Sr Examen
Otras calculadoras
- Gráfico de la función implícita
- Derivadas paso a paso
- Integrales paso a paso
- Gráfico de la función paramétrica
- Gráfico en coordenadas polares
- Superficie especificada paramétricamente
- Superficie dada por la ecuación
- Curva en el espacio especificada paramétricamente
- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
16*sqrt(2*x)
-
y=x^3-3x^2+5
-
x/x-5
-
y=4x^2-x^3-1
-
Expresiones idénticas
- - uno -x^ dos -cos(dos *x)-sin(dos *x)+x*(- uno -cos(dos *x)-sin(dos *x))
- menos 1 menos x al cuadrado menos coseno de (2 multiplicar por x) menos seno de (2 multiplicar por x) más x multiplicar por ( menos 1 menos coseno de (2 multiplicar por x) menos seno de (2 multiplicar por x))
- menos uno menos x en el grado dos menos coseno de (dos multiplicar por x) menos seno de (dos multiplicar por x) más x multiplicar por ( menos uno menos coseno de (dos multiplicar por x) menos seno de (dos multiplicar por x))
- -1-x2-cos(2*x)-sin(2*x)+x*(-1-cos(2*x)-sin(2*x))
- -1-x2-cos2*x-sin2*x+x*-1-cos2*x-sin2*x
- -1-x²-cos(2*x)-sin(2*x)+x*(-1-cos(2*x)-sin(2*x))
- -1-x en el grado 2-cos(2*x)-sin(2*x)+x*(-1-cos(2*x)-sin(2*x))
- -1-x^2-cos(2x)-sin(2x)+x(-1-cos(2x)-sin(2x))
- -1-x2-cos(2x)-sin(2x)+x(-1-cos(2x)-sin(2x))
- -1-x2-cos2x-sin2x+x-1-cos2x-sin2x
- -1-x^2-cos2x-sin2x+x-1-cos2x-sin2x
-
Expresiones semejantes
- -1-x^2-cos(2*x)-sin(2*x)+x*(-1-cos(2*x)+sin(2*x))
- -1-x^2-cos(2*x)+sin(2*x)+x*(-1-cos(2*x)-sin(2*x))
- 1-x^2-cos(2*x)-sin(2*x)+x*(-1-cos(2*x)-sin(2*x))
- -1+x^2-cos(2*x)-sin(2*x)+x*(-1-cos(2*x)-sin(2*x))
- -1-x^2-cos(2*x)-sin(2*x)-x*(-1-cos(2*x)-sin(2*x))
- -1-x^2+cos(2*x)-sin(2*x)+x*(-1-cos(2*x)-sin(2*x))
- -1-x^2-cos(2*x)-sin(2*x)+x*(1-cos(2*x)-sin(2*x))
- -1-x^2-cos(2*x)-sin(2*x)+x*(-1+cos(2*x)-sin(2*x))
-
Expresiones con funciones
- Coseno cos
- cos2x+x^3-4
- cos(x)^cos(1)*exp(-x*sin(1))
- cos(x+pi/4)+2
- cos(x)*((2*x^3-5)^3)^4
- cos2x-sinx
- Seno sin
- sin(2*x-pi/4)
- sin(5+6*x)^(2)+3^tan(x)
- sin(x)+arcsin(x)
- sin(2*x^3)^2/(x^3)
- sin(x)+sin(x+120)
- Coseno cos
- cos2x+x^3-4
- cos(x)^cos(1)*exp(-x*sin(1))
- cos(x+pi/4)+2
- cos(x)*((2*x^3-5)^3)^4
- cos2x-sinx
- Seno sin
- sin(2*x-pi/4)
- sin(5+6*x)^(2)+3^tan(x)
- sin(x)+arcsin(x)
- sin(2*x^3)^2/(x^3)
- sin(x)+sin(x+120)
Gráfico de la función y = -1-x^2-cos(2*x)-sin(2*x)+x*(-1-cos(2*x)-sin(2*x))
Solución
Ha introducido
[src]
2 f(x) = -1 - x - cos(2*x) - sin(2*x) + x*(-1 - cos(2*x) - sin(2*x))
$$f{\left(x \right)} = x \left(\left(- \cos{\left(2 x \right)} - 1\right) - \sin{\left(2 x \right)}\right) + \left(\left(\left(- x^{2} - 1\right) - \cos{\left(2 x \right)}\right) - \sin{\left(2 x \right)}\right)$$
f = x*(-cos(2*x) - 1 - sin(2*x)) - x^2 - 1 - cos(2*x) - sin(2*x)
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:
$$x \left(\left(- \cos{\left(2 x \right)} - 1\right) - \sin{\left(2 x \right)}\right) + \left(\left(\left(- x^{2} - 1\right) - \cos{\left(2 x \right)}\right) - \sin{\left(2 x \right)}\right) = 0$$
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
o sea hay que resolver la ecuación:
$$x \left(\left(- \cos{\left(2 x \right)} - 1\right) - \sin{\left(2 x \right)}\right) + \left(\left(\left(- x^{2} - 1\right) - \cos{\left(2 x \right)}\right) - \sin{\left(2 x \right)}\right) = 0$$
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
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
$$x \left(2 \sin{\left(2 x \right)} - 2 \cos{\left(2 x \right)}\right) - 2 x + \sin{\left(2 x \right)} - 3 \cos{\left(2 x \right)} - 1 = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 10.2101761241668$$
$$x_{2} = -2.02474021721027$$
$$x_{3} = -45.5643127462463$$
$$x_{4} = -7.9260478570743$$
$$x_{5} = 3.92699081698724$$
$$x_{6} = -36.1425423260911$$
$$x_{7} = -11.7809724509617$$
$$x_{8} = 23.5822819506365$$
$$x_{9} = 83.2581393952788$$
$$x_{10} = -29.8624520219236$$
$$x_{11} = 69.9004365423729$$
$$x_{12} = 98.174770424681$$
$$x_{13} = 67.5515357054677$$
$$x_{14} = 58.1279201318298$$
$$x_{15} = -23.5840807743632$$
$$x_{16} = -51.8461120608397$$
$$x_{17} = -33.7721210260903$$
$$x_{18} = 89.5409129365185$$
$$x_{19} = 14.1701145541021$$
$$x_{20} = -18.0641577581413$$
$$x_{21} = -89.5410376662227$$
$$x_{22} = -20.4460587354271$$
$$x_{23} = -14.1750991164082$$
$$x_{24} = 44.7676953136546$$
$$x_{25} = 20.4436649338304$$
$$x_{26} = -49.4800842940392$$
$$x_{27} = 29.8613302968488$$
$$x_{28} = 60.4756585816035$$
$$x_{29} = -77.7544181763474$$
$$x_{30} = -87.1791961371168$$
$$x_{31} = 39.2823199258332$$
$$x_{32} = 7.91003931549699$$
$$x_{33} = -93.4623814442964$$
$$x_{34} = -95.8238488207122$$
$$x_{35} = -43.1968989868597$$
$$x_{36} = -5.49778714378214$$
$$x_{37} = -42.4235706598293$$
$$x_{38} = -80.1169323420984$$
$$x_{39} = 80.1167765414997$$
$$x_{40} = -62.0464549083984$$
$$x_{41} = 17.3060661561232$$
$$x_{42} = 61.2690862390034$$
$$x_{43} = 54.1924732744239$$
$$x_{44} = 88.7499924639117$$
$$x_{45} = -99.7455667514759$$
$$x_{46} = 36.1417766332425$$
$$x_{47} = 82.4668071567321$$
$$x_{48} = -71.4712328691678$$
$$x_{49} = -21.2057504117311$$
$$x_{50} = 42.4230149457449$$
$$x_{51} = -64.4105343603388$$
$$x_{52} = -73.8342921497081$$
$$x_{53} = 64.4102933065518$$
$$x_{54} = 45.5638310124567$$
$$x_{55} = -27.4889357189107$$
$$x_{56} = -70.693008903436$$
$$x_{57} = -48.7051667918807$$
$$x_{58} = 66.7588438887831$$
$$x_{59} = -26.7229729855743$$
$$x_{60} = 95.8237399113812$$
$$x_{61} = 47.9092879672443$$
$$x_{62} = -55.7632696012188$$
$$x_{63} = -0.633079232665435$$
$$x_{64} = -65.1880475619882$$
$$x_{65} = 32.2013246992954$$
$$x_{66} = -102.887159405066$$
$$x_{67} = 22.776546738526$$
$$x_{68} = -67.5517548607303$$
$$x_{69} = -84.037603483527$$
$$x_{70} = 38.484510006475$$
$$x_{71} = 51.8457400025016$$
$$x_{72} = -58.1282161104251$$
$$x_{73} = 76.1836218495525$$
$$x_{74} = -86.3996527315662$$
$$x_{75} = -92.6824368340589$$
$$x_{76} = 25.9181393921158$$
$$x_{77} = 73.8341087051871$$
$$x_{78} = -40.0553063332699$$
$$x_{79} = 1.75061074716993$$
$$x_{80} = 16.4933614313464$$
$$x_{81} = 86.3995187666077$$
$$x_{82} = 91.8915851175014$$
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:
$$x_{1} = 10.2101761241668$$
$$x_{2} = -2.02474021721027$$
$$x_{3} = -45.5643127462463$$
$$x_{4} = -7.9260478570743$$
$$x_{5} = 3.92699081698724$$
$$x_{6} = -36.1425423260911$$
$$x_{7} = -29.8624520219236$$
$$x_{8} = 69.9004365423729$$
$$x_{9} = 98.174770424681$$
$$x_{10} = -23.5840807743632$$
$$x_{11} = -51.8461120608397$$
$$x_{12} = -89.5410376662227$$
$$x_{13} = -20.4460587354271$$
$$x_{14} = -14.1750991164082$$
$$x_{15} = 44.7676953136546$$
$$x_{16} = 60.4756585816035$$
$$x_{17} = -95.8238488207122$$
$$x_{18} = -42.4235706598293$$
$$x_{19} = -80.1169323420984$$
$$x_{20} = 54.1924732744239$$
$$x_{21} = 88.7499924639117$$
$$x_{22} = 82.4668071567321$$
$$x_{23} = -64.4105343603388$$
$$x_{24} = -73.8342921497081$$
$$x_{25} = -70.693008903436$$
$$x_{26} = -48.7051667918807$$
$$x_{27} = 66.7588438887831$$
$$x_{28} = -26.7229729855743$$
$$x_{29} = 47.9092879672443$$
$$x_{30} = 32.2013246992954$$
$$x_{31} = 22.776546738526$$
$$x_{32} = -67.5517548607303$$
$$x_{33} = 38.484510006475$$
$$x_{34} = -58.1282161104251$$
$$x_{35} = 76.1836218495525$$
$$x_{36} = -86.3996527315662$$
$$x_{37} = -92.6824368340589$$
$$x_{38} = 25.9181393921158$$
$$x_{39} = 16.4933614313464$$
$$x_{40} = 91.8915851175014$$
Puntos máximos de la función:
$$x_{40} = -11.7809724509617$$
$$x_{40} = 23.5822819506365$$
$$x_{40} = 83.2581393952788$$
$$x_{40} = 67.5515357054677$$
$$x_{40} = 58.1279201318298$$
$$x_{40} = -33.7721210260903$$
$$x_{40} = 89.5409129365185$$
$$x_{40} = 14.1701145541021$$
$$x_{40} = -18.0641577581413$$
$$x_{40} = 20.4436649338304$$
$$x_{40} = -49.4800842940392$$
$$x_{40} = 29.8613302968488$$
$$x_{40} = -77.7544181763474$$
$$x_{40} = -87.1791961371168$$
$$x_{40} = 39.2823199258332$$
$$x_{40} = 7.91003931549699$$
$$x_{40} = -93.4623814442964$$
$$x_{40} = -43.1968989868597$$
$$x_{40} = -5.49778714378214$$
$$x_{40} = 80.1167765414997$$
$$x_{40} = -62.0464549083984$$
$$x_{40} = 17.3060661561232$$
$$x_{40} = 61.2690862390034$$
$$x_{40} = -99.7455667514759$$
$$x_{40} = 36.1417766332425$$
$$x_{40} = -71.4712328691678$$
$$x_{40} = -21.2057504117311$$
$$x_{40} = 42.4230149457449$$
$$x_{40} = 64.4102933065518$$
$$x_{40} = 45.5638310124567$$
$$x_{40} = -27.4889357189107$$
$$x_{40} = 95.8237399113812$$
$$x_{40} = -55.7632696012188$$
$$x_{40} = -0.633079232665435$$
$$x_{40} = -65.1880475619882$$
$$x_{40} = -102.887159405066$$
$$x_{40} = -84.037603483527$$
$$x_{40} = 51.8457400025016$$
$$x_{40} = 73.8341087051871$$
$$x_{40} = -40.0553063332699$$
$$x_{40} = 1.75061074716993$$
$$x_{40} = 86.3995187666077$$
Decrece en los intervalos
$$\left[98.174770424681, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -95.8238488207122\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$x \left(2 \sin{\left(2 x \right)} - 2 \cos{\left(2 x \right)}\right) - 2 x + \sin{\left(2 x \right)} - 3 \cos{\left(2 x \right)} - 1 = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 10.2101761241668$$
$$x_{2} = -2.02474021721027$$
$$x_{3} = -45.5643127462463$$
$$x_{4} = -7.9260478570743$$
$$x_{5} = 3.92699081698724$$
$$x_{6} = -36.1425423260911$$
$$x_{7} = -11.7809724509617$$
$$x_{8} = 23.5822819506365$$
$$x_{9} = 83.2581393952788$$
$$x_{10} = -29.8624520219236$$
$$x_{11} = 69.9004365423729$$
$$x_{12} = 98.174770424681$$
$$x_{13} = 67.5515357054677$$
$$x_{14} = 58.1279201318298$$
$$x_{15} = -23.5840807743632$$
$$x_{16} = -51.8461120608397$$
$$x_{17} = -33.7721210260903$$
$$x_{18} = 89.5409129365185$$
$$x_{19} = 14.1701145541021$$
$$x_{20} = -18.0641577581413$$
$$x_{21} = -89.5410376662227$$
$$x_{22} = -20.4460587354271$$
$$x_{23} = -14.1750991164082$$
$$x_{24} = 44.7676953136546$$
$$x_{25} = 20.4436649338304$$
$$x_{26} = -49.4800842940392$$
$$x_{27} = 29.8613302968488$$
$$x_{28} = 60.4756585816035$$
$$x_{29} = -77.7544181763474$$
$$x_{30} = -87.1791961371168$$
$$x_{31} = 39.2823199258332$$
$$x_{32} = 7.91003931549699$$
$$x_{33} = -93.4623814442964$$
$$x_{34} = -95.8238488207122$$
$$x_{35} = -43.1968989868597$$
$$x_{36} = -5.49778714378214$$
$$x_{37} = -42.4235706598293$$
$$x_{38} = -80.1169323420984$$
$$x_{39} = 80.1167765414997$$
$$x_{40} = -62.0464549083984$$
$$x_{41} = 17.3060661561232$$
$$x_{42} = 61.2690862390034$$
$$x_{43} = 54.1924732744239$$
$$x_{44} = 88.7499924639117$$
$$x_{45} = -99.7455667514759$$
$$x_{46} = 36.1417766332425$$
$$x_{47} = 82.4668071567321$$
$$x_{48} = -71.4712328691678$$
$$x_{49} = -21.2057504117311$$
$$x_{50} = 42.4230149457449$$
$$x_{51} = -64.4105343603388$$
$$x_{52} = -73.8342921497081$$
$$x_{53} = 64.4102933065518$$
$$x_{54} = 45.5638310124567$$
$$x_{55} = -27.4889357189107$$
$$x_{56} = -70.693008903436$$
$$x_{57} = -48.7051667918807$$
$$x_{58} = 66.7588438887831$$
$$x_{59} = -26.7229729855743$$
$$x_{60} = 95.8237399113812$$
$$x_{61} = 47.9092879672443$$
$$x_{62} = -55.7632696012188$$
$$x_{63} = -0.633079232665435$$
$$x_{64} = -65.1880475619882$$
$$x_{65} = 32.2013246992954$$
$$x_{66} = -102.887159405066$$
$$x_{67} = 22.776546738526$$
$$x_{68} = -67.5517548607303$$
$$x_{69} = -84.037603483527$$
$$x_{70} = 38.484510006475$$
$$x_{71} = 51.8457400025016$$
$$x_{72} = -58.1282161104251$$
$$x_{73} = 76.1836218495525$$
$$x_{74} = -86.3996527315662$$
$$x_{75} = -92.6824368340589$$
$$x_{76} = 25.9181393921158$$
$$x_{77} = 73.8341087051871$$
$$x_{78} = -40.0553063332699$$
$$x_{79} = 1.75061074716993$$
$$x_{80} = 16.4933614313464$$
$$x_{81} = 86.3995187666077$$
$$x_{82} = 91.8915851175014$$
Signos de extremos en los puntos:
(10.210176124166829, -126.66804873484)
(-2.0247402172102715, -2.89776414817754)
(-45.56431274624626, -2075.0955035767)
(-7.926047857074296, -61.7556022254967)
(3.9269908169872414, -25.2752385106766)
(-36.14254232609107, -1305.26934329155)
(-11.780972450961725, -117.229366988396)
(23.582281950636478, -555.144766979483)
(83.25813939527882, -6930.92374471236)
(-29.862452021923605, -890.749022428737)
(69.9004365423729, -5027.87190189905)
(98.17477042468104, -9836.63508878819)
(67.55153570546773, -4562.21732275814)
(58.127920131829846, -3377.86362599263)
(-23.584080774363215, -555.187227246367)
(-51.8461120608397, -2687.00959987261)
(-33.772121026090275, -1075.01191654871)
(89.54091293651854, -8016.58064219772)
(14.17011455410207, -199.826155402871)
(-18.06415775814131, -292.185479994734)
(-89.54103766622266, -8016.59181131375)
(-20.44605873542713, -417.016283326842)
(-14.175099116408168, -199.896977320033)
(44.767695313654556, -2095.68193432352)
(20.443664933830423, -416.967283549325)
(-49.480084294039244, -2351.31857315715)
(29.86133029684883, -890.715506770883)
(60.47565858160352, -3780.25659804188)
(-77.75441817634739, -5892.24070958961)
(-87.17919613711676, -7427.85384683964)
(39.28231992583324, -1542.11322328059)
(7.910039315496987, -61.6278014618777)
(-93.46238144429635, -8550.29198235056)
(-95.82384882071216, -9181.20475782928)
(-43.19689898685966, -1781.57828410724)
(-5.497787143782138, -21.2300891907719)
(-42.42357065982932, -1798.74742453924)
(-80.1169323420984, -6417.71656833884)
(80.1167765414997, -6417.70408511231)
(-62.04645490839842, -3727.66965688312)
(17.306066156123187, -298.527964252438)
(61.26908623900338, -3752.90902218271)
(54.19247327442393, -3047.209106148)
(88.74999246391165, -8056.0611472722)
(-99.74556675147593, -9751.68695307019)
(36.14177663324248, -1305.24165888882)
(82.46680715673207, -6967.7078969391)
(-71.47123286916779, -4967.19466210048)
(-21.205750411731103, -409.272349701172)
(42.423014945744924, -1798.7238427607)
(-64.41053436033879, -4147.70911412078)
(-73.83429214970812, -5450.49587979838)
(64.41029330655181, -4147.69358587444)
(45.563831012456696, -2075.07354853205)
(-27.488935718910692, -702.663715520583)
(-70.69300890343597, -4996.49438533801)
(-48.70516679188066, -2371.18290217139)
(66.7588438887831, -4592.26092514448)
(-26.722972985574327, -713.098232339994)
(95.82373991138121, -9181.19432115554)
(47.909287967244346, -2393.11844946283)
(-55.76326960121883, -3000.01569741578)
(-0.6330792326654353, -0.527740999100943)
(-65.18804756198821, -4121.10544982006)
(32.201324699295384, -1103.32796178804)
(-102.88715940506573, -10381.9932516333)
(22.776546738526, -566.324174809311)
(-67.55175486073027, -4562.23212867957)
(-84.03760348352696, -6896.24359228745)
(38.48451000647497, -1560.02653045142)
(51.84574000250157, -2686.99030657264)
(-58.12821611042508, -3377.88083320407)
(76.18362184955248, -5958.31148181472)
(-86.39965273156625, -7463.89417178899)
(-92.68243683405893, -8589.02867379524)
(25.918139392115794, -725.586228333376)
(73.83410870518713, -5450.4823340761)
(-40.05530633326986, -1526.31695278555)
(1.75061074716993, -2.27258765470913)
(16.493361431346415, -307.017694167718)
(86.39951876660771, -7463.88259649658)
(91.89158511750145, -8629.84658564202)
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:
$$x_{1} = 10.2101761241668$$
$$x_{2} = -2.02474021721027$$
$$x_{3} = -45.5643127462463$$
$$x_{4} = -7.9260478570743$$
$$x_{5} = 3.92699081698724$$
$$x_{6} = -36.1425423260911$$
$$x_{7} = -29.8624520219236$$
$$x_{8} = 69.9004365423729$$
$$x_{9} = 98.174770424681$$
$$x_{10} = -23.5840807743632$$
$$x_{11} = -51.8461120608397$$
$$x_{12} = -89.5410376662227$$
$$x_{13} = -20.4460587354271$$
$$x_{14} = -14.1750991164082$$
$$x_{15} = 44.7676953136546$$
$$x_{16} = 60.4756585816035$$
$$x_{17} = -95.8238488207122$$
$$x_{18} = -42.4235706598293$$
$$x_{19} = -80.1169323420984$$
$$x_{20} = 54.1924732744239$$
$$x_{21} = 88.7499924639117$$
$$x_{22} = 82.4668071567321$$
$$x_{23} = -64.4105343603388$$
$$x_{24} = -73.8342921497081$$
$$x_{25} = -70.693008903436$$
$$x_{26} = -48.7051667918807$$
$$x_{27} = 66.7588438887831$$
$$x_{28} = -26.7229729855743$$
$$x_{29} = 47.9092879672443$$
$$x_{30} = 32.2013246992954$$
$$x_{31} = 22.776546738526$$
$$x_{32} = -67.5517548607303$$
$$x_{33} = 38.484510006475$$
$$x_{34} = -58.1282161104251$$
$$x_{35} = 76.1836218495525$$
$$x_{36} = -86.3996527315662$$
$$x_{37} = -92.6824368340589$$
$$x_{38} = 25.9181393921158$$
$$x_{39} = 16.4933614313464$$
$$x_{40} = 91.8915851175014$$
Puntos máximos de la función:
$$x_{40} = -11.7809724509617$$
$$x_{40} = 23.5822819506365$$
$$x_{40} = 83.2581393952788$$
$$x_{40} = 67.5515357054677$$
$$x_{40} = 58.1279201318298$$
$$x_{40} = -33.7721210260903$$
$$x_{40} = 89.5409129365185$$
$$x_{40} = 14.1701145541021$$
$$x_{40} = -18.0641577581413$$
$$x_{40} = 20.4436649338304$$
$$x_{40} = -49.4800842940392$$
$$x_{40} = 29.8613302968488$$
$$x_{40} = -77.7544181763474$$
$$x_{40} = -87.1791961371168$$
$$x_{40} = 39.2823199258332$$
$$x_{40} = 7.91003931549699$$
$$x_{40} = -93.4623814442964$$
$$x_{40} = -43.1968989868597$$
$$x_{40} = -5.49778714378214$$
$$x_{40} = 80.1167765414997$$
$$x_{40} = -62.0464549083984$$
$$x_{40} = 17.3060661561232$$
$$x_{40} = 61.2690862390034$$
$$x_{40} = -99.7455667514759$$
$$x_{40} = 36.1417766332425$$
$$x_{40} = -71.4712328691678$$
$$x_{40} = -21.2057504117311$$
$$x_{40} = 42.4230149457449$$
$$x_{40} = 64.4102933065518$$
$$x_{40} = 45.5638310124567$$
$$x_{40} = -27.4889357189107$$
$$x_{40} = 95.8237399113812$$
$$x_{40} = -55.7632696012188$$
$$x_{40} = -0.633079232665435$$
$$x_{40} = -65.1880475619882$$
$$x_{40} = -102.887159405066$$
$$x_{40} = -84.037603483527$$
$$x_{40} = 51.8457400025016$$
$$x_{40} = 73.8341087051871$$
$$x_{40} = -40.0553063332699$$
$$x_{40} = 1.75061074716993$$
$$x_{40} = 86.3995187666077$$
Decrece en los intervalos
$$\left[98.174770424681, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -95.8238488207122\right]$$
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
$$2 \left(2 x \left(\sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) + 4 \sin{\left(2 x \right)} - 1\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 5.98662612511467$$
$$x_{2} = 45.1673946790362$$
$$x_{3} = -23.9687095577235$$
$$x_{4} = 84.4382234024735$$
$$x_{5} = -63.2354255716521$$
$$x_{6} = -11.419223048863$$
$$x_{7} = -88.3650395443825$$
$$x_{8} = 100.144956746368$$
$$x_{9} = 18.4915445047263$$
$$x_{10} = -89.9317241032059$$
$$x_{11} = 40.4643237937789$$
$$x_{12} = -60.0944107474569$$
$$x_{13} = 62.4498193359286$$
$$x_{14} = 46.7453631236684$$
$$x_{15} = 57.7322679395137$$
$$x_{16} = -6.79147196864804$$
$$x_{17} = -97.7890633505603$$
$$x_{18} = 56.1678057612487$$
$$x_{19} = 71.8732183441602$$
$$x_{20} = -39.6709641291965$$
$$x_{21} = -9.89322279582156$$
$$x_{22} = -91.5063633665141$$
$$x_{23} = -47.5311309913532$$
$$x_{24} = 35.7444113127547$$
$$x_{25} = -77.3659514684403$$
$$x_{26} = 0.0971044674640491$$
$$x_{27} = 13.7663316668534$$
$$x_{28} = 26.3226643185088$$
$$x_{29} = -3.76785371331432$$
$$x_{30} = 4.37925880139647$$
$$x_{31} = -99.3561538376491$$
$$x_{32} = 76.5804870521455$$
$$x_{33} = 7.49917986705158$$
$$x_{34} = -41.2502139634449$$
$$x_{35} = 93.8622145441781$$
$$x_{36} = 54.5909861905112$$
$$x_{37} = -53.8125871721272$$
$$x_{38} = -22.4154243999115$$
$$x_{39} = -8.2908041153742$$
$$x_{40} = -33.3893988264775$$
$$x_{41} = 12.2247383012288$$
$$x_{42} = 64.0149215436354$$
$$x_{43} = 90.7208662023924$$
$$x_{44} = 79.7219175869372$$
$$x_{45} = -82.0824543747234$$
$$x_{46} = 27.9050381968332$$
$$x_{47} = -1.23722674404392$$
$$x_{48} = 86.0048137724282$$
$$x_{49} = -75.7999699876967$$
$$x_{50} = 20.0430045110297$$
$$x_{51} = -19.2792382629882$$
$$x_{52} = 1.3118112850912$$
$$x_{53} = 10.6306140273053$$
$$x_{54} = 68.7320439462207$$
$$x_{55} = 92.2877488973094$$
$$x_{56} = -31.830568528454$$
$$x_{57} = -66.3764958955262$$
$$x_{58} = 32.6036403210211$$
$$x_{59} = -80.5073769098954$$
$$x_{60} = -69.5176140911923$$
$$x_{61} = -58.5177822794283$$
$$x_{62} = 43.6047679802649$$
$$x_{63} = 70.2976688323528$$
$$x_{64} = 24.7662934366462$$
$$x_{65} = 49.8860813075943$$
$$x_{66} = -16.1452758563102$$
$$x_{67} = 103.286347865261$$
$$x_{68} = -2.19713035671063$$
$$x_{69} = 42.0263129371444$$
$$x_{70} = 98.5707155916995$$
$$x_{71} = -17.6908064293837$$
$$x_{72} = 21.6283361974761$$
$$x_{73} = 48.3085414833511$$
$$x_{74} = -38.1100430310972$$
$$x_{75} = 87.5795351850854$$
$$x_{76} = -45.9529819095161$$
$$x_{77} = -44.3905904706969$$
$$x_{78} = 65.5909089920649$$
$$x_{79} = 2.91763078715379$$
$$x_{80} = -107.213220962648$$
$$x_{81} = 34.1840496662439$$
$$x_{82} = -30.2488768108393$$
$$x_{83} = -61.6590839822113$$
$$x_{84} = -55.3765141809433$$
$$x_{85} = -96.2146696009281$$
$$x_{86} = -83.6488150605181$$
$$x_{87} = 78.1556666975032$$
$$x_{88} = -52.235285864066$$
$$x_{89} = -67.9417692855713$$
$$x_{90} = -25.5529872706575$$
$$x_{91} = -74.2245403713941$$
$$x_{92} = -119.778917552783$$
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
$$\left[103.286347865261, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -99.3561538376491\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$2 \left(2 x \left(\sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) + 4 \sin{\left(2 x \right)} - 1\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 5.98662612511467$$
$$x_{2} = 45.1673946790362$$
$$x_{3} = -23.9687095577235$$
$$x_{4} = 84.4382234024735$$
$$x_{5} = -63.2354255716521$$
$$x_{6} = -11.419223048863$$
$$x_{7} = -88.3650395443825$$
$$x_{8} = 100.144956746368$$
$$x_{9} = 18.4915445047263$$
$$x_{10} = -89.9317241032059$$
$$x_{11} = 40.4643237937789$$
$$x_{12} = -60.0944107474569$$
$$x_{13} = 62.4498193359286$$
$$x_{14} = 46.7453631236684$$
$$x_{15} = 57.7322679395137$$
$$x_{16} = -6.79147196864804$$
$$x_{17} = -97.7890633505603$$
$$x_{18} = 56.1678057612487$$
$$x_{19} = 71.8732183441602$$
$$x_{20} = -39.6709641291965$$
$$x_{21} = -9.89322279582156$$
$$x_{22} = -91.5063633665141$$
$$x_{23} = -47.5311309913532$$
$$x_{24} = 35.7444113127547$$
$$x_{25} = -77.3659514684403$$
$$x_{26} = 0.0971044674640491$$
$$x_{27} = 13.7663316668534$$
$$x_{28} = 26.3226643185088$$
$$x_{29} = -3.76785371331432$$
$$x_{30} = 4.37925880139647$$
$$x_{31} = -99.3561538376491$$
$$x_{32} = 76.5804870521455$$
$$x_{33} = 7.49917986705158$$
$$x_{34} = -41.2502139634449$$
$$x_{35} = 93.8622145441781$$
$$x_{36} = 54.5909861905112$$
$$x_{37} = -53.8125871721272$$
$$x_{38} = -22.4154243999115$$
$$x_{39} = -8.2908041153742$$
$$x_{40} = -33.3893988264775$$
$$x_{41} = 12.2247383012288$$
$$x_{42} = 64.0149215436354$$
$$x_{43} = 90.7208662023924$$
$$x_{44} = 79.7219175869372$$
$$x_{45} = -82.0824543747234$$
$$x_{46} = 27.9050381968332$$
$$x_{47} = -1.23722674404392$$
$$x_{48} = 86.0048137724282$$
$$x_{49} = -75.7999699876967$$
$$x_{50} = 20.0430045110297$$
$$x_{51} = -19.2792382629882$$
$$x_{52} = 1.3118112850912$$
$$x_{53} = 10.6306140273053$$
$$x_{54} = 68.7320439462207$$
$$x_{55} = 92.2877488973094$$
$$x_{56} = -31.830568528454$$
$$x_{57} = -66.3764958955262$$
$$x_{58} = 32.6036403210211$$
$$x_{59} = -80.5073769098954$$
$$x_{60} = -69.5176140911923$$
$$x_{61} = -58.5177822794283$$
$$x_{62} = 43.6047679802649$$
$$x_{63} = 70.2976688323528$$
$$x_{64} = 24.7662934366462$$
$$x_{65} = 49.8860813075943$$
$$x_{66} = -16.1452758563102$$
$$x_{67} = 103.286347865261$$
$$x_{68} = -2.19713035671063$$
$$x_{69} = 42.0263129371444$$
$$x_{70} = 98.5707155916995$$
$$x_{71} = -17.6908064293837$$
$$x_{72} = 21.6283361974761$$
$$x_{73} = 48.3085414833511$$
$$x_{74} = -38.1100430310972$$
$$x_{75} = 87.5795351850854$$
$$x_{76} = -45.9529819095161$$
$$x_{77} = -44.3905904706969$$
$$x_{78} = 65.5909089920649$$
$$x_{79} = 2.91763078715379$$
$$x_{80} = -107.213220962648$$
$$x_{81} = 34.1840496662439$$
$$x_{82} = -30.2488768108393$$
$$x_{83} = -61.6590839822113$$
$$x_{84} = -55.3765141809433$$
$$x_{85} = -96.2146696009281$$
$$x_{86} = -83.6488150605181$$
$$x_{87} = 78.1556666975032$$
$$x_{88} = -52.235285864066$$
$$x_{89} = -67.9417692855713$$
$$x_{90} = -25.5529872706575$$
$$x_{91} = -74.2245403713941$$
$$x_{92} = -119.778917552783$$
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
$$\left[103.286347865261, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -99.3561538376491\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(x \left(\left(- \cos{\left(2 x \right)} - 1\right) - \sin{\left(2 x \right)}\right) + \left(\left(\left(- x^{2} - 1\right) - \cos{\left(2 x \right)}\right) - \sin{\left(2 x \right)}\right)\right) = -\infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
$$\lim_{x \to \infty}\left(x \left(\left(- \cos{\left(2 x \right)} - 1\right) - \sin{\left(2 x \right)}\right) + \left(\left(\left(- x^{2} - 1\right) - \cos{\left(2 x \right)}\right) - \sin{\left(2 x \right)}\right)\right) = -\infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
$$\lim_{x \to -\infty}\left(x \left(\left(- \cos{\left(2 x \right)} - 1\right) - \sin{\left(2 x \right)}\right) + \left(\left(\left(- x^{2} - 1\right) - \cos{\left(2 x \right)}\right) - \sin{\left(2 x \right)}\right)\right) = -\infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
$$\lim_{x \to \infty}\left(x \left(\left(- \cos{\left(2 x \right)} - 1\right) - \sin{\left(2 x \right)}\right) + \left(\left(\left(- x^{2} - 1\right) - \cos{\left(2 x \right)}\right) - \sin{\left(2 x \right)}\right)\right) = -\infty$$
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 -1 - x^2 - cos(2*x) - sin(2*x) + x*(-1 - cos(2*x) - sin(2*x)), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{x \left(\left(- \cos{\left(2 x \right)} - 1\right) - \sin{\left(2 x \right)}\right) + \left(\left(\left(- x^{2} - 1\right) - \cos{\left(2 x \right)}\right) - \sin{\left(2 x \right)}\right)}{x}\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda
$$\lim_{x \to \infty}\left(\frac{x \left(\left(- \cos{\left(2 x \right)} - 1\right) - \sin{\left(2 x \right)}\right) + \left(\left(\left(- x^{2} - 1\right) - \cos{\left(2 x \right)}\right) - \sin{\left(2 x \right)}\right)}{x}\right) = -\infty$$
Tomamos como el límite
es decir,
no hay asíntota inclinada a la derecha
$$\lim_{x \to -\infty}\left(\frac{x \left(\left(- \cos{\left(2 x \right)} - 1\right) - \sin{\left(2 x \right)}\right) + \left(\left(\left(- x^{2} - 1\right) - \cos{\left(2 x \right)}\right) - \sin{\left(2 x \right)}\right)}{x}\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda
$$\lim_{x \to \infty}\left(\frac{x \left(\left(- \cos{\left(2 x \right)} - 1\right) - \sin{\left(2 x \right)}\right) + \left(\left(\left(- x^{2} - 1\right) - \cos{\left(2 x \right)}\right) - \sin{\left(2 x \right)}\right)}{x}\right) = -\infty$$
Tomamos como el límite
es decir,
no hay asíntota inclinada a la derecha
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:
$$x \left(\left(- \cos{\left(2 x \right)} - 1\right) - \sin{\left(2 x \right)}\right) + \left(\left(\left(- x^{2} - 1\right) - \cos{\left(2 x \right)}\right) - \sin{\left(2 x \right)}\right) = - x^{2} - x \left(\sin{\left(2 x \right)} - \cos{\left(2 x \right)} - 1\right) + \sin{\left(2 x \right)} - \cos{\left(2 x \right)} - 1$$
- No
$$x \left(\left(- \cos{\left(2 x \right)} - 1\right) - \sin{\left(2 x \right)}\right) + \left(\left(\left(- x^{2} - 1\right) - \cos{\left(2 x \right)}\right) - \sin{\left(2 x \right)}\right) = x^{2} + x \left(\sin{\left(2 x \right)} - \cos{\left(2 x \right)} - 1\right) - \sin{\left(2 x \right)} + \cos{\left(2 x \right)} + 1$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$x \left(\left(- \cos{\left(2 x \right)} - 1\right) - \sin{\left(2 x \right)}\right) + \left(\left(\left(- x^{2} - 1\right) - \cos{\left(2 x \right)}\right) - \sin{\left(2 x \right)}\right) = - x^{2} - x \left(\sin{\left(2 x \right)} - \cos{\left(2 x \right)} - 1\right) + \sin{\left(2 x \right)} - \cos{\left(2 x \right)} - 1$$
- No
$$x \left(\left(- \cos{\left(2 x \right)} - 1\right) - \sin{\left(2 x \right)}\right) + \left(\left(\left(- x^{2} - 1\right) - \cos{\left(2 x \right)}\right) - \sin{\left(2 x \right)}\right) = x^{2} + x \left(\sin{\left(2 x \right)} - \cos{\left(2 x \right)} - 1\right) - \sin{\left(2 x \right)} + \cos{\left(2 x \right)} + 1$$
- No
es decir, función
no es
par ni impar