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  • Gráfico de la función y =:
  • y=cosx y=cosx
  • y=3x^2-x^3 y=3x^2-x^3
  • 2*x^3-3*x 2*x^3-3*x
  • 3-x^2 3-x^2

  • Expresiones idénticas

  • y=cos2x-x^ uno / tres
  • y es igual a coseno de 2x menos x en el grado 1 dividir por 3
  • y es igual a coseno de 2x menos x en el grado uno dividir por tres
  • y=cos2x-x1/3
  • y=cos2x-x^1 dividir por 3

  • Expresiones semejantes

  • y=cos2x+x^1/3
Trazado el gráfico de la función/ y=cos2x-x^1/3

Gráfico de la función y = y=cos2x-x^1/3

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                  3 ___
f(x) = cos(2*x) - \/ x
f(x)=−x3+cos⁡(2x)f{\left(x \right)} = - \sqrt[3]{x} + \cos{\left(2 x \right)} 
f = -x^(1/3) + cos(2*x)
Gráfico de la función
02468-8-6-4-2-10105-5
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:
−x3+cos⁡(2x)=0- \sqrt[3]{x} + \cos{\left(2 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=0.380258446681609x_{1} = 0.380258446681609
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en cos(2*x) - x^(1/3).
−03+cos⁡(0⋅2)- \sqrt[3]{0} + \cos{\left(0 \cdot 2 \right)}
Resultado:
f(0)=1f{\left(0 \right)} = 1
Punto:
(0, 1)
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
−2sin⁡(2x)−13x23=0- 2 \sin{\left(2 x \right)} - \frac{1}{3 x^{\frac{2}{3}}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=36.1359399808004x_{1} = 36.1359399808004
x2=34.5496630820819x_{2} = 34.5496630820819
x3=29.8537900015697x_{3} = 29.8537900015697
x4=1.63108293582399x_{4} = 1.63108293582399
x5=15.6946678544559x_{5} = 15.6946678544559
x6=70.690709063599x_{6} = 70.690709063599
x7=48.7009350539313x_{7} = 48.7009350539313
x8=26.7128635890271x_{8} = 26.7128635890271
x9=42.4183524941555x_{9} = 42.4183524941555
x10=28.2653525829836x_{10} = 28.2653525829836
x11=6.25863784020032x_{11} = 6.25863784020032
x12=50.2593633942295x_{12} = 50.2593633942295
x13=12.5509380918273x_{13} = 12.5509380918273
x14=21.980527723609x_{14} = 21.980527723609
x15=7.8750409995985x_{15} = 7.8750409995985
x16=87.9603809024344x_{16} = 87.9603809024344
x17=45.5596264522288x_{17} = 45.5596264522288
x18=59.6848038675201x_{18} = 59.6848038675201
x19=78.5352722360833x_{19} = 78.5352722360833
x20=81.676982179018x_{20} = 81.676982179018
x21=64.4078358152586x_{21} = 64.4078358152586
x22=58.1250178323562x_{22} = 58.1250178323562
x23=20.4315034381183x_{23} = 20.4315034381183
x24=37.6916986691167x_{24} = 37.6916986691167
x25=86.3980620145129x_{25} = 86.3980620145129
x26=51.8422726424804x_{26} = 51.8422726424804
x27=43.9756082127529x_{27} = 43.9756082127529
x28=95.8225555841334x_{28} = 95.8225555841334
x29=100.527110418962x_{29} = 100.527110418962
x30=40.8336766750392x_{30} = 40.8336766750392
x31=72.2518271445003x_{31} = 72.2518271445003
x32=4.74193062444813x_{32} = 4.74193062444813
x33=14.1514124396753x_{33} = 14.1514124396753
x34=23.5720820675917x_{34} = 23.5720820675917
x35=89.5395543374455x_{35} = 89.5395543374455
x36=65.9683414285725x_{36} = 65.9683414285725
x37=94.2437556347455x_{37} = 94.2437556347455
x38=80.1150968323989x_{38} = 80.1150968323989
x39=92.6810523610412x_{39} = 92.6810523610412
x40=67.5492663916134x_{40} = 67.5492663916134
x41=139.803966763363x_{41} = 139.803966763363
x42=73.8321624487981x_{42} = 73.8321624487981
x43=56.5430109050366x_{43} = 56.5430109050366
Signos de extremos en los puntos:
(36.135939980800394, -4.30596191328923)

(34.54966308208195, -2.25709978822598)

(29.85379000156966, -4.10202643109194)

(1.6310829358239889, -2.16987234494791)

(15.694667854455876, -1.5040636254226)

(70.69070906359904, -5.1347487211907)

(48.70093505393132, -4.65176772117181)

(26.71286358902706, -3.98915345021126)

(42.41835249415547, -4.48743595101489)

(28.265352582983642, -2.04631261239431)

(6.258637840200322, -0.844068861587621)

(50.25936339422951, -2.69046542030057)

(12.550938091827291, -1.32441888108148)

(21.980527723608986, -1.80143798487504)

(7.8750409995984985, -2.98864519399288)

(87.96038090243437, -3.44732807038523)

(45.559626452228805, -4.57149192106365)

(59.6848038675201, -2.90805986498524)

(78.53527223608326, -3.28245136316207)

(81.67698217901798, -3.33880851753651)

(64.4078358152586, -5.00842479778946)

(58.12501783235618, -4.8735941561922)

(20.431503438118327, -3.73355153790667)

(37.691698669116704, -2.35296849659248)

(86.39806201451294, -5.42076838055218)

(51.842272642480445, -4.72866164241038)

(43.975608212752874, -2.52978533728692)

(95.82255558413335, -5.57600240032018)

(100.5271104189616, -3.64975969296203)

(40.83367667503917, -2.44364694423542)

(72.25182714450033, -3.16505835642445)

(4.7419306244481305, -2.67829035786212)

(14.151412439675303, -3.41839398961354)

(23.572082067591673, -3.86704725613626)

(89.53955433744547, -5.47371462077253)

(65.96834142857247, -3.04064586347855)

(94.24375563474554, -3.55079513771709)

(80.11509683239892, -5.31089457972184)

(92.68105236104117, -5.52543648831572)

(67.54926639161344, -5.07256628578308)

(139.80396676336287, -6.1900502533218)

(73.83216244879807, -5.19511516197194)

(56.54301090503657, -2.83825259155069)


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:
x1=36.1359399808004x_{1} = 36.1359399808004
x2=29.8537900015697x_{2} = 29.8537900015697
x3=1.63108293582399x_{3} = 1.63108293582399
x4=70.690709063599x_{4} = 70.690709063599
x5=48.7009350539313x_{5} = 48.7009350539313
x6=26.7128635890271x_{6} = 26.7128635890271
x7=42.4183524941555x_{7} = 42.4183524941555
x8=7.8750409995985x_{8} = 7.8750409995985
x9=45.5596264522288x_{9} = 45.5596264522288
x10=64.4078358152586x_{10} = 64.4078358152586
x11=58.1250178323562x_{11} = 58.1250178323562
x12=20.4315034381183x_{12} = 20.4315034381183
x13=86.3980620145129x_{13} = 86.3980620145129
x14=51.8422726424804x_{14} = 51.8422726424804
x15=95.8225555841334x_{15} = 95.8225555841334
x16=4.74193062444813x_{16} = 4.74193062444813
x17=14.1514124396753x_{17} = 14.1514124396753
x18=23.5720820675917x_{18} = 23.5720820675917
x19=89.5395543374455x_{19} = 89.5395543374455
x20=80.1150968323989x_{20} = 80.1150968323989
x21=92.6810523610412x_{21} = 92.6810523610412
x22=67.5492663916134x_{22} = 67.5492663916134
x23=139.803966763363x_{23} = 139.803966763363
x24=73.8321624487981x_{24} = 73.8321624487981
Puntos máximos de la función:
x24=34.5496630820819x_{24} = 34.5496630820819
x24=15.6946678544559x_{24} = 15.6946678544559
x24=28.2653525829836x_{24} = 28.2653525829836
x24=6.25863784020032x_{24} = 6.25863784020032
x24=50.2593633942295x_{24} = 50.2593633942295
x24=12.5509380918273x_{24} = 12.5509380918273
x24=21.980527723609x_{24} = 21.980527723609
x24=87.9603809024344x_{24} = 87.9603809024344
x24=59.6848038675201x_{24} = 59.6848038675201
x24=78.5352722360833x_{24} = 78.5352722360833
x24=81.676982179018x_{24} = 81.676982179018
x24=37.6916986691167x_{24} = 37.6916986691167
x24=43.9756082127529x_{24} = 43.9756082127529
x24=100.527110418962x_{24} = 100.527110418962
x24=40.8336766750392x_{24} = 40.8336766750392
x24=72.2518271445003x_{24} = 72.2518271445003
x24=65.9683414285725x_{24} = 65.9683414285725
x24=94.2437556347455x_{24} = 94.2437556347455
x24=56.5430109050366x_{24} = 56.5430109050366
Decrece en los intervalos
[139.803966763363,∞)\left[139.803966763363, \infty\right)
Crece en los intervalos
(−∞,1.63108293582399]\left(-\infty, 1.63108293582399\right]
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
2(−2cos⁡(2x)+19x53)=02 \left(- 2 \cos{\left(2 x \right)} + \frac{1}{9 x^{\frac{5}{3}}}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=8.64014333320374x_{1} = 8.64014333320374
x2=54.1924374812929x_{2} = 54.1924374812929
x3=60.47562876921x_{3} = 60.47562876921
x4=66.7588186045514x_{4} = 66.7588186045514
x5=19.6347597034675x_{5} = 19.6347597034675
x6=3.92414553924722x_{6} = 3.92414553924722
x7=85.6083831056862x_{7} = 85.6083831056862
x8=22.7763949557363x_{8} = 22.7763949557363
x9=49.4801259470283x_{9} = 49.4801259470283
x10=32.2012394724291x_{10} = 32.2012394724291
x11=69.9004131235752x_{11} = 69.9004131235752
x12=88.7499767331258x_{12} = 88.7499767331258
x13=90.3208040681672x_{13} = 90.3208040681672
x14=87.1792123431204x_{14} = 87.1792123431204
x15=30.63062100666x_{15} = 30.63062100666
x16=91.8915702728031x_{16} = 91.8915702728031
x17=52.6217145390875x_{17} = 52.6217145390875
x18=63.6172238358822x_{18} = 63.6172238358822
x19=10.209598006122x_{19} = 10.209598006122
x20=40.055365570717x_{20} = 40.055365570717
x21=25.9180170165952x_{21} = 25.9180170165952
x22=46.3385381055576x_{22} = 46.3385381055576
x23=84.0376207117804x_{23} = 84.0376207117804
x24=99.7455796996363x_{24} = 99.7455796996363
x25=18.0643811114387x_{25} = 18.0643811114387
x26=98.1747571293923x_{26} = 98.1747571293923
x27=38.4844466843585x_{27} = 38.4844466843585
x28=55.7633037296774x_{28} = 55.7633037296774
x29=47.9092440132208x_{29} = 47.9092440132208
x30=96.6039877554239x_{30} = 96.6039877554239
x31=44.7676460994579x_{31} = 44.7676460994579
x32=33.7721997485373x_{32} = 33.7721997485373
x33=77.754437786856x_{33} = 77.754437786856
x34=76.1836015605047x_{34} = 76.1836015605047
x35=24.3474788781643x_{35} = 24.3474788781643
x36=11.7814278243854x_{36} = 11.7814278243854
x37=16.4931014990166x_{37} = 16.4931014990166
x38=41.6260471009205x_{38} = 41.6260471009205
x39=68.3296645384779x_{39} = 68.3296645384779
x40=74.6128465286668x_{40} = 74.6128465286668
x41=2.36282161763644x_{41} = 2.36282161763644
x42=82.4667893780722x_{42} = 82.4667893780722
x43=27.4890466614338x_{43} = 27.4890466614338
x44=175.14379550411x_{44} = 175.14379550411
x45=62.046483473485x_{45} = 62.046483473485
x46=5.49940835019242x_{46} = 5.49940835019242

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
[98.1747571293923,∞)\left[98.1747571293923, \infty\right)
Convexa en los intervalos
(−∞,3.92414553924722]\left(-\infty, 3.92414553924722\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(−x3+cos⁡(2x))=⟨−1,1⟩−∞−13\lim_{x \to -\infty}\left(- \sqrt[3]{x} + \cos{\left(2 x \right)}\right) = \left\langle -1, 1\right\rangle - \infty \sqrt[3]{-1}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−1,1⟩−∞−13y = \left\langle -1, 1\right\rangle - \infty \sqrt[3]{-1}
lim⁡x→∞(−x3+cos⁡(2x))=−∞\lim_{x \to \infty}\left(- \sqrt[3]{x} + \cos{\left(2 x \right)}\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 cos(2*x) - x^(1/3), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(−x3+cos⁡(2x)x)=0\lim_{x \to -\infty}\left(\frac{- \sqrt[3]{x} + \cos{\left(2 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(−x3+cos⁡(2x)x)=0\lim_{x \to \infty}\left(\frac{- \sqrt[3]{x} + \cos{\left(2 x \right)}}{x}\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:
−x3+cos⁡(2x)=−−x3+cos⁡(2x)- \sqrt[3]{x} + \cos{\left(2 x \right)} = - \sqrt[3]{- x} + \cos{\left(2 x \right)}
- No
−x3+cos⁡(2x)=−x3−cos⁡(2x)- \sqrt[3]{x} + \cos{\left(2 x \right)} = \sqrt[3]{- x} - \cos{\left(2 x \right)}
- No
es decir, función
no es
par ni impar
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