3-atanx/3+x/3
- sin(x)^3*((-atan(x)) dividir por 3+x dividir por 3)
-
Expresiones semejantes
- sin(x)^3*((atan(x))/3+x/3)
- sin(x)^3*((-atan(x))/3-x/3)
- sin(x)^3*((-arctan(x))/3+x/3)
- sinx^3*((-arctanx)/3+x/3)
-
Expresiones con funciones
- Seno sin
- sin(x)^(1/3)
- sin(x)+3*cos(x)
- sin(x)^(2)*cos(x)^(2)
- sin(x)^4+cos(x)^4
- sin(x)+5*cos(x)
- Arcotangente arctan
- atan(5*x)/asin(6*x)
- atan((x+1)/((x*(x-1)^2)))
- atan(sqrt((4.00/x^2)-1))-atan(sqrt((1/x^2)-1))-pi/4
- atan(x^2-1/(x^2+1))
- atan((1-x)/(1+x))^-2
Gráfico de la función y = sin(x)^3*((-atan(x))/3+x/3)
Solución
Ha introducido
[src]
3 /-atan(x) x\
f(x) = sin (x)*|--------- + -|
\ 3 3/
$$f{\left(x \right)} = \left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)}$$
f = (x/3 + (-atan(x))/3)*sin(x)^3
Gráfico de la función
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:
$$\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 28.274327539771$$
$$x_{2} = 21.9911516425052$$
$$x_{3} = 43.9823032542865$$
$$x_{4} = -12.5664316443175$$
$$x_{5} = -59.690275753337$$
$$x_{6} = -56.5486761233879$$
$$x_{7} = -81.6813767358288$$
$$x_{8} = 72.2566292958367$$
$$x_{9} = -9.42485171603722$$
$$x_{10} = -78.5398074796132$$
$$x_{11} = -59.6902445763065$$
$$x_{12} = -15.7079741677484$$
$$x_{13} = -34.557546390117$$
$$x_{14} = 94.2477801894889$$
$$x_{15} = 9.42482919322591$$
$$x_{16} = 65.9734548151346$$
$$x_{17} = 87.9646063132303$$
$$x_{18} = 6.28325837390066$$
$$x_{19} = -37.6991249695058$$
$$x_{20} = 31.4158973414617$$
$$x_{21} = -3.14147637996428$$
$$x_{22} = -6.283115394968$$
$$x_{23} = -43.9823032321748$$
$$x_{24} = -21.9911516410072$$
$$x_{25} = 3.14163928523582$$
$$x_{26} = 0$$
$$x_{27} = 12.5663013873734$$
$$x_{28} = -81.6814265139205$$
$$x_{29} = -65.9734547046097$$
$$x_{30} = 50.2654784098211$$
$$x_{31} = 3.14157419664107$$
$$x_{32} = 6.28317671486296$$
$$x_{33} = -87.9646059653208$$
o sea hay que resolver la ecuación:
$$\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 28.274327539771$$
$$x_{2} = 21.9911516425052$$
$$x_{3} = 43.9823032542865$$
$$x_{4} = -12.5664316443175$$
$$x_{5} = -59.690275753337$$
$$x_{6} = -56.5486761233879$$
$$x_{7} = -81.6813767358288$$
$$x_{8} = 72.2566292958367$$
$$x_{9} = -9.42485171603722$$
$$x_{10} = -78.5398074796132$$
$$x_{11} = -59.6902445763065$$
$$x_{12} = -15.7079741677484$$
$$x_{13} = -34.557546390117$$
$$x_{14} = 94.2477801894889$$
$$x_{15} = 9.42482919322591$$
$$x_{16} = 65.9734548151346$$
$$x_{17} = 87.9646063132303$$
$$x_{18} = 6.28325837390066$$
$$x_{19} = -37.6991249695058$$
$$x_{20} = 31.4158973414617$$
$$x_{21} = -3.14147637996428$$
$$x_{22} = -6.283115394968$$
$$x_{23} = -43.9823032321748$$
$$x_{24} = -21.9911516410072$$
$$x_{25} = 3.14163928523582$$
$$x_{26} = 0$$
$$x_{27} = 12.5663013873734$$
$$x_{28} = -81.6814265139205$$
$$x_{29} = -65.9734547046097$$
$$x_{30} = 50.2654784098211$$
$$x_{31} = 3.14157419664107$$
$$x_{32} = 6.28317671486296$$
$$x_{33} = -87.9646059653208$$
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
$$\left(\frac{1}{3} - \frac{1}{3 \left(x^{2} + 1\right)}\right) \sin^{3}{\left(x \right)} + 3 \left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -47.1238897398428$$
$$x_{2} = 72.2566310277228$$
$$x_{3} = 15.7079634238123$$
$$x_{4} = 97.3893722386395$$
$$x_{5} = -42.4196515709562$$
$$x_{6} = 14.1633536976303$$
$$x_{7} = -43.982297174703$$
$$x_{8} = 95.8221117869534$$
$$x_{9} = 34.5575190521566$$
$$x_{10} = -36.1379432026438$$
$$x_{11} = 37.6991119965993$$
$$x_{12} = 40.8407044980859$$
$$x_{13} = -34.5575191251925$$
$$x_{14} = 12.5663704809964$$
$$x_{15} = -51.8429042612973$$
$$x_{16} = 36.1379432026438$$
$$x_{17} = -7.9047266518689$$
$$x_{18} = -78.5398162431523$$
$$x_{19} = 56.5486676281888$$
$$x_{20} = -59.6902604573859$$
$$x_{21} = -100.530964808226$$
$$x_{22} = -53.407075259837$$
$$x_{23} = 18.8495559923432$$
$$x_{24} = -29.8568869013277$$
$$x_{25} = -31.4159266845618$$
$$x_{26} = -12.5663706164231$$
$$x_{27} = -65.9734457652762$$
$$x_{28} = 20.4379300963978$$
$$x_{29} = 58.1253545010966$$
$$x_{30} = 6.28318528466365$$
$$x_{31} = -23.577034707719$$
$$x_{32} = -9.42477811595977$$
$$x_{33} = -3.14159285481255$$
$$x_{34} = 4.80501573129414$$
$$x_{35} = -95.8221117869534$$
$$x_{36} = -87.9645943592141$$
$$x_{37} = -45.5606634099038$$
$$x_{38} = 94.2477796093528$$
$$x_{39} = -80.1148552083624$$
$$x_{40} = -43.9822972046548$$
$$x_{41} = 73.8320385087492$$
$$x_{42} = 21.9911485851914$$
$$x_{43} = -15.7079632965462$$
$$x_{44} = 51.8429042612973$$
$$x_{45} = -6.28318518053504$$
$$x_{46} = -21.9911485864908$$
$$x_{47} = -89.5391788958977$$
$$x_{48} = -73.8320385087492$$
$$x_{49} = 62.8318530430108$$
$$x_{50} = 64.4079514776374$$
$$x_{51} = 75.398223692586$$
$$x_{52} = 53.4070751439253$$
$$x_{53} = 7.9047266518689$$
$$x_{54} = 0$$
$$x_{55} = -25.1327412189989$$
$$x_{56} = 80.1148552083624$$
$$x_{57} = 28.2743338652787$$
$$x_{58} = -67.549291920442$$
$$x_{59} = -14.1633536976303$$
$$x_{60} = 65.9734457527599$$
$$x_{61} = -69.1150382652533$$
$$x_{62} = -1.88455017054346$$
$$x_{63} = 1.88455017054346$$
$$x_{64} = 9.42477807889789$$
$$x_{65} = 92.680640992997$$
$$x_{66} = -87.9645942994217$$
$$x_{67} = -28.2743337405708$$
$$x_{68} = 84.8230015969263$$
$$x_{69} = -37.6991118770157$$
$$x_{70} = -81.6814090376207$$
$$x_{71} = -58.1253545010966$$
$$x_{72} = 87.9645943354964$$
$$x_{73} = -50.2654823161497$$
$$x_{74} = 43.9822971693753$$
$$x_{75} = 29.8568869013277$$
$$x_{76} = 31.41592659636$$
$$x_{77} = 86.397726460829$$
$$x_{78} = 42.4196515709562$$
$$x_{79} = -56.5486676809069$$
$$x_{80} = 50.2654824463717$$
$$x_{81} = 21.9911485108066$$
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:
$$x_{1} = -42.4196515709562$$
$$x_{2} = -36.1379432026438$$
$$x_{3} = 36.1379432026438$$
$$x_{4} = -29.8568869013277$$
$$x_{5} = -23.577034707719$$
$$x_{6} = 4.80501573129414$$
$$x_{7} = -80.1148552083624$$
$$x_{8} = 73.8320385087492$$
$$x_{9} = -73.8320385087492$$
$$x_{10} = 0$$
$$x_{11} = 80.1148552083624$$
$$x_{12} = -67.549291920442$$
$$x_{13} = 92.680640992997$$
$$x_{14} = 29.8568869013277$$
$$x_{15} = 86.397726460829$$
$$x_{16} = 42.4196515709562$$
Puntos máximos de la función:
$$x_{16} = 14.1633536976303$$
$$x_{16} = 95.8221117869534$$
$$x_{16} = -51.8429042612973$$
$$x_{16} = -7.9047266518689$$
$$x_{16} = 20.4379300963978$$
$$x_{16} = 58.1253545010966$$
$$x_{16} = -95.8221117869534$$
$$x_{16} = -45.5606634099038$$
$$x_{16} = 51.8429042612973$$
$$x_{16} = -89.5391788958977$$
$$x_{16} = 64.4079514776374$$
$$x_{16} = 7.9047266518689$$
$$x_{16} = -14.1633536976303$$
$$x_{16} = -1.88455017054346$$
$$x_{16} = 1.88455017054346$$
$$x_{16} = -58.1253545010966$$
Decrece en los intervalos
$$\left[92.680640992997, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -80.1148552083624\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\left(\frac{1}{3} - \frac{1}{3 \left(x^{2} + 1\right)}\right) \sin^{3}{\left(x \right)} + 3 \left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -47.1238897398428$$
$$x_{2} = 72.2566310277228$$
$$x_{3} = 15.7079634238123$$
$$x_{4} = 97.3893722386395$$
$$x_{5} = -42.4196515709562$$
$$x_{6} = 14.1633536976303$$
$$x_{7} = -43.982297174703$$
$$x_{8} = 95.8221117869534$$
$$x_{9} = 34.5575190521566$$
$$x_{10} = -36.1379432026438$$
$$x_{11} = 37.6991119965993$$
$$x_{12} = 40.8407044980859$$
$$x_{13} = -34.5575191251925$$
$$x_{14} = 12.5663704809964$$
$$x_{15} = -51.8429042612973$$
$$x_{16} = 36.1379432026438$$
$$x_{17} = -7.9047266518689$$
$$x_{18} = -78.5398162431523$$
$$x_{19} = 56.5486676281888$$
$$x_{20} = -59.6902604573859$$
$$x_{21} = -100.530964808226$$
$$x_{22} = -53.407075259837$$
$$x_{23} = 18.8495559923432$$
$$x_{24} = -29.8568869013277$$
$$x_{25} = -31.4159266845618$$
$$x_{26} = -12.5663706164231$$
$$x_{27} = -65.9734457652762$$
$$x_{28} = 20.4379300963978$$
$$x_{29} = 58.1253545010966$$
$$x_{30} = 6.28318528466365$$
$$x_{31} = -23.577034707719$$
$$x_{32} = -9.42477811595977$$
$$x_{33} = -3.14159285481255$$
$$x_{34} = 4.80501573129414$$
$$x_{35} = -95.8221117869534$$
$$x_{36} = -87.9645943592141$$
$$x_{37} = -45.5606634099038$$
$$x_{38} = 94.2477796093528$$
$$x_{39} = -80.1148552083624$$
$$x_{40} = -43.9822972046548$$
$$x_{41} = 73.8320385087492$$
$$x_{42} = 21.9911485851914$$
$$x_{43} = -15.7079632965462$$
$$x_{44} = 51.8429042612973$$
$$x_{45} = -6.28318518053504$$
$$x_{46} = -21.9911485864908$$
$$x_{47} = -89.5391788958977$$
$$x_{48} = -73.8320385087492$$
$$x_{49} = 62.8318530430108$$
$$x_{50} = 64.4079514776374$$
$$x_{51} = 75.398223692586$$
$$x_{52} = 53.4070751439253$$
$$x_{53} = 7.9047266518689$$
$$x_{54} = 0$$
$$x_{55} = -25.1327412189989$$
$$x_{56} = 80.1148552083624$$
$$x_{57} = 28.2743338652787$$
$$x_{58} = -67.549291920442$$
$$x_{59} = -14.1633536976303$$
$$x_{60} = 65.9734457527599$$
$$x_{61} = -69.1150382652533$$
$$x_{62} = -1.88455017054346$$
$$x_{63} = 1.88455017054346$$
$$x_{64} = 9.42477807889789$$
$$x_{65} = 92.680640992997$$
$$x_{66} = -87.9645942994217$$
$$x_{67} = -28.2743337405708$$
$$x_{68} = 84.8230015969263$$
$$x_{69} = -37.6991118770157$$
$$x_{70} = -81.6814090376207$$
$$x_{71} = -58.1253545010966$$
$$x_{72} = 87.9645943354964$$
$$x_{73} = -50.2654823161497$$
$$x_{74} = 43.9822971693753$$
$$x_{75} = 29.8568869013277$$
$$x_{76} = 31.41592659636$$
$$x_{77} = 86.397726460829$$
$$x_{78} = 42.4196515709562$$
$$x_{79} = -56.5486676809069$$
$$x_{80} = 50.2654824463717$$
$$x_{81} = 21.9911485108066$$
Signos de extremos en los puntos:
(-47.12388973984283, 3.98310346097366e-21)
(72.25663102772283, 2.67597069160318e-24)
(15.707963423812345, -1.79234694014091e-20)
(97.38937223863952, 3.70884132285647e-22)
(-42.419651570956205, -13.622783998703)
(14.163353697630292, 4.21667496713117)
(-43.982297174703014, 2.06639646233012e-22)
(95.82211178695336, 31.4199944574926)
(34.557519052156564, 2.85040058445483e-20)
(-36.13794320264376, -11.5300006037631)
(37.699111996599285, 4.36068792687663e-20)
(40.84070449808586, -3.73890091284254e-26)
(-34.55751912519251, 2.92505830313435e-21)
(12.566370480996397, -8.75640298167247e-21)
(-51.842904261297335, 16.7626944041327)
(36.13794320264376, -11.5300006037631)
(-7.904726651868903, 2.14495149966393)
(-78.53981624315229, 2.31258304446127e-20)
(56.54866762818882, -4.65491461069023e-20)
(-59.69026045738589, -1.16550125376709e-21)
(-100.5309648082264, -4.00155274285572e-20)
(-53.40707525983695, -5.69599789226226e-20)
(18.849555992343227, 2.05070614248904e-21)
(-29.85688690132773, -9.43790005122283)
(-31.41592668456182, 3.27213702463331e-20)
(-12.566370616423054, 3.24545686181302e-26)
(-65.97344576527618, -1.36299383969224e-21)
(20.43793009639784, 6.30241932128866)
(58.12535450109656, 18.8562721445462)
(6.283185284663652, -1.8530931632127e-23)
(-23.577034707718983, -7.34703244897591)
(-9.424778115959775, -9.91675067455694e-21)
(-3.1415928548125533, -5.10303885725562e-21)
(4.805015731294137, -1.13178788432727)
(-95.82211178695336, 31.4199944574926)
(-87.96459435921413, 5.82547059105447e-21)
(-45.560663409903775, 14.6693431242966)
(94.24777960935283, 1.41080617543143e-25)
(-80.1148552083624, -26.1848064691713)
(-43.98229720465481, 2.276863022088e-21)
(73.83203850874915, -24.0908268341739)
(21.991148585191436, -6.95143729038979e-24)
(-15.707963296546192, -1.1070354004941e-22)
(51.842904261297335, 16.7626944041327)
(-6.28318518053504, -3.29751523230015e-21)
(-21.991148586490784, -1.00068687626426e-23)
(-89.53917889589773, 29.3258855090146)
(-73.83203850874915, -24.0908268341739)
(62.83185304301077, -4.87168450060204e-22)
(64.40795147763735, 20.9500098681499)
(75.39822369258601, 6.54644327525439e-24)
(53.40707514392534, -6.15476868236516e-22)
(7.904726651868903, 2.14495149966393)
(0, 0)
(-25.132741218998916, -7.22344924354435e-24)
(80.1148552083624, -26.1848064691713)
(28.27433386527867, 4.40175054211864e-23)
(-67.54929192044199, -21.996924731538)
(-14.163353697630292, 4.21667496713117)
(65.97344575275994, -4.40466653095097e-22)
(-69.11503826525328, -3.31203997990776e-20)
(-1.8845501705434633, 0.22994789474801)
(1.8845501705434633, 0.22994789474801)
(9.424778078897885, -4.37359944257925e-21)
(92.68064099299698, -30.3729351221931)
(-87.96459429942173, -3.75539371192506e-26)
(-28.27433374057076, 2.53790472959541e-20)
(84.82300159692633, 3.46893455991417e-21)
(-37.699111877015746, 4.71099422940883e-22)
(-81.68140903762065, 2.31971907254417e-21)
(-58.12535450109656, 18.8562721445462)
(87.96459433549641, 1.23299113518815e-21)
(-50.26548231614965, -4.57978063086762e-20)
(43.98229716937529, 9.88405807000468e-23)
(29.85688690132773, -9.43790005122283)
(31.415926596359974, 2.20121989284558e-21)
(86.39772646082896, -28.2788466810656)
(42.419651570956205, -13.622783998703)
(-56.54866768090694, -1.07529612384756e-20)
(50.26548244637171, -2.19983144722745e-23)
(21.99114851080661, 1.81545208169053e-21)
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:
$$x_{1} = -42.4196515709562$$
$$x_{2} = -36.1379432026438$$
$$x_{3} = 36.1379432026438$$
$$x_{4} = -29.8568869013277$$
$$x_{5} = -23.577034707719$$
$$x_{6} = 4.80501573129414$$
$$x_{7} = -80.1148552083624$$
$$x_{8} = 73.8320385087492$$
$$x_{9} = -73.8320385087492$$
$$x_{10} = 0$$
$$x_{11} = 80.1148552083624$$
$$x_{12} = -67.549291920442$$
$$x_{13} = 92.680640992997$$
$$x_{14} = 29.8568869013277$$
$$x_{15} = 86.397726460829$$
$$x_{16} = 42.4196515709562$$
Puntos máximos de la función:
$$x_{16} = 14.1633536976303$$
$$x_{16} = 95.8221117869534$$
$$x_{16} = -51.8429042612973$$
$$x_{16} = -7.9047266518689$$
$$x_{16} = 20.4379300963978$$
$$x_{16} = 58.1253545010966$$
$$x_{16} = -95.8221117869534$$
$$x_{16} = -45.5606634099038$$
$$x_{16} = 51.8429042612973$$
$$x_{16} = -89.5391788958977$$
$$x_{16} = 64.4079514776374$$
$$x_{16} = 7.9047266518689$$
$$x_{16} = -14.1633536976303$$
$$x_{16} = -1.88455017054346$$
$$x_{16} = 1.88455017054346$$
$$x_{16} = -58.1253545010966$$
Decrece en los intervalos
$$\left[92.680640992997, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -80.1148552083624\right]$$
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
$$\left(\frac{2 x \sin^{2}{\left(x \right)}}{3 \left(x^{2} + 1\right)^{2}} + 2 \left(1 - \frac{1}{x^{2} + 1}\right) \sin{\left(x \right)} \cos{\left(x \right)} - \left(x - \operatorname{atan}{\left(x \right)}\right) \left(\sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right)\right) \sin{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -54.3686863927772$$
$$x_{2} = -50.2654824574367$$
$$x_{3} = -5.40936978846495$$
$$x_{4} = -2.4141722102196$$
$$x_{5} = 85.782269852414$$
$$x_{6} = -19.8229153148452$$
$$x_{7} = -55.599529764795$$
$$x_{8} = 76.35798871949$$
$$x_{9} = -39.8941003995144$$
$$x_{10} = 72.2566310325652$$
$$x_{11} = -83.8717397472183$$
$$x_{12} = -43.9822971502571$$
$$x_{13} = 0.000406608608637562$$
$$x_{14} = -63.7925138805732$$
$$x_{15} = -37.6991118430775$$
$$x_{16} = -12.5663706143592$$
$$x_{17} = -94.2477796076938$$
$$x_{18} = -68.1647337177157$$
$$x_{19} = -21.9911485751286$$
$$x_{20} = 50.2654824574367$$
$$x_{21} = -15.707963267949$$
$$x_{22} = 21.9911485751286$$
$$x_{23} = -87.9645943005142$$
$$x_{24} = -72.2566310325652$$
$$x_{25} = -24.1921794603074$$
$$x_{26} = 4.20190394940552$$
$$x_{27} = -4.20190394940552$$
$$x_{28} = 79.4994024460026$$
$$x_{29} = 84.8230016469244$$
$$x_{30} = 96.4375728840975$$
$$x_{31} = -92.0651813066377$$
$$x_{32} = 68.1647337177157$$
$$x_{33} = 54.3686863927772$$
$$x_{34} = -13.548906564136$$
$$x_{35} = 26.1015373766289$$
$$x_{36} = -33.6126232938398$$
$$x_{37} = 36.7532857268625$$
$$x_{38} = 30.4721625114502$$
$$x_{39} = -77.5888898474853$$
$$x_{40} = -82.6408300049229$$
$$x_{41} = 83.8717397472183$$
$$x_{42} = -57.5099267847335$$
$$x_{43} = -46.1760581313926$$
$$x_{44} = -65.9734457253857$$
$$x_{45} = -70.0752103496049$$
$$x_{46} = 15.707963267949$$
$$x_{47} = -17.9146341508486$$
$$x_{48} = 28.2743338823081$$
$$x_{49} = 94.2477796076938$$
$$x_{50} = 6.28318530717959$$
$$x_{51} = 32.3819978743262$$
$$x_{52} = -51.2274903560907$$
$$x_{53} = 41.8042715244945$$
$$x_{54} = 52.4583188066494$$
$$x_{55} = -79.4994024460026$$
$$x_{56} = 19.8229153148452$$
$$x_{57} = -75.398223686155$$
$$x_{58} = 70.0752103496049$$
$$x_{59} = -99.5790527132113$$
$$x_{60} = 0$$
$$x_{61} = 61.8820709651112$$
$$x_{62} = -41.8042715244945$$
$$x_{63} = -34.5575191894877$$
$$x_{64} = 98.348128249113$$
$$x_{65} = -348.716784548467$$
$$x_{66} = 22.9618974346408$$
$$x_{67} = -28.2743338823081$$
$$x_{68} = 39.8941003995144$$
$$x_{69} = 10.4165024727153$$
$$x_{70} = 46.1760581313926$$
$$x_{71} = -85.782269852414$$
$$x_{72} = -61.8820709651112$$
$$x_{73} = 8.51661058959281$$
$$x_{74} = 63.7925138805732$$
$$x_{75} = 57.5099267847335$$
$$x_{76} = -26.1015373766289$$
$$x_{77} = 90.1546372456078$$
$$x_{78} = -8.51661058959281$$
$$x_{79} = 12.5663706143592$$
$$x_{80} = 87.9645943005142$$
$$x_{81} = -100.530964914873$$
$$x_{82} = -11.6439446877948$$
$$x_{83} = 43.9822971502571$$
$$x_{84} = -81.6814089933346$$
$$x_{85} = -35.522602636666$$
$$x_{86} = 24.1921794603074$$
$$x_{87} = 48.086347606213$$
$$x_{88} = -59.6902604182061$$
$$x_{89} = -48.086347606213$$
$$x_{90} = 65.9734457253857$$
$$x_{91} = 74.4474865946004$$
$$x_{92} = 17.9146341508486$$
$$x_{93} = -90.1546372456078$$
$$x_{94} = 2.4141722102196$$
$$x_{95} = 92.0651813066377$$
$$x_{96} = -47.1238898038469$$
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
$$\left[98.348128249113, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -348.716784548467\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\left(\frac{2 x \sin^{2}{\left(x \right)}}{3 \left(x^{2} + 1\right)^{2}} + 2 \left(1 - \frac{1}{x^{2} + 1}\right) \sin{\left(x \right)} \cos{\left(x \right)} - \left(x - \operatorname{atan}{\left(x \right)}\right) \left(\sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right)\right) \sin{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -54.3686863927772$$
$$x_{2} = -50.2654824574367$$
$$x_{3} = -5.40936978846495$$
$$x_{4} = -2.4141722102196$$
$$x_{5} = 85.782269852414$$
$$x_{6} = -19.8229153148452$$
$$x_{7} = -55.599529764795$$
$$x_{8} = 76.35798871949$$
$$x_{9} = -39.8941003995144$$
$$x_{10} = 72.2566310325652$$
$$x_{11} = -83.8717397472183$$
$$x_{12} = -43.9822971502571$$
$$x_{13} = 0.000406608608637562$$
$$x_{14} = -63.7925138805732$$
$$x_{15} = -37.6991118430775$$
$$x_{16} = -12.5663706143592$$
$$x_{17} = -94.2477796076938$$
$$x_{18} = -68.1647337177157$$
$$x_{19} = -21.9911485751286$$
$$x_{20} = 50.2654824574367$$
$$x_{21} = -15.707963267949$$
$$x_{22} = 21.9911485751286$$
$$x_{23} = -87.9645943005142$$
$$x_{24} = -72.2566310325652$$
$$x_{25} = -24.1921794603074$$
$$x_{26} = 4.20190394940552$$
$$x_{27} = -4.20190394940552$$
$$x_{28} = 79.4994024460026$$
$$x_{29} = 84.8230016469244$$
$$x_{30} = 96.4375728840975$$
$$x_{31} = -92.0651813066377$$
$$x_{32} = 68.1647337177157$$
$$x_{33} = 54.3686863927772$$
$$x_{34} = -13.548906564136$$
$$x_{35} = 26.1015373766289$$
$$x_{36} = -33.6126232938398$$
$$x_{37} = 36.7532857268625$$
$$x_{38} = 30.4721625114502$$
$$x_{39} = -77.5888898474853$$
$$x_{40} = -82.6408300049229$$
$$x_{41} = 83.8717397472183$$
$$x_{42} = -57.5099267847335$$
$$x_{43} = -46.1760581313926$$
$$x_{44} = -65.9734457253857$$
$$x_{45} = -70.0752103496049$$
$$x_{46} = 15.707963267949$$
$$x_{47} = -17.9146341508486$$
$$x_{48} = 28.2743338823081$$
$$x_{49} = 94.2477796076938$$
$$x_{50} = 6.28318530717959$$
$$x_{51} = 32.3819978743262$$
$$x_{52} = -51.2274903560907$$
$$x_{53} = 41.8042715244945$$
$$x_{54} = 52.4583188066494$$
$$x_{55} = -79.4994024460026$$
$$x_{56} = 19.8229153148452$$
$$x_{57} = -75.398223686155$$
$$x_{58} = 70.0752103496049$$
$$x_{59} = -99.5790527132113$$
$$x_{60} = 0$$
$$x_{61} = 61.8820709651112$$
$$x_{62} = -41.8042715244945$$
$$x_{63} = -34.5575191894877$$
$$x_{64} = 98.348128249113$$
$$x_{65} = -348.716784548467$$
$$x_{66} = 22.9618974346408$$
$$x_{67} = -28.2743338823081$$
$$x_{68} = 39.8941003995144$$
$$x_{69} = 10.4165024727153$$
$$x_{70} = 46.1760581313926$$
$$x_{71} = -85.782269852414$$
$$x_{72} = -61.8820709651112$$
$$x_{73} = 8.51661058959281$$
$$x_{74} = 63.7925138805732$$
$$x_{75} = 57.5099267847335$$
$$x_{76} = -26.1015373766289$$
$$x_{77} = 90.1546372456078$$
$$x_{78} = -8.51661058959281$$
$$x_{79} = 12.5663706143592$$
$$x_{80} = 87.9645943005142$$
$$x_{81} = -100.530964914873$$
$$x_{82} = -11.6439446877948$$
$$x_{83} = 43.9822971502571$$
$$x_{84} = -81.6814089933346$$
$$x_{85} = -35.522602636666$$
$$x_{86} = 24.1921794603074$$
$$x_{87} = 48.086347606213$$
$$x_{88} = -59.6902604182061$$
$$x_{89} = -48.086347606213$$
$$x_{90} = 65.9734457253857$$
$$x_{91} = 74.4474865946004$$
$$x_{92} = 17.9146341508486$$
$$x_{93} = -90.1546372456078$$
$$x_{94} = 2.4141722102196$$
$$x_{95} = 92.0651813066377$$
$$x_{96} = -47.1238898038469$$
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
$$\left[98.348128249113, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -348.716784548467\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty}\left(\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x)^3*((-atan(x))/3 + x/3), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)}}{x}\right) = \left\langle - \frac{1}{3}, \frac{1}{3}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle - \frac{1}{3}, \frac{1}{3}\right\rangle x$$
$$\lim_{x \to \infty}\left(\frac{\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)}}{x}\right) = \left\langle - \frac{1}{3}, \frac{1}{3}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle - \frac{1}{3}, \frac{1}{3}\right\rangle x$$
$$\lim_{x \to -\infty}\left(\frac{\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)}}{x}\right) = \left\langle - \frac{1}{3}, \frac{1}{3}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle - \frac{1}{3}, \frac{1}{3}\right\rangle x$$
$$\lim_{x \to \infty}\left(\frac{\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)}}{x}\right) = \left\langle - \frac{1}{3}, \frac{1}{3}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle - \frac{1}{3}, \frac{1}{3}\right\rangle x$$
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:
$$\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)} = - \left(- \frac{x}{3} + \frac{\operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)}$$
- No
$$\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)} = \left(- \frac{x}{3} + \frac{\operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)} = - \left(- \frac{x}{3} + \frac{\operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)}$$
- No
$$\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)} = \left(- \frac{x}{3} + \frac{\operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)}$$
- No
es decir, función
no es
par ni impar