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- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x/(x^2-1)
-
x*exp(-x)
-
x^3/(3-x^2)
-
x^3-x^2+2
-
Expresiones idénticas
- (x*cos(x)-sin(x))/((dos *x^ tres))
- (x multiplicar por coseno de (x) menos seno de (x)) dividir por ((2 multiplicar por x al cubo ))
- (x multiplicar por coseno de (x) menos seno de (x)) dividir por ((dos multiplicar por x en el grado tres))
- (x*cos(x)-sin(x))/((2*x3))
- x*cosx-sinx/2*x3
- (x*cos(x)-sin(x))/((2*x³))
- (x*cos(x)-sin(x))/((2*x en el grado 3))
- (xcos(x)-sin(x))/((2x^3))
- (xcos(x)-sin(x))/((2x3))
- xcosx-sinx/2x3
- xcosx-sinx/2x^3
- (x*cos(x)-sin(x)) dividir por ((2*x^3))
-
Expresiones semejantes
- (x*cos(x)+sin(x))/((2*x^3))
- (x*cosx-sinx)/((2*x^3))
-
Expresiones con funciones
- Coseno cos
- cos^2(x)-cos(x)
- cosx-x
- cosx*cos2x
- cos(x)*2
- cos(x)^2-sin(x)
- Seno sin
- sin(x)/2
- sin(x)+sin(2*x)+sin(3*x)
- sin(x)+4
- sinx+cos^2x
- sinx+cos^(2)x
Gráfico de la función y = (x*cos(x)-sin(x))/((2*x^3))
Solución
Ha introducido
[src]
x*cos(x) - sin(x)
f(x) = -----------------
3
2*x
$$f{\left(x \right)} = \frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{2 x^{3}}$$
f = (x*cos(x) - sin(x))/((2*x^3))
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{2 x^{3}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = -4.49340945790906$$
$$x_{2} = 26.6660542588127$$
$$x_{3} = 86.3822220347287$$
$$x_{4} = -73.8138806006806$$
$$x_{5} = 95.8081387868617$$
$$x_{6} = -58.1022547544956$$
$$x_{7} = -86.3822220347287$$
$$x_{8} = -36.1006222443756$$
$$x_{9} = -95.8081387868617$$
$$x_{10} = -64.3871195905574$$
$$x_{11} = 4.49340945790906$$
$$x_{12} = -70.6716857116195$$
$$x_{13} = 64.3871195905574$$
$$x_{14} = -80.0981286289451$$
$$x_{15} = 567.055710479627$$
$$x_{16} = 36.1006222443756$$
$$x_{17} = -83.2401924707234$$
$$x_{18} = 70.6716857116195$$
$$x_{19} = -51.8169824872797$$
$$x_{20} = 23.519452498689$$
$$x_{21} = 10.9041216594289$$
$$x_{22} = -29.811598790893$$
$$x_{23} = 17.2207552719308$$
$$x_{24} = -48.6741442319544$$
$$x_{25} = 14.0661939128315$$
$$x_{26} = 39.2444323611642$$
$$x_{27} = -67.5294347771441$$
$$x_{28} = 158.644125673263$$
$$x_{29} = 76.9560263103312$$
$$x_{30} = 42.3879135681319$$
$$x_{31} = 83.2401924707234$$
$$x_{32} = -7.72525183693771$$
$$x_{33} = -61.2447302603744$$
$$x_{34} = -42.3879135681319$$
$$x_{35} = 92.6661922776228$$
$$x_{36} = 45.5311340139913$$
$$x_{37} = 7.72525183693771$$
$$x_{38} = -17.2207552719308$$
$$x_{39} = 32.9563890398225$$
$$x_{40} = 29.811598790893$$
$$x_{41} = -23.519452498689$$
$$x_{42} = -45.5311340139913$$
$$x_{43} = 58.1022547544956$$
$$x_{44} = -26.6660542588127$$
$$x_{45} = -98.9500628243319$$
$$x_{46} = -14.0661939128315$$
$$x_{47} = 61.2447302603744$$
$$x_{48} = 20.3713029592876$$
$$x_{49} = 54.9596782878889$$
$$x_{50} = -76.9560263103312$$
$$x_{51} = -39.2444323611642$$
$$x_{52} = -20.3713029592876$$
$$x_{53} = 51.8169824872797$$
$$x_{54} = 48.6741442319544$$
$$x_{55} = 98.9500628243319$$
$$x_{56} = -89.5242209304172$$
$$x_{57} = 89.5242209304172$$
$$x_{58} = -10.9041216594289$$
$$x_{59} = 73.8138806006806$$
$$x_{60} = -117.801235838224$$
$$x_{61} = 67.5294347771441$$
$$x_{62} = -92.6661922776228$$
$$x_{63} = -32.9563890398225$$
$$x_{64} = 80.0981286289451$$
$$x_{65} = -54.9596782878889$$
o sea hay que resolver la ecuación:
$$\frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{2 x^{3}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = -4.49340945790906$$
$$x_{2} = 26.6660542588127$$
$$x_{3} = 86.3822220347287$$
$$x_{4} = -73.8138806006806$$
$$x_{5} = 95.8081387868617$$
$$x_{6} = -58.1022547544956$$
$$x_{7} = -86.3822220347287$$
$$x_{8} = -36.1006222443756$$
$$x_{9} = -95.8081387868617$$
$$x_{10} = -64.3871195905574$$
$$x_{11} = 4.49340945790906$$
$$x_{12} = -70.6716857116195$$
$$x_{13} = 64.3871195905574$$
$$x_{14} = -80.0981286289451$$
$$x_{15} = 567.055710479627$$
$$x_{16} = 36.1006222443756$$
$$x_{17} = -83.2401924707234$$
$$x_{18} = 70.6716857116195$$
$$x_{19} = -51.8169824872797$$
$$x_{20} = 23.519452498689$$
$$x_{21} = 10.9041216594289$$
$$x_{22} = -29.811598790893$$
$$x_{23} = 17.2207552719308$$
$$x_{24} = -48.6741442319544$$
$$x_{25} = 14.0661939128315$$
$$x_{26} = 39.2444323611642$$
$$x_{27} = -67.5294347771441$$
$$x_{28} = 158.644125673263$$
$$x_{29} = 76.9560263103312$$
$$x_{30} = 42.3879135681319$$
$$x_{31} = 83.2401924707234$$
$$x_{32} = -7.72525183693771$$
$$x_{33} = -61.2447302603744$$
$$x_{34} = -42.3879135681319$$
$$x_{35} = 92.6661922776228$$
$$x_{36} = 45.5311340139913$$
$$x_{37} = 7.72525183693771$$
$$x_{38} = -17.2207552719308$$
$$x_{39} = 32.9563890398225$$
$$x_{40} = 29.811598790893$$
$$x_{41} = -23.519452498689$$
$$x_{42} = -45.5311340139913$$
$$x_{43} = 58.1022547544956$$
$$x_{44} = -26.6660542588127$$
$$x_{45} = -98.9500628243319$$
$$x_{46} = -14.0661939128315$$
$$x_{47} = 61.2447302603744$$
$$x_{48} = 20.3713029592876$$
$$x_{49} = 54.9596782878889$$
$$x_{50} = -76.9560263103312$$
$$x_{51} = -39.2444323611642$$
$$x_{52} = -20.3713029592876$$
$$x_{53} = 51.8169824872797$$
$$x_{54} = 48.6741442319544$$
$$x_{55} = 98.9500628243319$$
$$x_{56} = -89.5242209304172$$
$$x_{57} = 89.5242209304172$$
$$x_{58} = -10.9041216594289$$
$$x_{59} = 73.8138806006806$$
$$x_{60} = -117.801235838224$$
$$x_{61} = 67.5294347771441$$
$$x_{62} = -92.6661922776228$$
$$x_{63} = -32.9563890398225$$
$$x_{64} = 80.0981286289451$$
$$x_{65} = -54.9596782878889$$
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
$$- \frac{1}{2 x^{3}} x \sin{\left(x \right)} - \frac{3 \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)}{2 x^{4}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -47.0601416127605$$
$$x_{2} = 84.7876191237855$$
$$x_{3} = -28.1678297079936$$
$$x_{4} = -53.3508435852932$$
$$x_{5} = -15.5146030108867$$
$$x_{6} = 94.2159378620236$$
$$x_{7} = 28.1678297079936$$
$$x_{8} = 59.6399585795582$$
$$x_{9} = 47.0601416127605$$
$$x_{10} = -50.205728336738$$
$$x_{11} = 62.7840702561801$$
$$x_{12} = -188.47964237706$$
$$x_{13} = -12.3229409705666$$
$$x_{14} = -69.0716051946096$$
$$x_{15} = -87.9304764379571$$
$$x_{16} = 81.6446644013823$$
$$x_{17} = 97.3585583298596$$
$$x_{18} = -100.501114500159$$
$$x_{19} = -84.7876191237855$$
$$x_{20} = -59.6399585795582$$
$$x_{21} = 78.5016005602391$$
$$x_{22} = 34.470488331285$$
$$x_{23} = -78.5016005602391$$
$$x_{24} = -5.76345919689455$$
$$x_{25} = -75.3584139333214$$
$$x_{26} = -43.9139818113646$$
$$x_{27} = 15.5146030108867$$
$$x_{28} = 9.09501133047636$$
$$x_{29} = -91.0732464360163$$
$$x_{30} = 69.0716051946096$$
$$x_{31} = 21.8538742227098$$
$$x_{32} = -37.6193657535884$$
$$x_{33} = -72.2150884704073$$
$$x_{34} = 5.76345919689455$$
$$x_{35} = 50.205728336738$$
$$x_{36} = 75.3584139333214$$
$$x_{37} = -97.3585583298596$$
$$x_{38} = 31.3201417074472$$
$$x_{39} = 25.0128032022896$$
$$x_{40} = -40.7671158214068$$
$$x_{41} = 18.6890363553628$$
$$x_{42} = 100.501114500159$$
$$x_{43} = -34.470488331285$$
$$x_{44} = -21.8538742227098$$
$$x_{45} = -31.3201417074472$$
$$x_{46} = -56.495566261812$$
$$x_{47} = -65.9279415029586$$
$$x_{48} = -9.09501133047636$$
$$x_{49} = -81.6446644013823$$
$$x_{50} = -62.7840702561801$$
$$x_{51} = 56.495566261812$$
$$x_{52} = 116.213113540404$$
$$x_{53} = 40.7671158214068$$
$$x_{54} = -94.2159378620236$$
$$x_{55} = 43.9139818113646$$
$$x_{56} = 53.3508435852932$$
$$x_{57} = 65.9279415029586$$
$$x_{58} = 87.9304764379571$$
$$x_{59} = 91.0732464360163$$
$$x_{60} = -18.6890363553628$$
$$x_{61} = 72.2150884704073$$
$$x_{62} = 37.6193657535884$$
$$x_{63} = 12.3229409705666$$
$$x_{64} = -25.0128032022896$$
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:
$$x_{1} = -47.0601416127605$$
$$x_{2} = 84.7876191237855$$
$$x_{3} = -28.1678297079936$$
$$x_{4} = -53.3508435852932$$
$$x_{5} = -15.5146030108867$$
$$x_{6} = 28.1678297079936$$
$$x_{7} = 59.6399585795582$$
$$x_{8} = 47.0601416127605$$
$$x_{9} = 97.3585583298596$$
$$x_{10} = -84.7876191237855$$
$$x_{11} = -59.6399585795582$$
$$x_{12} = 78.5016005602391$$
$$x_{13} = 34.470488331285$$
$$x_{14} = -78.5016005602391$$
$$x_{15} = 15.5146030108867$$
$$x_{16} = 9.09501133047636$$
$$x_{17} = -91.0732464360163$$
$$x_{18} = 21.8538742227098$$
$$x_{19} = -72.2150884704073$$
$$x_{20} = -97.3585583298596$$
$$x_{21} = -40.7671158214068$$
$$x_{22} = -34.470488331285$$
$$x_{23} = -21.8538742227098$$
$$x_{24} = -65.9279415029586$$
$$x_{25} = -9.09501133047636$$
$$x_{26} = 116.213113540404$$
$$x_{27} = 40.7671158214068$$
$$x_{28} = 53.3508435852932$$
$$x_{29} = 65.9279415029586$$
$$x_{30} = 91.0732464360163$$
$$x_{31} = 72.2150884704073$$
Puntos máximos de la función:
$$x_{31} = 94.2159378620236$$
$$x_{31} = -50.205728336738$$
$$x_{31} = 62.7840702561801$$
$$x_{31} = -188.47964237706$$
$$x_{31} = -12.3229409705666$$
$$x_{31} = -69.0716051946096$$
$$x_{31} = -87.9304764379571$$
$$x_{31} = 81.6446644013823$$
$$x_{31} = -100.501114500159$$
$$x_{31} = -5.76345919689455$$
$$x_{31} = -75.3584139333214$$
$$x_{31} = -43.9139818113646$$
$$x_{31} = 69.0716051946096$$
$$x_{31} = -37.6193657535884$$
$$x_{31} = 5.76345919689455$$
$$x_{31} = 50.205728336738$$
$$x_{31} = 75.3584139333214$$
$$x_{31} = 31.3201417074472$$
$$x_{31} = 25.0128032022896$$
$$x_{31} = 18.6890363553628$$
$$x_{31} = 100.501114500159$$
$$x_{31} = -31.3201417074472$$
$$x_{31} = -56.495566261812$$
$$x_{31} = -81.6446644013823$$
$$x_{31} = -62.7840702561801$$
$$x_{31} = 56.495566261812$$
$$x_{31} = -94.2159378620236$$
$$x_{31} = 43.9139818113646$$
$$x_{31} = 87.9304764379571$$
$$x_{31} = -18.6890363553628$$
$$x_{31} = 37.6193657535884$$
$$x_{31} = 12.3229409705666$$
$$x_{31} = -25.0128032022896$$
Decrece en los intervalos
$$\left[116.213113540404, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -97.3585583298596\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- \frac{1}{2 x^{3}} x \sin{\left(x \right)} - \frac{3 \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)}{2 x^{4}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -47.0601416127605$$
$$x_{2} = 84.7876191237855$$
$$x_{3} = -28.1678297079936$$
$$x_{4} = -53.3508435852932$$
$$x_{5} = -15.5146030108867$$
$$x_{6} = 94.2159378620236$$
$$x_{7} = 28.1678297079936$$
$$x_{8} = 59.6399585795582$$
$$x_{9} = 47.0601416127605$$
$$x_{10} = -50.205728336738$$
$$x_{11} = 62.7840702561801$$
$$x_{12} = -188.47964237706$$
$$x_{13} = -12.3229409705666$$
$$x_{14} = -69.0716051946096$$
$$x_{15} = -87.9304764379571$$
$$x_{16} = 81.6446644013823$$
$$x_{17} = 97.3585583298596$$
$$x_{18} = -100.501114500159$$
$$x_{19} = -84.7876191237855$$
$$x_{20} = -59.6399585795582$$
$$x_{21} = 78.5016005602391$$
$$x_{22} = 34.470488331285$$
$$x_{23} = -78.5016005602391$$
$$x_{24} = -5.76345919689455$$
$$x_{25} = -75.3584139333214$$
$$x_{26} = -43.9139818113646$$
$$x_{27} = 15.5146030108867$$
$$x_{28} = 9.09501133047636$$
$$x_{29} = -91.0732464360163$$
$$x_{30} = 69.0716051946096$$
$$x_{31} = 21.8538742227098$$
$$x_{32} = -37.6193657535884$$
$$x_{33} = -72.2150884704073$$
$$x_{34} = 5.76345919689455$$
$$x_{35} = 50.205728336738$$
$$x_{36} = 75.3584139333214$$
$$x_{37} = -97.3585583298596$$
$$x_{38} = 31.3201417074472$$
$$x_{39} = 25.0128032022896$$
$$x_{40} = -40.7671158214068$$
$$x_{41} = 18.6890363553628$$
$$x_{42} = 100.501114500159$$
$$x_{43} = -34.470488331285$$
$$x_{44} = -21.8538742227098$$
$$x_{45} = -31.3201417074472$$
$$x_{46} = -56.495566261812$$
$$x_{47} = -65.9279415029586$$
$$x_{48} = -9.09501133047636$$
$$x_{49} = -81.6446644013823$$
$$x_{50} = -62.7840702561801$$
$$x_{51} = 56.495566261812$$
$$x_{52} = 116.213113540404$$
$$x_{53} = 40.7671158214068$$
$$x_{54} = -94.2159378620236$$
$$x_{55} = 43.9139818113646$$
$$x_{56} = 53.3508435852932$$
$$x_{57} = 65.9279415029586$$
$$x_{58} = 87.9304764379571$$
$$x_{59} = 91.0732464360163$$
$$x_{60} = -18.6890363553628$$
$$x_{61} = 72.2150884704073$$
$$x_{62} = 37.6193657535884$$
$$x_{63} = 12.3229409705666$$
$$x_{64} = -25.0128032022896$$
Signos de extremos en los puntos:
(-47.06014161276053, -0.000225615636669507)
(84.78761912378548, -6.95367658389369e-5)
(-28.167829707993622, -0.000628985506814357)
(-53.35084358529321, -0.000175573282983773)
(-15.514603010886749, -0.00206426604519197)
(94.21593786202358, 5.63180815586409e-5)
(28.167829707993622, -0.000628985506814357)
(59.639958579558154, -0.000140511578744414)
(47.06014161276053, -0.000225615636669507)
(-50.205728336738034, 0.00019824619552509)
(62.78407025618014, 0.000126796055545467)
(-188.4796423770602, 1.40741691940626e-5)
(-12.322940970566583, 0.00325994373348013)
(-69.0716051946096, 0.000104769366606048)
(-87.93047643795707, 6.46557092537797e-5)
(81.64466440138234, 7.49922938134965e-5)
(97.35855832985965, -5.27415632271757e-5)
(-100.50111450015908, 4.9495275149513e-5)
(-84.78761912378548, -6.95367658389369e-5)
(-59.639958579558154, -0.000140511578744414)
(78.50160056023911, -8.11161341505881e-5)
(34.47048833128499, -0.000420267871095791)
(-78.50160056023911, -8.11161341505881e-5)
(-5.76345919689455, 0.0143618156936512)
(-75.3584139333214, 8.8022107503451e-5)
(-43.91398181136465, 0.000259075472442818)
(15.514603010886749, -0.00206426604519197)
(9.095011330476355, -0.00593406337793414)
(-91.07324643601635, -6.02711965903649e-5)
(69.0716051946096, 0.000104769366606048)
(21.853874222709766, -0.00104362593723816)
(-37.619365753588426, 0.000352928123452401)
(-72.21508847040727, -9.58493466083015e-5)
(5.76345919689455, 0.0143618156936512)
(50.205728336738034, 0.00019824619552509)
(75.3584139333214, 8.8022107503451e-5)
(-97.35855832985965, -5.27415632271757e-5)
(31.320141707447174, 0.000508929311415012)
(25.01280320228961, 0.000797262893691229)
(-40.767115821406804, -0.000300578355805196)
(18.689036355362823, 0.00142535541473414)
(100.50111450015908, 4.9495275149513e-5)
(-34.47048833128499, -0.000420267871095791)
(-21.853874222709766, -0.00104362593723816)
(-31.320141707447174, 0.000508929311415012)
(-56.49556626181198, 0.00015658028339586)
(-65.92794150295865, -0.000114995552337877)
(-9.095011330476355, -0.00593406337793414)
(-81.64466440138234, 7.49922938134965e-5)
(-62.78407025618014, 0.000126796055545467)
(56.49556626181198, 0.00015658028339586)
(116.21311354040374, -3.70178754642193e-5)
(40.767115821406804, -0.000300578355805196)
(-94.21593786202358, 5.63180815586409e-5)
(43.91398181136465, 0.000259075472442818)
(53.35084358529321, -0.000175573282983773)
(65.92794150295865, -0.000114995552337877)
(87.93047643795707, 6.46557092537797e-5)
(91.07324643601635, -6.02711965903649e-5)
(-18.689036355362823, 0.00142535541473414)
(72.21508847040727, -9.58493466083015e-5)
(37.619365753588426, 0.000352928123452401)
(12.322940970566583, 0.00325994373348013)
(-25.01280320228961, 0.000797262893691229)
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:
$$x_{1} = -47.0601416127605$$
$$x_{2} = 84.7876191237855$$
$$x_{3} = -28.1678297079936$$
$$x_{4} = -53.3508435852932$$
$$x_{5} = -15.5146030108867$$
$$x_{6} = 28.1678297079936$$
$$x_{7} = 59.6399585795582$$
$$x_{8} = 47.0601416127605$$
$$x_{9} = 97.3585583298596$$
$$x_{10} = -84.7876191237855$$
$$x_{11} = -59.6399585795582$$
$$x_{12} = 78.5016005602391$$
$$x_{13} = 34.470488331285$$
$$x_{14} = -78.5016005602391$$
$$x_{15} = 15.5146030108867$$
$$x_{16} = 9.09501133047636$$
$$x_{17} = -91.0732464360163$$
$$x_{18} = 21.8538742227098$$
$$x_{19} = -72.2150884704073$$
$$x_{20} = -97.3585583298596$$
$$x_{21} = -40.7671158214068$$
$$x_{22} = -34.470488331285$$
$$x_{23} = -21.8538742227098$$
$$x_{24} = -65.9279415029586$$
$$x_{25} = -9.09501133047636$$
$$x_{26} = 116.213113540404$$
$$x_{27} = 40.7671158214068$$
$$x_{28} = 53.3508435852932$$
$$x_{29} = 65.9279415029586$$
$$x_{30} = 91.0732464360163$$
$$x_{31} = 72.2150884704073$$
Puntos máximos de la función:
$$x_{31} = 94.2159378620236$$
$$x_{31} = -50.205728336738$$
$$x_{31} = 62.7840702561801$$
$$x_{31} = -188.47964237706$$
$$x_{31} = -12.3229409705666$$
$$x_{31} = -69.0716051946096$$
$$x_{31} = -87.9304764379571$$
$$x_{31} = 81.6446644013823$$
$$x_{31} = -100.501114500159$$
$$x_{31} = -5.76345919689455$$
$$x_{31} = -75.3584139333214$$
$$x_{31} = -43.9139818113646$$
$$x_{31} = 69.0716051946096$$
$$x_{31} = -37.6193657535884$$
$$x_{31} = 5.76345919689455$$
$$x_{31} = 50.205728336738$$
$$x_{31} = 75.3584139333214$$
$$x_{31} = 31.3201417074472$$
$$x_{31} = 25.0128032022896$$
$$x_{31} = 18.6890363553628$$
$$x_{31} = 100.501114500159$$
$$x_{31} = -31.3201417074472$$
$$x_{31} = -56.495566261812$$
$$x_{31} = -81.6446644013823$$
$$x_{31} = -62.7840702561801$$
$$x_{31} = 56.495566261812$$
$$x_{31} = -94.2159378620236$$
$$x_{31} = 43.9139818113646$$
$$x_{31} = 87.9304764379571$$
$$x_{31} = -18.6890363553628$$
$$x_{31} = 37.6193657535884$$
$$x_{31} = 12.3229409705666$$
$$x_{31} = -25.0128032022896$$
Decrece en los intervalos
$$\left[116.213113540404, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -97.3585583298596\right]$$
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
$$\frac{- \frac{x \cos{\left(x \right)}}{2} + \frac{5 \sin{\left(x \right)}}{2} + \frac{6 \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)}{x^{2}}}{x^{3}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 89.4795030680389$$
$$x_{2} = -92.6229930341506$$
$$x_{3} = 80.0481379038761$$
$$x_{4} = -80.0481379038761$$
$$x_{5} = -98.9096108380069$$
$$x_{6} = 51.7395952684481$$
$$x_{7} = -89.4795030680389$$
$$x_{8} = -70.6150102433405$$
$$x_{9} = 64.3248951996227$$
$$x_{10} = -51.7395952684481$$
$$x_{11} = 86.3358747789945$$
$$x_{12} = -73.7596238120155$$
$$x_{13} = 32.8342639763876$$
$$x_{14} = 70.6150102433405$$
$$x_{15} = -67.4701145463391$$
$$x_{16} = 54.8867364434332$$
$$x_{17} = 83.1920924539639$$
$$x_{18} = 61.1793020994601$$
$$x_{19} = -29.6764038225906$$
$$x_{20} = -48.5917327642285$$
$$x_{21} = 26.514622544008$$
$$x_{22} = 76.903989955402$$
$$x_{23} = 35.9892492569686$$
$$x_{24} = -61.1793020994601$$
$$x_{25} = 23.3472895319727$$
$$x_{26} = 67.4701145463391$$
$$x_{27} = 42.293194769059$$
$$x_{28} = -13.7716779871269$$
$$x_{29} = 39.1420627323401$$
$$x_{30} = 7.13600879219012$$
$$x_{31} = -488.507422319867$$
$$x_{32} = -7.13600879219012$$
$$x_{33} = -20.1717085381322$$
$$x_{34} = -64.3248951996227$$
$$x_{35} = 20.1717085381322$$
$$x_{36} = -26.514622544008$$
$$x_{37} = -35.9892492569686$$
$$x_{38} = -45.4429980316939$$
$$x_{39} = 95.7663583220911$$
$$x_{40} = -58.0332742172009$$
$$x_{41} = -83.1920924539639$$
$$x_{42} = -2.5011326204094$$
$$x_{43} = -95.7663583220911$$
$$x_{44} = 45.4429980316939$$
$$x_{45} = -10.5146010554263$$
$$x_{46} = -42.293194769059$$
$$x_{47} = 16.9830552416253$$
$$x_{48} = -86.3358747789945$$
$$x_{49} = 10.5146010554263$$
$$x_{50} = -76.903989955402$$
$$x_{51} = 29.6764038225906$$
$$x_{52} = 48.5917327642285$$
$$x_{53} = -32.8342639763876$$
$$x_{54} = -23.3472895319727$$
$$x_{55} = 98.9096108380069$$
$$x_{56} = -39.1420627323401$$
$$x_{57} = 2.5011326204094$$
$$x_{58} = 92.6229930341506$$
$$x_{59} = -16.9830552416253$$
$$x_{60} = -54.8867364434332$$
$$x_{61} = 58.0332742172009$$
$$x_{62} = 13.7716779871269$$
$$x_{63} = 73.7596238120155$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{- \frac{x \cos{\left(x \right)}}{2} + \frac{5 \sin{\left(x \right)}}{2} + \frac{6 \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)}{x^{2}}}{x^{3}}\right) = \frac{1}{30}$$
$$\lim_{x \to 0^+}\left(\frac{- \frac{x \cos{\left(x \right)}}{2} + \frac{5 \sin{\left(x \right)}}{2} + \frac{6 \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)}{x^{2}}}{x^{3}}\right) = \frac{1}{30}$$
- los límites son iguales, es decir omitimos el punto correspondiente
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
$$\left[95.7663583220911, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -488.507422319867\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{- \frac{x \cos{\left(x \right)}}{2} + \frac{5 \sin{\left(x \right)}}{2} + \frac{6 \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)}{x^{2}}}{x^{3}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 89.4795030680389$$
$$x_{2} = -92.6229930341506$$
$$x_{3} = 80.0481379038761$$
$$x_{4} = -80.0481379038761$$
$$x_{5} = -98.9096108380069$$
$$x_{6} = 51.7395952684481$$
$$x_{7} = -89.4795030680389$$
$$x_{8} = -70.6150102433405$$
$$x_{9} = 64.3248951996227$$
$$x_{10} = -51.7395952684481$$
$$x_{11} = 86.3358747789945$$
$$x_{12} = -73.7596238120155$$
$$x_{13} = 32.8342639763876$$
$$x_{14} = 70.6150102433405$$
$$x_{15} = -67.4701145463391$$
$$x_{16} = 54.8867364434332$$
$$x_{17} = 83.1920924539639$$
$$x_{18} = 61.1793020994601$$
$$x_{19} = -29.6764038225906$$
$$x_{20} = -48.5917327642285$$
$$x_{21} = 26.514622544008$$
$$x_{22} = 76.903989955402$$
$$x_{23} = 35.9892492569686$$
$$x_{24} = -61.1793020994601$$
$$x_{25} = 23.3472895319727$$
$$x_{26} = 67.4701145463391$$
$$x_{27} = 42.293194769059$$
$$x_{28} = -13.7716779871269$$
$$x_{29} = 39.1420627323401$$
$$x_{30} = 7.13600879219012$$
$$x_{31} = -488.507422319867$$
$$x_{32} = -7.13600879219012$$
$$x_{33} = -20.1717085381322$$
$$x_{34} = -64.3248951996227$$
$$x_{35} = 20.1717085381322$$
$$x_{36} = -26.514622544008$$
$$x_{37} = -35.9892492569686$$
$$x_{38} = -45.4429980316939$$
$$x_{39} = 95.7663583220911$$
$$x_{40} = -58.0332742172009$$
$$x_{41} = -83.1920924539639$$
$$x_{42} = -2.5011326204094$$
$$x_{43} = -95.7663583220911$$
$$x_{44} = 45.4429980316939$$
$$x_{45} = -10.5146010554263$$
$$x_{46} = -42.293194769059$$
$$x_{47} = 16.9830552416253$$
$$x_{48} = -86.3358747789945$$
$$x_{49} = 10.5146010554263$$
$$x_{50} = -76.903989955402$$
$$x_{51} = 29.6764038225906$$
$$x_{52} = 48.5917327642285$$
$$x_{53} = -32.8342639763876$$
$$x_{54} = -23.3472895319727$$
$$x_{55} = 98.9096108380069$$
$$x_{56} = -39.1420627323401$$
$$x_{57} = 2.5011326204094$$
$$x_{58} = 92.6229930341506$$
$$x_{59} = -16.9830552416253$$
$$x_{60} = -54.8867364434332$$
$$x_{61} = 58.0332742172009$$
$$x_{62} = 13.7716779871269$$
$$x_{63} = 73.7596238120155$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{- \frac{x \cos{\left(x \right)}}{2} + \frac{5 \sin{\left(x \right)}}{2} + \frac{6 \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)}{x^{2}}}{x^{3}}\right) = \frac{1}{30}$$
$$\lim_{x \to 0^+}\left(\frac{- \frac{x \cos{\left(x \right)}}{2} + \frac{5 \sin{\left(x \right)}}{2} + \frac{6 \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)}{x^{2}}}{x^{3}}\right) = \frac{1}{30}$$
- los límites son iguales, es decir omitimos el punto correspondiente
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
$$\left[95.7663583220911, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -488.507422319867\right]$$
Asíntotas verticales
Hay:
$$x_{1} = 0$$
$$x_{1} = 0$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \lim_{x \to -\infty}\left(\frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{2 x^{3}}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(\frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{2 x^{3}}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \lim_{x \to -\infty}\left(\frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{2 x^{3}}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(\frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{2 x^{3}}\right)$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (x*cos(x) - sin(x))/((2*x^3)), dividida por x con x->+oo y x ->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{\frac{1}{2 x^{3}} \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)}{x}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\frac{1}{2 x^{3}} \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)}{x}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{\frac{1}{2 x^{3}} \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)}{x}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\frac{1}{2 x^{3}} \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)}{x}\right)$$
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:
$$\frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{2 x^{3}} = - \frac{- x \cos{\left(x \right)} + \sin{\left(x \right)}}{2 x^{3}}$$
- No
$$\frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{2 x^{3}} = \frac{- x \cos{\left(x \right)} + \sin{\left(x \right)}}{2 x^{3}}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{2 x^{3}} = - \frac{- x \cos{\left(x \right)} + \sin{\left(x \right)}}{2 x^{3}}$$
- No
$$\frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{2 x^{3}} = \frac{- x \cos{\left(x \right)} + \sin{\left(x \right)}}{2 x^{3}}$$
- No
es decir, función
no es
par ni impar