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- Integrales paso a paso
- Gráfico de la función paramétrica
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- 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 =:
-
-x+4
-
e^(-x^2)
-
x*(x-4)
-
x^3-6x+5
-
Expresiones idénticas
- log(x)*cos(tres *x)
- logaritmo de (x) multiplicar por coseno de (3 multiplicar por x)
- logaritmo de (x) multiplicar por coseno de (tres multiplicar por x)
- log(x)cos(3x)
- logxcos3x
-
Expresiones con funciones
- Logaritmo log
- log(x^2-5*x+6)+3
- log(sin(sqrt(x)))
- log(x^2/(x-1))
- log((x-1)/(x-2))
- log(2*x+3)
- Coseno cos
- cos(4*x)/sin(8*x)
- cos(x)/8
- cos(x)^(2)+cos(x)
- cos(x)-(1/cos(x))
- cos(0.5*x)
Gráfico de la función y = log(x)*cos(3*x)
Solución
Ha introducido
[src]
f(x) = log(x)*cos(3*x)
$$f{\left(x \right)} = \log{\left(x \right)} \cos{\left(3 x \right)}$$
f = log(x)*cos(3*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:
$$\log{\left(x \right)} \cos{\left(3 x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 1$$
$$x_{2} = - \frac{5 \pi}{6}$$
$$x_{3} = - \frac{\pi}{2}$$
$$x_{4} = - \frac{\pi}{6}$$
$$x_{5} = \frac{\pi}{6}$$
$$x_{6} = \frac{\pi}{2}$$
$$x_{7} = \frac{5 \pi}{6}$$
Solución numérica
$$x_{1} = -62.3082542961976$$
$$x_{2} = -73.8274273593601$$
$$x_{3} = -23.5619449019235$$
$$x_{4} = -65.4498469497874$$
$$x_{5} = 42.4115008234622$$
$$x_{6} = -91.6297857297023$$
$$x_{7} = 22.5147473507269$$
$$x_{8} = 93.7241808320955$$
$$x_{9} = -42.4115008234622$$
$$x_{10} = -108.384946548848$$
$$x_{11} = -97.9129710368819$$
$$x_{12} = -82.2050077689329$$
$$x_{13} = 66.497044500984$$
$$x_{14} = -36.1283155162826$$
$$x_{15} = 86.3937979737193$$
$$x_{16} = 60.2138591938044$$
$$x_{17} = -84.2994028713261$$
$$x_{18} = 97.9129710368819$$
$$x_{19} = 58.1194640914112$$
$$x_{20} = -31.9395253114962$$
$$x_{21} = 34.0339204138894$$
$$x_{22} = 100.007366139275$$
$$x_{23} = -51.8362787842316$$
$$x_{24} = -45.553093477052$$
$$x_{25} = -27.7507351067098$$
$$x_{26} = 12.0427718387609$$
$$x_{27} = 75.9218224617533$$
$$x_{28} = 62.3082542961976$$
$$x_{29} = 71.733032256967$$
$$x_{30} = 18.3259571459405$$
$$x_{31} = -56.025068989018$$
$$x_{32} = -58.1194640914112$$
$$x_{33} = -14.1371669411541$$
$$x_{34} = 1.5707963267949$$
$$x_{35} = 36.1283155162826$$
$$x_{36} = -7.85398163397448$$
$$x_{37} = 38.2227106186758$$
$$x_{38} = -3.66519142918809$$
$$x_{39} = 31.9395253114962$$
$$x_{40} = -5.75958653158129$$
$$x_{41} = -9.94837673636768$$
$$x_{42} = 73.8274273593601$$
$$x_{43} = 68.5914396033772$$
$$x_{44} = -100.007366139275$$
$$x_{45} = 44.5058959258554$$
$$x_{46} = -34.0339204138894$$
$$x_{47} = 699.004365423729$$
$$x_{48} = 20.4203522483337$$
$$x_{49} = -88.4881930761125$$
$$x_{50} = -43.4586983746588$$
$$x_{51} = -93.7241808320955$$
$$x_{52} = 82.2050077689329$$
$$x_{53} = -12.0427718387609$$
$$x_{54} = 84.2994028713261$$
$$x_{55} = 95.8185759344887$$
$$x_{56} = -16.2315620435473$$
$$x_{57} = -71.733032256967$$
$$x_{58} = -25.6563400043166$$
$$x_{59} = -38.2227106186758$$
$$x_{60} = -69.6386371545737$$
$$x_{61} = -89.5353906273091$$
$$x_{62} = 3.66519142918809$$
$$x_{63} = 80.1106126665397$$
$$x_{64} = -6.80678408277789$$
$$x_{65} = 49.7418836818384$$
$$x_{66} = 7.85398163397448$$
$$x_{67} = 40.317105721069$$
$$x_{68} = -95.8185759344887$$
$$x_{69} = 29.845130209103$$
$$x_{70} = -78.0162175641465$$
$$x_{71} = 14.1371669411541$$
$$x_{72} = 2.61799387799149$$
$$x_{73} = -1.5707963267949$$
$$x_{74} = 27.7507351067098$$
$$x_{75} = -75.9218224617533$$
$$x_{76} = 88.4881930761125$$
$$x_{77} = -19.3731546971371$$
$$x_{78} = -80.1106126665397$$
$$x_{79} = 51.8362787842316$$
$$x_{80} = 56.025068989018$$
$$x_{81} = 91.6297857297023$$
$$x_{82} = 78.0162175641465$$
$$x_{83} = -47.6474885794452$$
$$x_{84} = -67.5442420521806$$
$$x_{85} = 64.4026493985908$$
$$x_{86} = -49.7418836818384$$
$$x_{87} = -21.4675497995303$$
$$x_{88} = -15.1843644923507$$
$$x_{89} = -53.9306738866248$$
$$x_{90} = 16.2315620435473$$
$$x_{91} = -29.845130209103$$
$$x_{92} = 5.75958653158129$$
$$x_{93} = 9.94837673636768$$
$$x_{94} = -60.2138591938044$$
$$x_{95} = 53.9306738866248$$
o sea hay que resolver la ecuación:
$$\log{\left(x \right)} \cos{\left(3 x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 1$$
$$x_{2} = - \frac{5 \pi}{6}$$
$$x_{3} = - \frac{\pi}{2}$$
$$x_{4} = - \frac{\pi}{6}$$
$$x_{5} = \frac{\pi}{6}$$
$$x_{6} = \frac{\pi}{2}$$
$$x_{7} = \frac{5 \pi}{6}$$
Solución numérica
$$x_{1} = -62.3082542961976$$
$$x_{2} = -73.8274273593601$$
$$x_{3} = -23.5619449019235$$
$$x_{4} = -65.4498469497874$$
$$x_{5} = 42.4115008234622$$
$$x_{6} = -91.6297857297023$$
$$x_{7} = 22.5147473507269$$
$$x_{8} = 93.7241808320955$$
$$x_{9} = -42.4115008234622$$
$$x_{10} = -108.384946548848$$
$$x_{11} = -97.9129710368819$$
$$x_{12} = -82.2050077689329$$
$$x_{13} = 66.497044500984$$
$$x_{14} = -36.1283155162826$$
$$x_{15} = 86.3937979737193$$
$$x_{16} = 60.2138591938044$$
$$x_{17} = -84.2994028713261$$
$$x_{18} = 97.9129710368819$$
$$x_{19} = 58.1194640914112$$
$$x_{20} = -31.9395253114962$$
$$x_{21} = 34.0339204138894$$
$$x_{22} = 100.007366139275$$
$$x_{23} = -51.8362787842316$$
$$x_{24} = -45.553093477052$$
$$x_{25} = -27.7507351067098$$
$$x_{26} = 12.0427718387609$$
$$x_{27} = 75.9218224617533$$
$$x_{28} = 62.3082542961976$$
$$x_{29} = 71.733032256967$$
$$x_{30} = 18.3259571459405$$
$$x_{31} = -56.025068989018$$
$$x_{32} = -58.1194640914112$$
$$x_{33} = -14.1371669411541$$
$$x_{34} = 1.5707963267949$$
$$x_{35} = 36.1283155162826$$
$$x_{36} = -7.85398163397448$$
$$x_{37} = 38.2227106186758$$
$$x_{38} = -3.66519142918809$$
$$x_{39} = 31.9395253114962$$
$$x_{40} = -5.75958653158129$$
$$x_{41} = -9.94837673636768$$
$$x_{42} = 73.8274273593601$$
$$x_{43} = 68.5914396033772$$
$$x_{44} = -100.007366139275$$
$$x_{45} = 44.5058959258554$$
$$x_{46} = -34.0339204138894$$
$$x_{47} = 699.004365423729$$
$$x_{48} = 20.4203522483337$$
$$x_{49} = -88.4881930761125$$
$$x_{50} = -43.4586983746588$$
$$x_{51} = -93.7241808320955$$
$$x_{52} = 82.2050077689329$$
$$x_{53} = -12.0427718387609$$
$$x_{54} = 84.2994028713261$$
$$x_{55} = 95.8185759344887$$
$$x_{56} = -16.2315620435473$$
$$x_{57} = -71.733032256967$$
$$x_{58} = -25.6563400043166$$
$$x_{59} = -38.2227106186758$$
$$x_{60} = -69.6386371545737$$
$$x_{61} = -89.5353906273091$$
$$x_{62} = 3.66519142918809$$
$$x_{63} = 80.1106126665397$$
$$x_{64} = -6.80678408277789$$
$$x_{65} = 49.7418836818384$$
$$x_{66} = 7.85398163397448$$
$$x_{67} = 40.317105721069$$
$$x_{68} = -95.8185759344887$$
$$x_{69} = 29.845130209103$$
$$x_{70} = -78.0162175641465$$
$$x_{71} = 14.1371669411541$$
$$x_{72} = 2.61799387799149$$
$$x_{73} = -1.5707963267949$$
$$x_{74} = 27.7507351067098$$
$$x_{75} = -75.9218224617533$$
$$x_{76} = 88.4881930761125$$
$$x_{77} = -19.3731546971371$$
$$x_{78} = -80.1106126665397$$
$$x_{79} = 51.8362787842316$$
$$x_{80} = 56.025068989018$$
$$x_{81} = 91.6297857297023$$
$$x_{82} = 78.0162175641465$$
$$x_{83} = -47.6474885794452$$
$$x_{84} = -67.5442420521806$$
$$x_{85} = 64.4026493985908$$
$$x_{86} = -49.7418836818384$$
$$x_{87} = -21.4675497995303$$
$$x_{88} = -15.1843644923507$$
$$x_{89} = -53.9306738866248$$
$$x_{90} = 16.2315620435473$$
$$x_{91} = -29.845130209103$$
$$x_{92} = 5.75958653158129$$
$$x_{93} = 9.94837673636768$$
$$x_{94} = -60.2138591938044$$
$$x_{95} = 53.9306738866248$$
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
$$- 3 \log{\left(x \right)} \sin{\left(3 x \right)} + \frac{\cos{\left(3 x \right)}}{x} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 92.1536510526427$$
$$x_{2} = 74.3513729610722$$
$$x_{3} = 76.445756400651$$
$$x_{4} = 100.531204638865$$
$$x_{5} = 96.3424271874236$$
$$x_{6} = 43.9829647927364$$
$$x_{7} = 41.8886122281726$$
$$x_{8} = 6.29278191723044$$
$$x_{9} = 15.7105309536845$$
$$x_{10} = 19.8986205587611$$
$$x_{11} = 65.9738477459929$$
$$x_{12} = 56.5491547048483$$
$$x_{13} = 78.5401405447888$$
$$x_{14} = 98.4368157600501$$
$$x_{15} = 85.8704897852191$$
$$x_{16} = 46.0773218089678$$
$$x_{17} = 2.16029784897009$$
$$x_{18} = 68.0682275933998$$
$$x_{19} = 24.0869935064117$$
$$x_{20} = 79.5873328619807$$
$$x_{21} = 26.1812385788794$$
$$x_{22} = 10.4764899884892$$
$$x_{23} = 39.7942649107512$$
$$x_{24} = 21.9927831881195$$
$$x_{25} = 30.3698008017668$$
$$x_{26} = 61.7850916250074$$
$$x_{27} = 13.6166929275579$$
$$x_{28} = 48.171682632084$$
$$x_{29} = 63.8794690465696$$
$$x_{30} = 52.3604136794562$$
$$x_{31} = 94.2480389434111$$
$$x_{32} = 4.20715294221441$$
$$x_{33} = 32.4641075469997$$
$$x_{34} = 50.2660467342498$$
$$x_{35} = 90.0592635422263$$
$$x_{36} = 28.2755096981709$$
$$x_{37} = 70.1626084756746$$
$$x_{38} = 8.38381259282155$$
$$x_{39} = 83.7761036083348$$
$$x_{40} = 59.6907156311258$$
$$x_{41} = 17.8045256303582$$
$$x_{42} = 87.9648764420645$$
$$x_{43} = 54.4547831045677$$
$$x_{44} = 33.511265754867$$
$$x_{45} = 37.6999238326345$$
$$x_{46} = 72.2569902938691$$
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:
$$x_{1} = 74.3513729610722$$
$$x_{2} = 76.445756400651$$
$$x_{3} = 15.7105309536845$$
$$x_{4} = 19.8986205587611$$
$$x_{5} = 65.9738477459929$$
$$x_{6} = 78.5401405447888$$
$$x_{7} = 68.0682275933998$$
$$x_{8} = 24.0869935064117$$
$$x_{9} = 26.1812385788794$$
$$x_{10} = 21.9927831881195$$
$$x_{11} = 30.3698008017668$$
$$x_{12} = 61.7850916250074$$
$$x_{13} = 13.6166929275579$$
$$x_{14} = 63.8794690465696$$
$$x_{15} = 32.4641075469997$$
$$x_{16} = 28.2755096981709$$
$$x_{17} = 70.1626084756746$$
$$x_{18} = 59.6907156311258$$
$$x_{19} = 17.8045256303582$$
$$x_{20} = 72.2569902938691$$
Puntos máximos de la función:
$$x_{20} = 92.1536510526427$$
$$x_{20} = 100.531204638865$$
$$x_{20} = 96.3424271874236$$
$$x_{20} = 43.9829647927364$$
$$x_{20} = 41.8886122281726$$
$$x_{20} = 6.29278191723044$$
$$x_{20} = 56.5491547048483$$
$$x_{20} = 98.4368157600501$$
$$x_{20} = 85.8704897852191$$
$$x_{20} = 46.0773218089678$$
$$x_{20} = 2.16029784897009$$
$$x_{20} = 79.5873328619807$$
$$x_{20} = 10.4764899884892$$
$$x_{20} = 39.7942649107512$$
$$x_{20} = 48.171682632084$$
$$x_{20} = 52.3604136794562$$
$$x_{20} = 94.2480389434111$$
$$x_{20} = 4.20715294221441$$
$$x_{20} = 50.2660467342498$$
$$x_{20} = 90.0592635422263$$
$$x_{20} = 8.38381259282155$$
$$x_{20} = 83.7761036083348$$
$$x_{20} = 87.9648764420645$$
$$x_{20} = 54.4547831045677$$
$$x_{20} = 33.511265754867$$
$$x_{20} = 37.6999238326345$$
Decrece en los intervalos
$$\left[78.5401405447888, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, 13.6166929275579\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- 3 \log{\left(x \right)} \sin{\left(3 x \right)} + \frac{\cos{\left(3 x \right)}}{x} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 92.1536510526427$$
$$x_{2} = 74.3513729610722$$
$$x_{3} = 76.445756400651$$
$$x_{4} = 100.531204638865$$
$$x_{5} = 96.3424271874236$$
$$x_{6} = 43.9829647927364$$
$$x_{7} = 41.8886122281726$$
$$x_{8} = 6.29278191723044$$
$$x_{9} = 15.7105309536845$$
$$x_{10} = 19.8986205587611$$
$$x_{11} = 65.9738477459929$$
$$x_{12} = 56.5491547048483$$
$$x_{13} = 78.5401405447888$$
$$x_{14} = 98.4368157600501$$
$$x_{15} = 85.8704897852191$$
$$x_{16} = 46.0773218089678$$
$$x_{17} = 2.16029784897009$$
$$x_{18} = 68.0682275933998$$
$$x_{19} = 24.0869935064117$$
$$x_{20} = 79.5873328619807$$
$$x_{21} = 26.1812385788794$$
$$x_{22} = 10.4764899884892$$
$$x_{23} = 39.7942649107512$$
$$x_{24} = 21.9927831881195$$
$$x_{25} = 30.3698008017668$$
$$x_{26} = 61.7850916250074$$
$$x_{27} = 13.6166929275579$$
$$x_{28} = 48.171682632084$$
$$x_{29} = 63.8794690465696$$
$$x_{30} = 52.3604136794562$$
$$x_{31} = 94.2480389434111$$
$$x_{32} = 4.20715294221441$$
$$x_{33} = 32.4641075469997$$
$$x_{34} = 50.2660467342498$$
$$x_{35} = 90.0592635422263$$
$$x_{36} = 28.2755096981709$$
$$x_{37} = 70.1626084756746$$
$$x_{38} = 8.38381259282155$$
$$x_{39} = 83.7761036083348$$
$$x_{40} = 59.6907156311258$$
$$x_{41} = 17.8045256303582$$
$$x_{42} = 87.9648764420645$$
$$x_{43} = 54.4547831045677$$
$$x_{44} = 33.511265754867$$
$$x_{45} = 37.6999238326345$$
$$x_{46} = 72.2569902938691$$
Signos de extremos en los puntos:
(92.15365105264269, 4.52345585787558)
(74.35137296107223, -4.30879980657959)
(76.445756400651, -4.33657923050441)
(100.53120463886512, 4.6104669809387)
(96.34242718742355, 4.56790748454639)
(43.98296479273636, 3.7837948053774)
(41.888612228172626, 3.73500552846426)
(6.292781917230444, 1.83864100311292)
(15.710530953684493, -2.75424953227859)
(19.898620558761056, -2.9906034963246)
(65.97384774599286, -4.18925537041119)
(56.549154704848284, 4.03510594924633)
(78.54014054478876, -4.36360777467183)
(98.43681576005011, 4.58941362872088)
(85.8704897852191, 4.45283853645973)
(46.07732180896777, 3.83031406272333)
(2.160297848970094, 0.755241176694825)
(68.06822759339975, -4.220507708109)
(24.086993506411748, -3.18164191083909)
(79.5873328619807, 4.37685294138224)
(26.181238578879434, -3.26501824653837)
(10.476489988489204, 2.34891825700492)
(39.79426491075122, 3.6837132806588)
(21.992783188119542, -3.09067720052523)
(30.369800801766818, -3.41343107396627)
(61.785091625007404, -4.12365857031863)
(13.6166929275579, -2.61118172530022)
(48.171682632084014, 3.87476517245795)
(63.8794690465696, -4.15699473648039)
(52.36041367945622, 3.95814572218027)
(94.24803894341115, 4.54592864333061)
(4.207152942214409, 1.43460659397474)
(32.464107546999664, -3.48011994906059)
(50.26604673424984, 3.91732422106114)
(90.0592635422263, 4.50046641543387)
(28.275509698170943, -3.34197525628304)
(70.1626084756746, -4.25081287145983)
(8.383812592821554, 2.12593114896635)
(83.77610360833481, 4.42814601943997)
(59.690715631125755, -4.0891726781444)
(17.80452563035817, -2.87939181233473)
(87.96487644206448, 4.47693599974731)
(54.45478310456767, 3.99736600265705)
(33.51126575486701, 3.51186758697996)
(37.69992383263446, 3.62964730501)
(72.25699029386908, -4.28022658778899)
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:
$$x_{1} = 74.3513729610722$$
$$x_{2} = 76.445756400651$$
$$x_{3} = 15.7105309536845$$
$$x_{4} = 19.8986205587611$$
$$x_{5} = 65.9738477459929$$
$$x_{6} = 78.5401405447888$$
$$x_{7} = 68.0682275933998$$
$$x_{8} = 24.0869935064117$$
$$x_{9} = 26.1812385788794$$
$$x_{10} = 21.9927831881195$$
$$x_{11} = 30.3698008017668$$
$$x_{12} = 61.7850916250074$$
$$x_{13} = 13.6166929275579$$
$$x_{14} = 63.8794690465696$$
$$x_{15} = 32.4641075469997$$
$$x_{16} = 28.2755096981709$$
$$x_{17} = 70.1626084756746$$
$$x_{18} = 59.6907156311258$$
$$x_{19} = 17.8045256303582$$
$$x_{20} = 72.2569902938691$$
Puntos máximos de la función:
$$x_{20} = 92.1536510526427$$
$$x_{20} = 100.531204638865$$
$$x_{20} = 96.3424271874236$$
$$x_{20} = 43.9829647927364$$
$$x_{20} = 41.8886122281726$$
$$x_{20} = 6.29278191723044$$
$$x_{20} = 56.5491547048483$$
$$x_{20} = 98.4368157600501$$
$$x_{20} = 85.8704897852191$$
$$x_{20} = 46.0773218089678$$
$$x_{20} = 2.16029784897009$$
$$x_{20} = 79.5873328619807$$
$$x_{20} = 10.4764899884892$$
$$x_{20} = 39.7942649107512$$
$$x_{20} = 48.171682632084$$
$$x_{20} = 52.3604136794562$$
$$x_{20} = 94.2480389434111$$
$$x_{20} = 4.20715294221441$$
$$x_{20} = 50.2660467342498$$
$$x_{20} = 90.0592635422263$$
$$x_{20} = 8.38381259282155$$
$$x_{20} = 83.7761036083348$$
$$x_{20} = 87.9648764420645$$
$$x_{20} = 54.4547831045677$$
$$x_{20} = 33.511265754867$$
$$x_{20} = 37.6999238326345$$
Decrece en los intervalos
$$\left[78.5401405447888, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, 13.6166929275579\right]$$
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
$$- (9 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{6 \sin{\left(3 x \right)}}{x} + \frac{\cos{\left(3 x \right)}}{x^{2}}) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 3.71031523340446$$
$$x_{2} = 82.2056208554033$$
$$x_{3} = 71.7337572470289$$
$$x_{4} = 53.9317071553088$$
$$x_{5} = 60.214759762845$$
$$x_{6} = 22.5179158502217$$
$$x_{7} = 5.78141981025447$$
$$x_{8} = 56.0260542206658$$
$$x_{9} = 64.403477802222$$
$$x_{10} = 86.3943748334274$$
$$x_{11} = 16.2364713628348$$
$$x_{12} = 95.8190842516045$$
$$x_{13} = 25.6590087988046$$
$$x_{14} = 91.6303225429824$$
$$x_{15} = 80.1112454848742$$
$$x_{16} = 20.4239585238892$$
$$x_{17} = 19.3770232316731$$
$$x_{18} = 34.035771298825$$
$$x_{19} = 88.4887532733364$$
$$x_{20} = 75.9224984752237$$
$$x_{21} = 14.1430960173866$$
$$x_{22} = 36.130030099593$$
$$x_{23} = 84.2999973343513$$
$$x_{24} = 100.007848641105$$
$$x_{25} = 49.7430271381783$$
$$x_{26} = 9.95807859605126$$
$$x_{27} = 51.8373645824136$$
$$x_{28} = 66.4978406970204$$
$$x_{29} = 40.3185966045144$$
$$x_{30} = 29.8473224084856$$
$$x_{31} = 78.0168713213474$$
$$x_{32} = 97.9134661347637$$
$$x_{33} = 27.7531443070156$$
$$x_{34} = 12.0501772700338$$
$$x_{35} = 62.3091173945134$$
$$x_{36} = 58.1204052417674$$
$$x_{37} = 18.3301246319229$$
$$x_{38} = 7.86765489287871$$
$$x_{39} = 93.7247030375304$$
$$x_{40} = 1.75852871774408$$
$$x_{41} = 31.9415336946425$$
$$x_{42} = 73.8281270705241$$
$$x_{43} = 44.5072113388584$$
$$x_{44} = 38.2243062042622$$
$$x_{45} = 42.412898940997$$
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
$$\left[88.4887532733364, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, 19.3770232316731\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- (9 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{6 \sin{\left(3 x \right)}}{x} + \frac{\cos{\left(3 x \right)}}{x^{2}}) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 3.71031523340446$$
$$x_{2} = 82.2056208554033$$
$$x_{3} = 71.7337572470289$$
$$x_{4} = 53.9317071553088$$
$$x_{5} = 60.214759762845$$
$$x_{6} = 22.5179158502217$$
$$x_{7} = 5.78141981025447$$
$$x_{8} = 56.0260542206658$$
$$x_{9} = 64.403477802222$$
$$x_{10} = 86.3943748334274$$
$$x_{11} = 16.2364713628348$$
$$x_{12} = 95.8190842516045$$
$$x_{13} = 25.6590087988046$$
$$x_{14} = 91.6303225429824$$
$$x_{15} = 80.1112454848742$$
$$x_{16} = 20.4239585238892$$
$$x_{17} = 19.3770232316731$$
$$x_{18} = 34.035771298825$$
$$x_{19} = 88.4887532733364$$
$$x_{20} = 75.9224984752237$$
$$x_{21} = 14.1430960173866$$
$$x_{22} = 36.130030099593$$
$$x_{23} = 84.2999973343513$$
$$x_{24} = 100.007848641105$$
$$x_{25} = 49.7430271381783$$
$$x_{26} = 9.95807859605126$$
$$x_{27} = 51.8373645824136$$
$$x_{28} = 66.4978406970204$$
$$x_{29} = 40.3185966045144$$
$$x_{30} = 29.8473224084856$$
$$x_{31} = 78.0168713213474$$
$$x_{32} = 97.9134661347637$$
$$x_{33} = 27.7531443070156$$
$$x_{34} = 12.0501772700338$$
$$x_{35} = 62.3091173945134$$
$$x_{36} = 58.1204052417674$$
$$x_{37} = 18.3301246319229$$
$$x_{38} = 7.86765489287871$$
$$x_{39} = 93.7247030375304$$
$$x_{40} = 1.75852871774408$$
$$x_{41} = 31.9415336946425$$
$$x_{42} = 73.8281270705241$$
$$x_{43} = 44.5072113388584$$
$$x_{44} = 38.2243062042622$$
$$x_{45} = 42.412898940997$$
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
$$\left[88.4887532733364, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, 19.3770232316731\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\log{\left(x \right)} \cos{\left(3 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\log{\left(x \right)} \cos{\left(3 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty}\left(\log{\left(x \right)} \cos{\left(3 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\log{\left(x \right)} \cos{\left(3 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función log(x)*cos(3*x), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)} \cos{\left(3 x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)} \cos{\left(3 x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)} \cos{\left(3 x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)} \cos{\left(3 x \right)}}{x}\right) = 0$$
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:
$$\log{\left(x \right)} \cos{\left(3 x \right)} = \log{\left(- x \right)} \cos{\left(3 x \right)}$$
- No
$$\log{\left(x \right)} \cos{\left(3 x \right)} = - \log{\left(- x \right)} \cos{\left(3 x \right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\log{\left(x \right)} \cos{\left(3 x \right)} = \log{\left(- x \right)} \cos{\left(3 x \right)}$$
- No
$$\log{\left(x \right)} \cos{\left(3 x \right)} = - \log{\left(- x \right)} \cos{\left(3 x \right)}$$
- No
es decir, función
no es
par ni impar