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tan(2)^(6)*x*cos(7*x)^2
Trazado el gráfico de la función/ tan(2)^(6)*x*cos(7*x)^2

Gráfico de la función y = tan(2)^(6)*x*cos(7*x)^2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          6         2     
f(x) = tan (2)*x*cos (7*x)
$$f{\left(x \right)} = x \tan^{6}{\left(2 \right)} \cos^{2}{\left(7 x \right)}$$ 
f = (x*tan(2)^6)*cos(7*x)^2
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:
$$x \tan^{6}{\left(2 \right)} \cos^{2}{\left(7 x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{2}$$
$$x_{3} = - \frac{3 \pi}{14}$$
$$x_{4} = - \frac{\pi}{14}$$
$$x_{5} = \frac{\pi}{14}$$
$$x_{6} = \frac{3 \pi}{14}$$
$$x_{7} = \frac{\pi}{2}$$
$$x_{8} = \frac{11 \pi}{14}$$
$$x_{9} = \frac{13 \pi}{14}$$
$$x_{10} = - i \log{\left(- \sqrt[14]{-1} \right)}$$
$$x_{11} = - i \log{\left(- \left(-1\right)^{\frac{3}{14}} \right)}$$
Solución numérica
$$x_{1} = 0$$
$$x_{2} = -1.5707963267949$$
$$x_{3} = -0.673198425769241$$
$$x_{4} = -0.224399475256414$$
$$x_{5} = 0.224399475256414$$
$$x_{6} = 0.673198425769241$$
$$x_{7} = 1.5707963267949$$
$$x_{8} = 2.46839422782055$$
$$x_{9} = 2.91719317833338$$
$$x_{10} = -2.01959527730772$$
$$x_{11} = 2.01959527730772$$
$$x_{12} = -1.12199737628207$$
$$x_{13} = 1.12199737628207$$
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 (tan(2)^6*x)*cos(7*x)^2.
$$0 \tan^{6}{\left(2 \right)} \cos^{2}{\left(0 \cdot 7 \right)}$$
Resultado:
$$f{\left(0 \right)} = 0$$
Punto:
(0, 0)
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
$$- 14 x \sin{\left(7 x \right)} \cos{\left(7 x \right)} \tan^{6}{\left(2 \right)} + \cos^{2}{\left(7 x \right)} \tan^{6}{\left(2 \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -9.87461025876599$$
$$x_{2} = -19.7476705436182$$
$$x_{3} = 96.2673748850015$$
$$x_{4} = -25.8059396544876$$
$$x_{5} = 38.1481782787015$$
$$x_{6} = 60.1392290429398$$
$$x_{7} = 48.245887180129$$
$$x_{8} = 33.211429583558$$
$$x_{9} = -89.5353906273091$$
$$x_{10} = -41.7385468736615$$
$$x_{11} = 24.0107438524363$$
$$x_{12} = 55.2024557613023$$
$$x_{13} = 62.1586546460266$$
$$x_{14} = -79.8863409237441$$
$$x_{15} = -75.8471571714217$$
$$x_{16} = -53.856063530932$$
$$x_{17} = 14.1371669411541$$
$$x_{18} = 64.1784089188219$$
$$x_{19} = 74.276226309873$$
$$x_{20} = -11.8931721885899$$
$$x_{21} = 52.2850777347444$$
$$x_{22} = 71.583432606796$$
$$x_{23} = -3.14483680523185$$
$$x_{24} = 12.1184136852306$$
$$x_{25} = -9.20037848551297$$
$$x_{26} = 40.1675060708981$$
$$x_{27} = 36.1283155162826$$
$$x_{28} = 42.1873432234011$$
$$x_{29} = -1.80085906346792$$
$$x_{30} = 2.24853133657212$$
$$x_{31} = -39.7187071203852$$
$$x_{32} = -46.0018924275648$$
$$x_{33} = 16.1573937568963$$
$$x_{34} = -27.8259016426157$$
$$x_{35} = -61.7098556955138$$
$$x_{36} = -29.845130209103$$
$$x_{37} = -33.8843207637185$$
$$x_{38} = 0.0933244552992004$$
$$x_{39} = -31.8650457139301$$
$$x_{40} = -17.7275585452567$$
$$x_{41} = 86.1695169171295$$
$$x_{42} = -93.7990894437344$$
$$x_{43} = -69.7882368047447$$
$$x_{44} = -82.1303321863628$$
$$x_{45} = -65.7490462501292$$
$$x_{46} = -0.224399475256414$$
$$x_{47} = -63.7296110879891$$
$$x_{48} = -87.7401948252578$$
$$x_{49} = -15.708612848622$$
$$x_{50} = 20.1964580121188$$
$$x_{51} = 66.1978452006421$$
$$x_{52} = -51.8362787842316$$
$$x_{53} = -47.7970882296161$$
$$x_{54} = -92.0038957643417$$
$$x_{55} = -71.8079741843521$$
$$x_{56} = -35.9042002436571$$
$$x_{57} = -85.7207185866077$$
$$x_{58} = 98.2870739814527$$
$$x_{59} = 58.1194640914112$$
$$x_{60} = 88.1889937757706$$
$$x_{61} = 0.470330003040298$$
$$x_{62} = -0.0933244552992004$$
$$x_{63} = 94.2478878762175$$
$$x_{64} = 90.2087021694325$$
$$x_{65} = -95.8185759344887$$
$$x_{66} = -99.8577664891041$$
$$x_{67} = 18.1763574957695$$
$$x_{68} = -23.7867733572188$$
$$x_{69} = -97.8382755071733$$
$$x_{70} = 6.28480884777756$$
$$x_{71} = -49.8168883385579$$
$$x_{72} = 28.2746947724966$$
$$x_{73} = -57.89524086685$$
$$x_{74} = 82.1303321863628$$
$$x_{75} = -55.875469338847$$
$$x_{76} = -7.85398163397448$$
$$x_{77} = 72.256772252246$$
$$x_{78} = 80.1106126665397$$
$$x_{79} = 84.1498032211552$$
$$x_{80} = 76.5202210624371$$
$$x_{81} = 26.2547386050004$$
$$x_{82} = -5.83613470085334$$
$$x_{83} = 4.26359002987186$$
$$x_{84} = -73.8274273593601$$
$$x_{85} = 92.4526941764618$$
$$x_{86} = 50.2656854602341$$
$$x_{87} = 56.100050704779$$
$$x_{88} = -3.81479107935903$$
$$x_{89} = 44.2066966255135$$
$$x_{90} = 30.2939291596159$$
$$x_{91} = 34.1090193993406$$
$$x_{92} = 10.0979763865386$$
$$x_{93} = -77.8666179139756$$
Signos de extremos en los puntos:
                                          6    
(-9.874610258765987, -9.87409360304636*tan (2))

                                          6    
(-19.74767054361817, -19.7474121853447*tan (2))

                                            6    
(96.26737488500152, 8.78839398920626e-25*tan (2))

                                               6    
(-25.805939654487588, -7.50489962228377e-28*tan (2))

                                        6    
(38.14817827870152, 38.1480445364585*tan (2))

                                        6    
(60.13922904293982, 60.1391442059091*tan (2))

                                            6    
(48.24588718012897, 2.34503295865519e-26*tan (2))

                                        6    
(33.21142958355798, 33.2112759612273*tan (2))

                                              6    
(-89.53539062730911, -3.91939002728315e-25*tan (2))

                                          6    
(-41.73854687366148, -41.7384246359159*tan (2))

                                             6    
(24.010743852436278, 1.66536603249196e-27*tan (2))

                                        6    
(55.20245576130234, 55.2023633372932*tan (2))

                                             6    
(62.158654646026626, 1.07871605619822e-25*tan (2))

                                          6    
(-79.88634092374414, -79.8862770575478*tan (2))

                                          6    
(-75.84715717142166, -75.8470899040845*tan (2))

                                        6    
(-53.856063530932, -53.8559687963468*tan (2))

                                             6    
(14.137166941154069, 3.06085974379874e-29*tan (2))

                                        6    
(64.17840891882194, 64.1783294211438*tan (2))

                                          6    
(74.27622630987297, 3.609449200233e-26*tan (2))

                                               6    
(-11.893172188589931, -2.56635054906443e-29*tan (2))

                                            6    
(52.28507773474442, 1.38447369493553e-25*tan (2))

                                            6    
(71.58343260679601, 1.30146993039844e-25*tan (2))

                                           6    
(-3.1448368052318516, -3.14321528702457*tan (2))

                                         6    
(12.118413685230559, 12.1179926842893*tan (2))

                                              6    
(-9.200378485512966, -5.65448569364571e-28*tan (2))

                                            6    
(40.16750607089807, 8.72602038708157e-29*tan (2))

                                            6    
(36.12831551628262, 8.33333793772587e-27*tan (2))

                                         6    
(42.187343223401115, 42.1872222860346*tan (2))

                                          6    
(-1.800859063467916, -1.79803039853426*tan (2))

                                         6    
(2.2485313365721242, 2.24626456923199*tan (2))

                                              6    
(-39.71870712038524, -7.89675566599354e-26*tan (2))

                                              6    
(-46.00189242756483, -4.66837920492084e-26*tan (2))

                                        6    
(16.15739375689631, 16.1570779917931*tan (2))

                                          6    
(-27.825901642615715, -27.825718288011*tan (2))

                                               6    
(-61.709855695513795, -1.19709102156758e-27*tan (2))

                                              6    
(-29.845130209103036, -1.2100893490981e-27*tan (2))

                                               6    
(-33.884320763718485, -3.59049041614397e-27*tan (2))

                                            6    
(0.09332445529920044, 0.0588498971202244*tan (2))

                                           6    
(-31.865045713930073, -31.8648856007068*tan (2))

                                              6    
(-17.72755854525669, -4.28624518877433e-30*tan (2))

                                        6    
(86.16951691712951, 86.1694577078233*tan (2))

                                          6    
(-93.79908944373445, -93.7990350504787*tan (2))

                                             6    
(-69.7882368047447, -5.83619849852646e-26*tan (2))

                                          6    
(-82.13033218636284, -82.1302700651365*tan (2))

                                              6    
(-65.74904625012924, -1.30705788845543e-25*tan (2))

                                               6    
(-0.2243994752564138, -8.41363270599985e-34*tan (2))

                                          6    
(-63.72961108798911, -63.7295310304724*tan (2))

                                             6    
(-87.7401948252578, -1.70013271407891e-27*tan (2))

                                           6    
(-15.708612848622005, -15.7082880627623*tan (2))

                                         6    
(20.196458012118764, 20.1962053947045*tan (2))

                                            6    
(66.19784520064208, 2.25152079060782e-25*tan (2))

                                              6    
(-51.83627878423159, -1.50889048851708e-27*tan (2))

                                              6    
(-47.79708822961614, -9.92883711869596e-26*tan (2))

                                          6    
(-92.00389576434168, -92.0038403097579*tan (2))

                                         6    
(-71.8079741843521, -71.8079031332491*tan (2))

                                           6    
(-35.904200243657144, -35.9040581427166*tan (2))

                                          6    
(-85.72071858660769, -85.7206590673064*tan (2))

                                        6    
(98.28707398145274, 98.2870220718993*tan (2))

                                             6    
(58.119464091411174, 3.63160384824741e-26*tan (2))

                                            6    
(88.18899377577063, 2.03283636195362e-26*tan (2))

                                          6    
(0.4703300030402981, 0.459726768858255*tan (2))

                                              6    
(-0.09332445529920044, -0.0588498971202244*tan (2))

                                        6    
(94.24788787621746, 94.2478337419764*tan (2))

                                        6    
(90.20870216943248, 90.2086456112791*tan (2))

                                              6    
(-95.81857593448869, -8.01441641500049e-26*tan (2))

                                              6    
(-99.85776648910414, -4.63163922232588e-25*tan (2))

                                            6    
(18.17635749576952, 2.79585219288648e-28*tan (2))

                                           6    
(-23.786773357218845, -23.7865588684887*tan (2))

                                          6    
(-97.83827550717332, -97.8382233595034*tan (2))

                                        6    
(6.284808847777558, 6.28399714737469*tan (2))

                                          6    
(-49.81688833855788, -49.8167859228812*tan (2))

                                        6    
(28.27469477249662, 28.2745143281701*tan (2))

                                           6    
(-57.895240866850024, -57.8951527415918*tan (2))

                                        6    
(82.13033218636284, 82.1302700651365*tan (2))

                                              6    
(-55.87546933884704, -2.71831594338562e-26*tan (2))

                                              6    
(-7.853981633974483, -1.20655944359504e-28*tan (2))

                                        6    
(72.25677225224598, 72.2567016424516*tan (2))

                                            6    
(80.11061266653972, 1.88481359874362e-25*tan (2))

                                            6    
(84.14980322115518, 5.63657643207218e-25*tan (2))

                                           6    
(76.5202210624371, 4.41367934949078e-25*tan (2))

                                             6    
(26.254738605000416, 7.62140381506604e-28*tan (2))

                                          6    
(-5.836134700853338, -5.83526061604565*tan (2))

                                            6    
(4.263590029871862, 3.68470334815878e-29*tan (2))

                                              6    
(-73.82742735936014, -9.36648117450064e-27*tan (2))

                                        6    
(92.45269417646178, 92.4526389910741*tan (2))

                                         6    
(50.265685460234145, 50.2655839589721*tan (2))

                                        6    
(56.10005070477896, 56.0999597595395*tan (2))

                                               6    
(-3.8147910793590345, -2.06312883147019e-30*tan (2))

                                             6    
(44.206696625513516, 6.62357259545183e-26*tan (2))

                                             6    
(30.293929159615864, 3.85015058508972e-27*tan (2))

                                        6    
(34.109019399340575, 34.108869819595*tan (2))

                                            6    
(10.09797638653862, 9.68026139285446e-30*tan (2))

                                              6    
(-77.86661791397559, -2.26868252765451e-27*tan (2))


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:
$$x_{1} = -9.87461025876599$$
$$x_{2} = -19.7476705436182$$
$$x_{3} = 96.2673748850015$$
$$x_{4} = 48.245887180129$$
$$x_{5} = -41.7385468736615$$
$$x_{6} = 24.0107438524363$$
$$x_{7} = 62.1586546460266$$
$$x_{8} = -79.8863409237441$$
$$x_{9} = -75.8471571714217$$
$$x_{10} = -53.856063530932$$
$$x_{11} = 14.1371669411541$$
$$x_{12} = 74.276226309873$$
$$x_{13} = 52.2850777347444$$
$$x_{14} = 71.583432606796$$
$$x_{15} = -3.14483680523185$$
$$x_{16} = 40.1675060708981$$
$$x_{17} = 36.1283155162826$$
$$x_{18} = -1.80085906346792$$
$$x_{19} = -27.8259016426157$$
$$x_{20} = -31.8650457139301$$
$$x_{21} = -93.7990894437344$$
$$x_{22} = -82.1303321863628$$
$$x_{23} = -63.7296110879891$$
$$x_{24} = -15.708612848622$$
$$x_{25} = 66.1978452006421$$
$$x_{26} = -92.0038957643417$$
$$x_{27} = -71.8079741843521$$
$$x_{28} = -35.9042002436571$$
$$x_{29} = -85.7207185866077$$
$$x_{30} = 58.1194640914112$$
$$x_{31} = 88.1889937757706$$
$$x_{32} = -0.0933244552992004$$
$$x_{33} = 18.1763574957695$$
$$x_{34} = -23.7867733572188$$
$$x_{35} = -97.8382755071733$$
$$x_{36} = -49.8168883385579$$
$$x_{37} = -57.89524086685$$
$$x_{38} = 80.1106126665397$$
$$x_{39} = 84.1498032211552$$
$$x_{40} = 76.5202210624371$$
$$x_{41} = 26.2547386050004$$
$$x_{42} = -5.83613470085334$$
$$x_{43} = 4.26359002987186$$
$$x_{44} = 44.2066966255135$$
$$x_{45} = 30.2939291596159$$
$$x_{46} = 10.0979763865386$$
Puntos máximos de la función:
$$x_{46} = -25.8059396544876$$
$$x_{46} = 38.1481782787015$$
$$x_{46} = 60.1392290429398$$
$$x_{46} = 33.211429583558$$
$$x_{46} = -89.5353906273091$$
$$x_{46} = 55.2024557613023$$
$$x_{46} = 64.1784089188219$$
$$x_{46} = -11.8931721885899$$
$$x_{46} = 12.1184136852306$$
$$x_{46} = -9.20037848551297$$
$$x_{46} = 42.1873432234011$$
$$x_{46} = 2.24853133657212$$
$$x_{46} = -39.7187071203852$$
$$x_{46} = -46.0018924275648$$
$$x_{46} = 16.1573937568963$$
$$x_{46} = -61.7098556955138$$
$$x_{46} = -29.845130209103$$
$$x_{46} = -33.8843207637185$$
$$x_{46} = 0.0933244552992004$$
$$x_{46} = -17.7275585452567$$
$$x_{46} = 86.1695169171295$$
$$x_{46} = -69.7882368047447$$
$$x_{46} = -65.7490462501292$$
$$x_{46} = -0.224399475256414$$
$$x_{46} = -87.7401948252578$$
$$x_{46} = 20.1964580121188$$
$$x_{46} = -51.8362787842316$$
$$x_{46} = -47.7970882296161$$
$$x_{46} = 98.2870739814527$$
$$x_{46} = 0.470330003040298$$
$$x_{46} = 94.2478878762175$$
$$x_{46} = 90.2087021694325$$
$$x_{46} = -95.8185759344887$$
$$x_{46} = -99.8577664891041$$
$$x_{46} = 6.28480884777756$$
$$x_{46} = 28.2746947724966$$
$$x_{46} = 82.1303321863628$$
$$x_{46} = -55.875469338847$$
$$x_{46} = -7.85398163397448$$
$$x_{46} = 72.256772252246$$
$$x_{46} = -73.8274273593601$$
$$x_{46} = 92.4526941764618$$
$$x_{46} = 50.2656854602341$$
$$x_{46} = 56.100050704779$$
$$x_{46} = -3.81479107935903$$
$$x_{46} = 34.1090193993406$$
$$x_{46} = -77.8666179139756$$
Decrece en los intervalos
$$\left[96.2673748850015, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -97.8382755071733\right]$$
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
$$14 \left(7 x \left(\sin^{2}{\left(7 x \right)} - \cos^{2}{\left(7 x \right)}\right) - 2 \sin{\left(7 x \right)} \cos{\left(7 x \right)}\right) \tan^{6}{\left(2 \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -75.7349576329835$$
$$x_{2} = 40.2799591364608$$
$$x_{3} = 30.1820675037122$$
$$x_{4} = 62.2710182490294$$
$$x_{5} = -83.8133257558764$$
$$x_{6} = 22.1038099498946$$
$$x_{7} = 94.1356882675712$$
$$x_{8} = -7.74309957591203$$
$$x_{9} = 18.2891151539985$$
$$x_{10} = 66.3101988220774$$
$$x_{11} = 90.0965025725999$$
$$x_{12} = 48.1338994355408$$
$$x_{13} = 92.1160953668133$$
$$x_{14} = 10.2111753642745$$
$$x_{15} = 64.2906083787768$$
$$x_{16} = 40.0555610803785$$
$$x_{17} = 26.1429291824741$$
$$x_{18} = -10.43555335728$$
$$x_{19} = -27.713703387155$$
$$x_{20} = -85.832918168548$$
$$x_{21} = 82.2425317543994$$
$$x_{22} = -80.2229396005895$$
$$x_{23} = 16.2695891273924$$
$$x_{24} = -33.099230885982$$
$$x_{25} = -53.7438641885171$$
$$x_{26} = 52.1730735780056$$
$$x_{27} = -97.7260758892311$$
$$x_{28} = -39.8311630402863$$
$$x_{29} = 4.15384586276036$$
$$x_{30} = 84.2621240579364$$
$$x_{31} = -59.8026307848198$$
$$x_{32} = -37.8115814459875$$
$$x_{33} = -91.8916961620663$$
$$x_{34} = -31.3040527627982$$
$$x_{35} = 50.1534861763339$$
$$x_{36} = 8.1918263624443$$
$$x_{37} = 79.9985404821234$$
$$x_{38} = -0.363356078150162$$
$$x_{39} = 0$$
$$x_{40} = -51.7242763244855$$
$$x_{41} = -23.8989710771266$$
$$x_{42} = -99.7456690524054$$
$$x_{43} = 76.1837557897812$$
$$x_{44} = -41.8507459551568$$
$$x_{45} = -61.8222204880913$$
$$x_{46} = 72.1445727341044$$
$$x_{47} = -11.7818384942038$$
$$x_{48} = 65.6370019745912$$
$$x_{49} = -29.7332736561249$$
$$x_{50} = -95.7064828152812$$
$$x_{51} = -63.8418105440214$$
$$x_{52} = -15.820807966589$$
$$x_{53} = -17.8403302377427$$
$$x_{54} = -13.8013070580432$$
$$x_{55} = -69.9005825220923$$
$$x_{56} = -19.8598673554491$$
$$x_{57} = 54.1926615666706$$
$$x_{58} = 74.1641641598402$$
$$x_{59} = -45.889915049221$$
$$x_{60} = -89.8721033801316$$
$$x_{61} = 32.2016415780069$$
$$x_{62} = 23.8989710771266$$
$$x_{63} = 86.2817165006867$$
$$x_{64} = -50.1534861763339$$
$$x_{65} = -58.0074402633035$$
$$x_{66} = 28.1624964703041$$
$$x_{67} = -87.8525107129261$$
$$x_{68} = -3.70534386095906$$
$$x_{69} = -31.7528471061951$$
$$x_{70} = 88.7501074392409$$
$$x_{71} = 3.48112148535039$$
$$x_{72} = 98.1748743624197$$
$$x_{73} = -70.7981785723577$$
$$x_{74} = 56.2122500790616$$
$$x_{75} = 70.1249815302169$$
$$x_{76} = 14.2500827265421$$
$$x_{77} = 96.1552812681603$$
$$x_{78} = -73.7153660469939$$
$$x_{79} = -94.3600874850462$$
$$x_{80} = 60.2514284643652$$
$$x_{81} = 38.2603772309992$$
$$x_{82} = -43.8703300082251$$
$$x_{83} = -81.7937334846673$$
$$x_{84} = -65.8614009201684$$
$$x_{85} = -67.8809915879513$$
$$x_{86} = 33.9968206476951$$
$$x_{87} = -77.7545494107282$$
$$x_{88} = -21.8794152090433$$
$$x_{89} = -46.3387118461673$$
$$x_{90} = 46.1143134424544$$
$$x_{91} = 100.194467544685$$
$$x_{92} = -5.72396894243389$$
$$x_{93} = 68.1053905679135$$
$$x_{94} = 6.17263838957976$$
$$x_{95} = -35.7920013957943$$
$$x_{96} = 24.1233665808737$$
$$x_{97} = 2.13656384782513$$
$$x_{98} = 78.2033476081068$$

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
$$\left[100.194467544685, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -99.7456690524054\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(x \tan^{6}{\left(2 \right)} \cos^{2}{\left(7 x \right)}\right) = \left\langle -\infty, 0\right\rangle \tan^{6}{\left(2 \right)}$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, 0\right\rangle \tan^{6}{\left(2 \right)}$$
$$\lim_{x \to \infty}\left(x \tan^{6}{\left(2 \right)} \cos^{2}{\left(7 x \right)}\right) = \left\langle 0, \infty\right\rangle \tan^{6}{\left(2 \right)}$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle 0, \infty\right\rangle \tan^{6}{\left(2 \right)}$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (tan(2)^6*x)*cos(7*x)^2, dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\cos^{2}{\left(7 x \right)} \tan^{6}{\left(2 \right)}\right) = \left\langle 0, 1\right\rangle \tan^{6}{\left(2 \right)}$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle 0, 1\right\rangle x \tan^{6}{\left(2 \right)}$$
$$\lim_{x \to \infty}\left(\cos^{2}{\left(7 x \right)} \tan^{6}{\left(2 \right)}\right) = \left\langle 0, 1\right\rangle \tan^{6}{\left(2 \right)}$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle 0, 1\right\rangle x \tan^{6}{\left(2 \right)}$$
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:
$$x \tan^{6}{\left(2 \right)} \cos^{2}{\left(7 x \right)} = - x \cos^{2}{\left(7 x \right)} \tan^{6}{\left(2 \right)}$$
- No
$$x \tan^{6}{\left(2 \right)} \cos^{2}{\left(7 x \right)} = x \cos^{2}{\left(7 x \right)} \tan^{6}{\left(2 \right)}$$
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = tan(2)^(6)*x*cos(7*x)^2
    © Sr Examen – Калькулятор