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- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
3/(x^2+1)
-
(1/3)^x
-
x/(x^3+2)
-
y=2x-3
-
Expresiones idénticas
- (sin3x)/ dos x+x- uno /x^2- cuatro
- ( seno de 3x) dividir por 2x más x menos 1 dividir por x al cuadrado menos 4
- ( seno de 3x) dividir por dos x más x menos uno dividir por x al cuadrado menos cuatro
- (sin3x)/2x+x-1/x2-4
- sin3x/2x+x-1/x2-4
- (sin3x)/2x+x-1/x²-4
- (sin3x)/2x+x-1/x en el grado 2-4
- sin3x/2x+x-1/x^2-4
- (sin3x) dividir por 2x+x-1 dividir por x^2-4
-
Expresiones semejantes
- (sin3x)/2x-x-1/x^2-4
- (sin3x)/2x+x+1/x^2-4
- (sin3x)/2x+x-1/x^2+4
Gráfico de la función y = (sin3x)/2x+x-1/x^2-4
Solución
Ha introducido
[src]
sin(3*x) 1
f(x) = --------*x + x - -- - 4
2 2
x
$$f{\left(x \right)} = \left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4$$
f = x*(sin(3*x)/2) + x - 1/x^2 - 4
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:
$$\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4 = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 5.41431902897203$$
$$x_{2} = 7.76635597363129$$
$$x_{3} = 7.91207856548297$$
$$x_{4} = 5.41431902897203$$
$$x_{5} = 6.04004815600269$$
$$x_{6} = 4.17068707471415$$
o sea hay que resolver la ecuación:
$$\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4 = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 5.41431902897203$$
$$x_{2} = 7.76635597363129$$
$$x_{3} = 7.91207856548297$$
$$x_{4} = 5.41431902897203$$
$$x_{5} = 6.04004815600269$$
$$x_{6} = 4.17068707471415$$
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
$$\frac{3 x \cos{\left(3 x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{2} + 1 + \frac{2}{x^{3}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 89.5341496175094$$
$$x_{2} = 73.8319421320736$$
$$x_{3} = -29.8414070727572$$
$$x_{4} = 7.8396736012599$$
$$x_{5} = 67.5491767500035$$
$$x_{6} = -78.0147933275031$$
$$x_{7} = -3.75177195803661$$
$$x_{8} = -84.2980847924199$$
$$x_{9} = -16.2520658835412$$
$$x_{10} = 86.3976561110523$$
$$x_{11} = 57.0781065328832$$
$$x_{12} = 35.0779499321882$$
$$x_{13} = -67.5425969944273$$
$$x_{14} = 3.63183452315228$$
$$x_{15} = -38.2198035188413$$
$$x_{16} = -51.842708427932$$
$$x_{17} = -73.8259223119028$$
$$x_{18} = 12.0335124171734$$
$$x_{19} = -23.5572290697416$$
$$x_{20} = -87.4448074455823$$
$$x_{21} = -69.6370415742658$$
$$x_{22} = -34.0306555064577$$
$$x_{23} = 97.9118362155217$$
$$x_{24} = 44.5133844234738$$
$$x_{25} = 22.5098087774546$$
$$x_{26} = -80.1092256661263$$
$$x_{27} = -91.6334234062544$$
$$x_{28} = -53.9368539004158$$
$$x_{29} = -12.0703667254341$$
$$x_{30} = 53.9286134447172$$
$$x_{31} = -41.3723600294113$$
$$x_{32} = 51.8341350739919$$
$$x_{33} = -89.5391133882524$$
$$x_{34} = 38.2314296566913$$
$$x_{35} = 40.3253719836236$$
$$x_{36} = 78.020489958096$$
$$x_{37} = -17.2723300965849$$
$$x_{38} = -49.7485839674963$$
$$x_{39} = -58.1251987998863$$
$$x_{40} = -47.6544832880973$$
$$x_{41} = 80.1147733747798$$
$$x_{42} = 75.9262127024484$$
$$x_{43} = -27.7467309956061$$
$$x_{44} = 93.7229952896613$$
$$x_{45} = 27.7627423539192$$
$$x_{46} = 36.137539795388$$
$$x_{47} = 42.4193590081437$$
$$x_{48} = 16.2247054307724$$
$$x_{49} = 92.6805798692254$$
$$x_{50} = -71.7314832647885$$
$$x_{51} = 60.212013791028$$
$$x_{52} = 29.856295359853$$
$$x_{53} = -5.81650479334164$$
$$x_{54} = -45.5604096674964$$
$$x_{55} = -25.6520090653707$$
$$x_{56} = -1.71221607813122$$
$$x_{57} = 20.414906074639$$
$$x_{58} = 5.7397730916088$$
$$x_{59} = -14.160695216171$$
$$x_{60} = 82.2090624816772$$
$$x_{61} = -36.1252398830002$$
$$x_{62} = 88.4919599039137$$
$$x_{63} = 58.1175521765418$$
$$x_{64} = 71.7376788324169$$
$$x_{65} = 14.1292889745525$$
$$x_{66} = -43.4663670146734$$
$$x_{67} = -60.2193944752711$$
$$x_{68} = 84.3033568547452$$
$$x_{69} = -7.89608208530915$$
$$x_{70} = -82.2036561072884$$
$$x_{71} = -93.7277372264291$$
$$x_{72} = 66.4953734902289$$
$$x_{73} = 31.9499587197976$$
$$x_{74} = -0.95989719783345$$
$$x_{75} = -21.4623740434466$$
$$x_{76} = 100.006255084988$$
$$x_{77} = -75.9203589343812$$
$$x_{78} = -97.9163752974962$$
$$x_{79} = 64.4009240388803$$
$$x_{80} = -56.0310180297196$$
$$x_{81} = 48.7015306215084$$
$$x_{82} = -95.8220545994953$$
$$x_{83} = 18.3198868045554$$
$$x_{84} = -31.9360463105638$$
$$x_{85} = -100.010699111567$$
$$x_{86} = 34.0437120782042$$
$$x_{87} = 56.0230855877076$$
$$x_{88} = -9.98172626669137$$
$$x_{89} = -65.448149247889$$
$$x_{90} = 24.60462482455$$
$$x_{91} = 95.8174163069689$$
$$x_{92} = 62.3064709329742$$
$$x_{93} = 9.93714128227479$$
$$x_{94} = 49.7396496884211$$
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:
$$x_{1} = 89.5341496175094$$
$$x_{2} = 7.8396736012599$$
$$x_{3} = -3.75177195803661$$
$$x_{4} = -16.2520658835412$$
$$x_{5} = 35.0779499321882$$
$$x_{6} = 3.63183452315228$$
$$x_{7} = -51.842708427932$$
$$x_{8} = 12.0335124171734$$
$$x_{9} = -87.4448074455823$$
$$x_{10} = 97.9118362155217$$
$$x_{11} = 22.5098087774546$$
$$x_{12} = -91.6334234062544$$
$$x_{13} = -53.9368539004158$$
$$x_{14} = -12.0703667254341$$
$$x_{15} = 53.9286134447172$$
$$x_{16} = -41.3723600294113$$
$$x_{17} = 51.8341350739919$$
$$x_{18} = -89.5391133882524$$
$$x_{19} = -49.7485839674963$$
$$x_{20} = -58.1251987998863$$
$$x_{21} = -47.6544832880973$$
$$x_{22} = 93.7229952896613$$
$$x_{23} = 16.2247054307724$$
$$x_{24} = 60.212013791028$$
$$x_{25} = -5.81650479334164$$
$$x_{26} = -45.5604096674964$$
$$x_{27} = -1.71221607813122$$
$$x_{28} = 20.414906074639$$
$$x_{29} = 5.7397730916088$$
$$x_{30} = -14.160695216171$$
$$x_{31} = 58.1175521765418$$
$$x_{32} = 14.1292889745525$$
$$x_{33} = -43.4663670146734$$
$$x_{34} = -60.2193944752711$$
$$x_{35} = -7.89608208530915$$
$$x_{36} = -93.7277372264291$$
$$x_{37} = 66.4953734902289$$
$$x_{38} = 100.006255084988$$
$$x_{39} = -97.9163752974962$$
$$x_{40} = 64.4009240388803$$
$$x_{41} = -56.0310180297196$$
$$x_{42} = -95.8220545994953$$
$$x_{43} = 18.3198868045554$$
$$x_{44} = -100.010699111567$$
$$x_{45} = 56.0230855877076$$
$$x_{46} = -9.98172626669137$$
$$x_{47} = 24.60462482455$$
$$x_{48} = 95.8174163069689$$
$$x_{49} = 62.3064709329742$$
$$x_{50} = 9.93714128227479$$
$$x_{51} = 49.7396496884211$$
Puntos máximos de la función:
$$x_{51} = 73.8319421320736$$
$$x_{51} = -29.8414070727572$$
$$x_{51} = 67.5491767500035$$
$$x_{51} = -78.0147933275031$$
$$x_{51} = -84.2980847924199$$
$$x_{51} = 86.3976561110523$$
$$x_{51} = 57.0781065328832$$
$$x_{51} = -67.5425969944273$$
$$x_{51} = -38.2198035188413$$
$$x_{51} = -73.8259223119028$$
$$x_{51} = -23.5572290697416$$
$$x_{51} = -69.6370415742658$$
$$x_{51} = -34.0306555064577$$
$$x_{51} = 44.5133844234738$$
$$x_{51} = -80.1092256661263$$
$$x_{51} = 38.2314296566913$$
$$x_{51} = 40.3253719836236$$
$$x_{51} = 78.020489958096$$
$$x_{51} = -17.2723300965849$$
$$x_{51} = 80.1147733747798$$
$$x_{51} = 75.9262127024484$$
$$x_{51} = -27.7467309956061$$
$$x_{51} = 27.7627423539192$$
$$x_{51} = 36.137539795388$$
$$x_{51} = 42.4193590081437$$
$$x_{51} = 92.6805798692254$$
$$x_{51} = -71.7314832647885$$
$$x_{51} = 29.856295359853$$
$$x_{51} = -25.6520090653707$$
$$x_{51} = 82.2090624816772$$
$$x_{51} = -36.1252398830002$$
$$x_{51} = 88.4919599039137$$
$$x_{51} = 71.7376788324169$$
$$x_{51} = 84.3033568547452$$
$$x_{51} = -82.2036561072884$$
$$x_{51} = 31.9499587197976$$
$$x_{51} = -0.95989719783345$$
$$x_{51} = -21.4623740434466$$
$$x_{51} = -75.9203589343812$$
$$x_{51} = 48.7015306215084$$
$$x_{51} = -31.9360463105638$$
$$x_{51} = 34.0437120782042$$
$$x_{51} = -65.448149247889$$
Decrece en los intervalos
$$\left[100.006255084988, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100.010699111567\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{3 x \cos{\left(3 x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{2} + 1 + \frac{2}{x^{3}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 89.5341496175094$$
$$x_{2} = 73.8319421320736$$
$$x_{3} = -29.8414070727572$$
$$x_{4} = 7.8396736012599$$
$$x_{5} = 67.5491767500035$$
$$x_{6} = -78.0147933275031$$
$$x_{7} = -3.75177195803661$$
$$x_{8} = -84.2980847924199$$
$$x_{9} = -16.2520658835412$$
$$x_{10} = 86.3976561110523$$
$$x_{11} = 57.0781065328832$$
$$x_{12} = 35.0779499321882$$
$$x_{13} = -67.5425969944273$$
$$x_{14} = 3.63183452315228$$
$$x_{15} = -38.2198035188413$$
$$x_{16} = -51.842708427932$$
$$x_{17} = -73.8259223119028$$
$$x_{18} = 12.0335124171734$$
$$x_{19} = -23.5572290697416$$
$$x_{20} = -87.4448074455823$$
$$x_{21} = -69.6370415742658$$
$$x_{22} = -34.0306555064577$$
$$x_{23} = 97.9118362155217$$
$$x_{24} = 44.5133844234738$$
$$x_{25} = 22.5098087774546$$
$$x_{26} = -80.1092256661263$$
$$x_{27} = -91.6334234062544$$
$$x_{28} = -53.9368539004158$$
$$x_{29} = -12.0703667254341$$
$$x_{30} = 53.9286134447172$$
$$x_{31} = -41.3723600294113$$
$$x_{32} = 51.8341350739919$$
$$x_{33} = -89.5391133882524$$
$$x_{34} = 38.2314296566913$$
$$x_{35} = 40.3253719836236$$
$$x_{36} = 78.020489958096$$
$$x_{37} = -17.2723300965849$$
$$x_{38} = -49.7485839674963$$
$$x_{39} = -58.1251987998863$$
$$x_{40} = -47.6544832880973$$
$$x_{41} = 80.1147733747798$$
$$x_{42} = 75.9262127024484$$
$$x_{43} = -27.7467309956061$$
$$x_{44} = 93.7229952896613$$
$$x_{45} = 27.7627423539192$$
$$x_{46} = 36.137539795388$$
$$x_{47} = 42.4193590081437$$
$$x_{48} = 16.2247054307724$$
$$x_{49} = 92.6805798692254$$
$$x_{50} = -71.7314832647885$$
$$x_{51} = 60.212013791028$$
$$x_{52} = 29.856295359853$$
$$x_{53} = -5.81650479334164$$
$$x_{54} = -45.5604096674964$$
$$x_{55} = -25.6520090653707$$
$$x_{56} = -1.71221607813122$$
$$x_{57} = 20.414906074639$$
$$x_{58} = 5.7397730916088$$
$$x_{59} = -14.160695216171$$
$$x_{60} = 82.2090624816772$$
$$x_{61} = -36.1252398830002$$
$$x_{62} = 88.4919599039137$$
$$x_{63} = 58.1175521765418$$
$$x_{64} = 71.7376788324169$$
$$x_{65} = 14.1292889745525$$
$$x_{66} = -43.4663670146734$$
$$x_{67} = -60.2193944752711$$
$$x_{68} = 84.3033568547452$$
$$x_{69} = -7.89608208530915$$
$$x_{70} = -82.2036561072884$$
$$x_{71} = -93.7277372264291$$
$$x_{72} = 66.4953734902289$$
$$x_{73} = 31.9499587197976$$
$$x_{74} = -0.95989719783345$$
$$x_{75} = -21.4623740434466$$
$$x_{76} = 100.006255084988$$
$$x_{77} = -75.9203589343812$$
$$x_{78} = -97.9163752974962$$
$$x_{79} = 64.4009240388803$$
$$x_{80} = -56.0310180297196$$
$$x_{81} = 48.7015306215084$$
$$x_{82} = -95.8220545994953$$
$$x_{83} = 18.3198868045554$$
$$x_{84} = -31.9360463105638$$
$$x_{85} = -100.010699111567$$
$$x_{86} = 34.0437120782042$$
$$x_{87} = 56.0230855877076$$
$$x_{88} = -9.98172626669137$$
$$x_{89} = -65.448149247889$$
$$x_{90} = 24.60462482455$$
$$x_{91} = 95.8174163069689$$
$$x_{92} = 62.3064709329742$$
$$x_{93} = 9.93714128227479$$
$$x_{94} = 49.7396496884211$$
Signos de extremos en los puntos:
(89.53414961750944, 40.767260320607)
(73.83194213207365, 106.744343711783)
(-29.841407072757196, -18.9227572006474)
(7.839673601259899, -0.0928232644072904)
(67.54917675000348, 97.3198449939322)
(-78.0147933275031, -43.0079170269249)
(-3.7517719580366107, -9.63577788418861)
(-84.2980847924199, -46.1495126395795)
(-16.2520658835412, -28.3665165897607)
(86.39765611105234, 125.593456623441)
(57.07810653288318, 81.6124729402231)
(35.077949932188154, 13.538954386956)
(-67.54259699442734, -37.7719289641448)
(3.631834523152284, -2.25081162630326)
(-38.21980351884126, -23.111313092809)
(-51.842708427932045, -81.7596126747617)
(-73.82592231190284, -40.9135208958062)
(12.033512417173416, 2.01217160130921)
(-23.557229069741553, -15.7815952565633)
(-87.44480744558226, -135.164483041905)
(-69.63704157426584, -38.8191258973999)
(-34.03065550645767, -21.0170074370133)
(97.91183621552172, 44.956097505348)
(44.513384423473845, 62.7639557301852)
(22.509808777454612, 7.2541660356287)
(-80.10922566612635, -44.0551154081801)
(-91.63342340625437, -141.447525978401)
(-53.936853900415834, -84.9009897514051)
(-12.070366725434083, -22.0917451476389)
(53.92861344471723, 22.9644780134213)
(-41.37236002941131, -66.0530821082106)
(51.834135073991895, 21.9172312998043)
(-89.53911338825245, -138.306002776883)
(38.231429656691304, 53.3399212613474)
(40.32537198362361, 56.4812435153022)
(78.02048995809595, 113.027366397324)
(-17.272330096584913, -12.6411234786029)
(-49.74858396749628, -78.6182550138179)
(-58.12519879988633, -91.1837932897095)
(-47.65448328809729, -75.4769195021771)
(80.11477337477979, 116.168883760784)
(75.92621270244842, 109.885852943855)
(-27.746730995606125, -17.8756653242941)
(93.72299528966128, 42.8616801907127)
(27.762742353919162, 37.6338111038007)
(36.13753979538799, 50.1986259870911)
(42.419359008143665, 59.6225893035084)
(16.224705430772445, 4.11027009772762)
(92.68057986922537, 135.018055965606)
(-71.73148326478852, -39.8663232292551)
(60.212013791027964, 26.1061924378668)
(29.856295359853007, 40.774947707108)
(-5.816504793341635, -12.7120199168973)
(-45.56040966749643, -72.3356094094135)
(-25.652009065370727, -16.8286068152483)
(-1.712216078131216, -6.83352581855356)
(20.41490607463904, 6.20641601866254)
(5.739773091608804, -1.15539869871637)
(-14.160695216171039, -25.2283991322338)
(82.20906248167715, 119.310404752608)
(-36.12523988300024, -22.0641550869415)
(88.49195990391365, 128.734987059214)
(58.11755217654183, 25.0589580220582)
(71.73767883241692, 103.602839046315)
(14.129288974552527, 3.06160831731661)
(-43.46636701467342, -69.1943286785897)
(-60.219394475271066, -94.3252161299858)
(84.30335685474525, 122.451929117488)
(-7.896082085309151, -15.8287143054271)
(-82.20365610728835, -45.1023139548754)
(-93.72773722642908, -144.589052405349)
(66.49537349022893, 29.2478783489101)
(31.949958719797575, 43.9161337183775)
(-0.9598971978334502, -5.9209320980479)
(-21.462374043446612, -14.7346515424909)
(100.00625508498813, 46.0033053214858)
(-75.92035893438117, -41.9607188434179)
(-97.91637529749619, -150.872114077811)
(64.40092403888028, 28.2006522622469)
(-56.031018029719625, -88.0423839408958)
(48.70153062150844, 69.0467410714497)
(-95.82205459949529, -147.730581838443)
(18.319886804555388, 5.1584826928003)
(-31.936046310563775, -19.9698733316274)
(-100.01069911156709, -154.013648940895)
(34.0437120782042, 47.0573618170311)
(56.023085587707556, 24.0117200508985)
(-9.981726266691371, -18.9576683040401)
(-65.44814924788905, -36.7247325059037)
(24.604624824550005, 8.30179040932915)
(95.81741630696887, 43.9088891428627)
(62.30647093297416, 27.1534237293258)
(9.937141282274794, 0.961265904515388)
(49.73964968842108, 20.8699791769845)
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:
$$x_{1} = 89.5341496175094$$
$$x_{2} = 7.8396736012599$$
$$x_{3} = -3.75177195803661$$
$$x_{4} = -16.2520658835412$$
$$x_{5} = 35.0779499321882$$
$$x_{6} = 3.63183452315228$$
$$x_{7} = -51.842708427932$$
$$x_{8} = 12.0335124171734$$
$$x_{9} = -87.4448074455823$$
$$x_{10} = 97.9118362155217$$
$$x_{11} = 22.5098087774546$$
$$x_{12} = -91.6334234062544$$
$$x_{13} = -53.9368539004158$$
$$x_{14} = -12.0703667254341$$
$$x_{15} = 53.9286134447172$$
$$x_{16} = -41.3723600294113$$
$$x_{17} = 51.8341350739919$$
$$x_{18} = -89.5391133882524$$
$$x_{19} = -49.7485839674963$$
$$x_{20} = -58.1251987998863$$
$$x_{21} = -47.6544832880973$$
$$x_{22} = 93.7229952896613$$
$$x_{23} = 16.2247054307724$$
$$x_{24} = 60.212013791028$$
$$x_{25} = -5.81650479334164$$
$$x_{26} = -45.5604096674964$$
$$x_{27} = -1.71221607813122$$
$$x_{28} = 20.414906074639$$
$$x_{29} = 5.7397730916088$$
$$x_{30} = -14.160695216171$$
$$x_{31} = 58.1175521765418$$
$$x_{32} = 14.1292889745525$$
$$x_{33} = -43.4663670146734$$
$$x_{34} = -60.2193944752711$$
$$x_{35} = -7.89608208530915$$
$$x_{36} = -93.7277372264291$$
$$x_{37} = 66.4953734902289$$
$$x_{38} = 100.006255084988$$
$$x_{39} = -97.9163752974962$$
$$x_{40} = 64.4009240388803$$
$$x_{41} = -56.0310180297196$$
$$x_{42} = -95.8220545994953$$
$$x_{43} = 18.3198868045554$$
$$x_{44} = -100.010699111567$$
$$x_{45} = 56.0230855877076$$
$$x_{46} = -9.98172626669137$$
$$x_{47} = 24.60462482455$$
$$x_{48} = 95.8174163069689$$
$$x_{49} = 62.3064709329742$$
$$x_{50} = 9.93714128227479$$
$$x_{51} = 49.7396496884211$$
Puntos máximos de la función:
$$x_{51} = 73.8319421320736$$
$$x_{51} = -29.8414070727572$$
$$x_{51} = 67.5491767500035$$
$$x_{51} = -78.0147933275031$$
$$x_{51} = -84.2980847924199$$
$$x_{51} = 86.3976561110523$$
$$x_{51} = 57.0781065328832$$
$$x_{51} = -67.5425969944273$$
$$x_{51} = -38.2198035188413$$
$$x_{51} = -73.8259223119028$$
$$x_{51} = -23.5572290697416$$
$$x_{51} = -69.6370415742658$$
$$x_{51} = -34.0306555064577$$
$$x_{51} = 44.5133844234738$$
$$x_{51} = -80.1092256661263$$
$$x_{51} = 38.2314296566913$$
$$x_{51} = 40.3253719836236$$
$$x_{51} = 78.020489958096$$
$$x_{51} = -17.2723300965849$$
$$x_{51} = 80.1147733747798$$
$$x_{51} = 75.9262127024484$$
$$x_{51} = -27.7467309956061$$
$$x_{51} = 27.7627423539192$$
$$x_{51} = 36.137539795388$$
$$x_{51} = 42.4193590081437$$
$$x_{51} = 92.6805798692254$$
$$x_{51} = -71.7314832647885$$
$$x_{51} = 29.856295359853$$
$$x_{51} = -25.6520090653707$$
$$x_{51} = 82.2090624816772$$
$$x_{51} = -36.1252398830002$$
$$x_{51} = 88.4919599039137$$
$$x_{51} = 71.7376788324169$$
$$x_{51} = 84.3033568547452$$
$$x_{51} = -82.2036561072884$$
$$x_{51} = 31.9499587197976$$
$$x_{51} = -0.95989719783345$$
$$x_{51} = -21.4623740434466$$
$$x_{51} = -75.9203589343812$$
$$x_{51} = 48.7015306215084$$
$$x_{51} = -31.9360463105638$$
$$x_{51} = 34.0437120782042$$
$$x_{51} = -65.448149247889$$
Decrece en los intervalos
$$\left[100.006255084988, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100.010699111567\right]$$
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
$$3 \left(- \frac{3 x \sin{\left(3 x \right)}}{2} + \cos{\left(3 x \right)} - \frac{2}{x^{4}}\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -19.907911952782$$
$$x_{2} = -39.799090019458$$
$$x_{3} = 24.0947642225796$$
$$x_{4} = -37.705004929418$$
$$x_{5} = 90.0614568086612$$
$$x_{6} = 17.8148268039322$$
$$x_{7} = 48.1756998040024$$
$$x_{8} = -50.2699027788224$$
$$x_{9} = -2.18458133232718$$
$$x_{10} = 83.7784565380389$$
$$x_{11} = -43.987348718435$$
$$x_{12} = -53.4112354854412$$
$$x_{13} = 70.1654029547235$$
$$x_{14} = -61.7882518940721$$
$$x_{15} = 4.24044984746248$$
$$x_{16} = 22.0012460097448$$
$$x_{17} = 26.1884224975902$$
$$x_{18} = -70.1654029547235$$
$$x_{19} = 63.8825291042832$$
$$x_{20} = -95.2973090044054$$
$$x_{21} = -35.6109562808109$$
$$x_{22} = 2.18458133232718$$
$$x_{23} = 37.705004929418$$
$$x_{24} = -68.071105283931$$
$$x_{25} = -15.7220896627602$$
$$x_{26} = 76.4483279929109$$
$$x_{27} = -33.5169508979714$$
$$x_{28} = -1.30898709351237$$
$$x_{29} = -77.4954862681777$$
$$x_{30} = -4.24044984746248$$
$$x_{31} = -24.0947642225796$$
$$x_{32} = 46.0815142865079$$
$$x_{33} = -59.6939829545149$$
$$x_{34} = 68.071105283931$$
$$x_{35} = 98.4388272430731$$
$$x_{36} = 100.533175319149$$
$$x_{37} = -26.1884224975902$$
$$x_{38} = -57.5997231874848$$
$$x_{39} = -90.0614568086612$$
$$x_{40} = -28.2821897723174$$
$$x_{41} = 85.8727869533063$$
$$x_{42} = -94.250137360338$$
$$x_{43} = 65.9768137977467$$
$$x_{44} = 87.9671204484587$$
$$x_{45} = -6.31818359713174$$
$$x_{46} = -55.5054736309118$$
$$x_{47} = -81.6841294395199$$
$$x_{48} = -72.259706272491$$
$$x_{49} = -22.0012460097448$$
$$x_{50} = -85.8727869533063$$
$$x_{51} = -41.8932060898102$$
$$x_{52} = 9.44826483019524$$
$$x_{53} = 8.4039571519597$$
$$x_{54} = -99.4860010337234$$
$$x_{55} = -87.9671204484587$$
$$x_{56} = -46.0815142865079$$
$$x_{57} = -30.3760435342236$$
$$x_{58} = -83.7784565380389$$
$$x_{59} = -11.5384131843255$$
$$x_{60} = 15.7220896627602$$
$$x_{61} = 28.2821897723174$$
$$x_{62} = 74.3540147601572$$
$$x_{63} = 6.31818359713174$$
$$x_{64} = 50.2699027788224$$
$$x_{65} = -3.21110226651524$$
$$x_{66} = 92.1557958386049$$
$$x_{67} = 59.6939829545149$$
$$x_{68} = -13.6298601979387$$
$$x_{69} = -65.9768137977467$$
$$x_{70} = -63.8825291042832$$
$$x_{71} = 94.250137360338$$
$$x_{72} = -17.8148268039322$$
$$x_{73} = 30.3760435342236$$
$$x_{74} = 19.907911952782$$
$$x_{75} = 78.5426455912395$$
$$x_{76} = 54.4583530476818$$
$$x_{77} = 52.3641211173087$$
$$x_{78} = 12.5840118070877$$
$$x_{79} = -9.44826483019524$$
$$x_{80} = 39.799090019458$$
$$x_{81} = 41.8932060898102$$
$$x_{82} = -79.5898059195386$$
$$x_{83} = -92.1557958386049$$
$$x_{84} = 32.4699670691998$$
$$x_{85} = 56.5525970604809$$
$$x_{86} = 43.987348718435$$
$$x_{87} = 10.4931214936956$$
$$x_{88} = 61.7882518940721$$
$$x_{89} = -48.1756998040024$$
$$x_{90} = 72.259706272491$$
$$x_{91} = 96.3444812113793$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(3 \left(- \frac{3 x \sin{\left(3 x \right)}}{2} + \cos{\left(3 x \right)} - \frac{2}{x^{4}}\right)\right) = -\infty$$
$$\lim_{x \to 0^+}\left(3 \left(- \frac{3 x \sin{\left(3 x \right)}}{2} + \cos{\left(3 x \right)} - \frac{2}{x^{4}}\right)\right) = -\infty$$
- 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
$$\left[78.5426455912395, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -94.250137360338\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$3 \left(- \frac{3 x \sin{\left(3 x \right)}}{2} + \cos{\left(3 x \right)} - \frac{2}{x^{4}}\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -19.907911952782$$
$$x_{2} = -39.799090019458$$
$$x_{3} = 24.0947642225796$$
$$x_{4} = -37.705004929418$$
$$x_{5} = 90.0614568086612$$
$$x_{6} = 17.8148268039322$$
$$x_{7} = 48.1756998040024$$
$$x_{8} = -50.2699027788224$$
$$x_{9} = -2.18458133232718$$
$$x_{10} = 83.7784565380389$$
$$x_{11} = -43.987348718435$$
$$x_{12} = -53.4112354854412$$
$$x_{13} = 70.1654029547235$$
$$x_{14} = -61.7882518940721$$
$$x_{15} = 4.24044984746248$$
$$x_{16} = 22.0012460097448$$
$$x_{17} = 26.1884224975902$$
$$x_{18} = -70.1654029547235$$
$$x_{19} = 63.8825291042832$$
$$x_{20} = -95.2973090044054$$
$$x_{21} = -35.6109562808109$$
$$x_{22} = 2.18458133232718$$
$$x_{23} = 37.705004929418$$
$$x_{24} = -68.071105283931$$
$$x_{25} = -15.7220896627602$$
$$x_{26} = 76.4483279929109$$
$$x_{27} = -33.5169508979714$$
$$x_{28} = -1.30898709351237$$
$$x_{29} = -77.4954862681777$$
$$x_{30} = -4.24044984746248$$
$$x_{31} = -24.0947642225796$$
$$x_{32} = 46.0815142865079$$
$$x_{33} = -59.6939829545149$$
$$x_{34} = 68.071105283931$$
$$x_{35} = 98.4388272430731$$
$$x_{36} = 100.533175319149$$
$$x_{37} = -26.1884224975902$$
$$x_{38} = -57.5997231874848$$
$$x_{39} = -90.0614568086612$$
$$x_{40} = -28.2821897723174$$
$$x_{41} = 85.8727869533063$$
$$x_{42} = -94.250137360338$$
$$x_{43} = 65.9768137977467$$
$$x_{44} = 87.9671204484587$$
$$x_{45} = -6.31818359713174$$
$$x_{46} = -55.5054736309118$$
$$x_{47} = -81.6841294395199$$
$$x_{48} = -72.259706272491$$
$$x_{49} = -22.0012460097448$$
$$x_{50} = -85.8727869533063$$
$$x_{51} = -41.8932060898102$$
$$x_{52} = 9.44826483019524$$
$$x_{53} = 8.4039571519597$$
$$x_{54} = -99.4860010337234$$
$$x_{55} = -87.9671204484587$$
$$x_{56} = -46.0815142865079$$
$$x_{57} = -30.3760435342236$$
$$x_{58} = -83.7784565380389$$
$$x_{59} = -11.5384131843255$$
$$x_{60} = 15.7220896627602$$
$$x_{61} = 28.2821897723174$$
$$x_{62} = 74.3540147601572$$
$$x_{63} = 6.31818359713174$$
$$x_{64} = 50.2699027788224$$
$$x_{65} = -3.21110226651524$$
$$x_{66} = 92.1557958386049$$
$$x_{67} = 59.6939829545149$$
$$x_{68} = -13.6298601979387$$
$$x_{69} = -65.9768137977467$$
$$x_{70} = -63.8825291042832$$
$$x_{71} = 94.250137360338$$
$$x_{72} = -17.8148268039322$$
$$x_{73} = 30.3760435342236$$
$$x_{74} = 19.907911952782$$
$$x_{75} = 78.5426455912395$$
$$x_{76} = 54.4583530476818$$
$$x_{77} = 52.3641211173087$$
$$x_{78} = 12.5840118070877$$
$$x_{79} = -9.44826483019524$$
$$x_{80} = 39.799090019458$$
$$x_{81} = 41.8932060898102$$
$$x_{82} = -79.5898059195386$$
$$x_{83} = -92.1557958386049$$
$$x_{84} = 32.4699670691998$$
$$x_{85} = 56.5525970604809$$
$$x_{86} = 43.987348718435$$
$$x_{87} = 10.4931214936956$$
$$x_{88} = 61.7882518940721$$
$$x_{89} = -48.1756998040024$$
$$x_{90} = 72.259706272491$$
$$x_{91} = 96.3444812113793$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(3 \left(- \frac{3 x \sin{\left(3 x \right)}}{2} + \cos{\left(3 x \right)} - \frac{2}{x^{4}}\right)\right) = -\infty$$
$$\lim_{x \to 0^+}\left(3 \left(- \frac{3 x \sin{\left(3 x \right)}}{2} + \cos{\left(3 x \right)} - \frac{2}{x^{4}}\right)\right) = -\infty$$
- 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
$$\left[78.5426455912395, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -94.250137360338\right]$$
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
$$\lim_{x \to -\infty}\left(\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4\right) = -\infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
$$\lim_{x \to \infty}\left(\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
$$\lim_{x \to -\infty}\left(\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4\right) = -\infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
$$\lim_{x \to \infty}\left(\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (sin(3*x)/2)*x + x - 1/x^2 - 4, dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4}{x}\right) = 1$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x$$
$$\lim_{x \to \infty}\left(\frac{\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4}{x}\right) = 1$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x$$
$$\lim_{x \to -\infty}\left(\frac{\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4}{x}\right) = 1$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x$$
$$\lim_{x \to \infty}\left(\frac{\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4}{x}\right) = 1$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x$$
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:
$$\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4 = \frac{x \sin{\left(3 x \right)}}{2} - x - 4 - \frac{1}{x^{2}}$$
- No
$$\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4 = - \frac{x \sin{\left(3 x \right)}}{2} + x + 4 + \frac{1}{x^{2}}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4 = \frac{x \sin{\left(3 x \right)}}{2} - x - 4 - \frac{1}{x^{2}}$$
- No
$$\left(\left(x \frac{\sin{\left(3 x \right)}}{2} + x\right) - \frac{1}{x^{2}}\right) - 4 = - \frac{x \sin{\left(3 x \right)}}{2} + x + 4 + \frac{1}{x^{2}}$$
- No
es decir, función
no es
par ni impar