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- Gráfico de la función implícita
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- Integrales paso a paso
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- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
e^3*x+10*e^2*x
-
x^11
-
(-cos(2*x)-sin(2*x))*exp(x)
-
5/(x^2-16)
-
Expresiones idénticas
- lnx*sin(x)^ dos
- lnx multiplicar por seno de (x) al cuadrado
- lnx multiplicar por seno de (x) en el grado dos
- lnx*sin(x)2
- lnx*sinx2
- lnx*sin(x)²
- lnx*sin(x) en el grado 2
- lnxsin(x)^2
- lnxsin(x)2
- lnxsinx2
- lnxsinx^2
-
Expresiones semejantes
- lnx*sinx^2
-
Expresiones con funciones
- lnx
- lnx-x+1
- lnx^2+4
- lnx-4,5x^2
- lnx((x^2)-2x+4)
- lnx−3x
- Seno sin
- sin(x)^2/(x^2*cos(x)^2)
- sin(x)/2+(1+x)*exp(-x)
- sin(x)^(2)+2
- sin(3*x)+x^4
- sin(pi*x)/sin(pi*x)
Gráfico de la función y = lnx*sin(x)^2
Solución
Ha introducido
[src]
2 f(x) = log(x)*sin (x)
$$f{\left(x \right)} = \log{\left(x \right)} \sin^{2}{\left(x \right)}$$
f = log(x)*sin(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:
$$\log{\left(x \right)} \sin^{2}{\left(x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 1$$
$$x_{2} = \pi$$
Solución numérica
$$x_{1} = 100.530964766714$$
$$x_{2} = 56.5486676093942$$
$$x_{3} = -12.5663703688317$$
$$x_{4} = 62.8318528334018$$
$$x_{5} = 65.9734457529035$$
$$x_{6} = -87.9645943587859$$
$$x_{7} = -62.8318528390238$$
$$x_{8} = -75.398223862187$$
$$x_{9} = 75.3982242072123$$
$$x_{10} = 69.1150385904649$$
$$x_{11} = -21.9911485864536$$
$$x_{12} = -53.4070752838512$$
$$x_{13} = -56.5486675197118$$
$$x_{14} = -91.1061872054414$$
$$x_{15} = 37.699112019774$$
$$x_{16} = 53.4070753637887$$
$$x_{17} = 59.6902605980227$$
$$x_{18} = 40.8407042574134$$
$$x_{19} = -91.1061872017085$$
$$x_{20} = 47.1238895923028$$
$$x_{21} = -84.8230018275441$$
$$x_{22} = 47.1238900212775$$
$$x_{23} = -43.9822971745836$$
$$x_{24} = 91.1061867326316$$
$$x_{25} = -40.8407042678506$$
$$x_{26} = -97.3893724404962$$
$$x_{27} = -47.1238900499755$$
$$x_{28} = -25.1327414743941$$
$$x_{29} = -18.8495556986729$$
$$x_{30} = -59.6902604576489$$
$$x_{31} = 21.9911485851996$$
$$x_{32} = 18.8495556834211$$
$$x_{33} = 34.5575190309907$$
$$x_{34} = -69.1150386258394$$
$$x_{35} = 25.1327414538853$$
$$x_{36} = 3.14159296679403$$
$$x_{37} = -9.42477812775273$$
$$x_{38} = -65.9734457650256$$
$$x_{39} = 84.8230014098108$$
$$x_{40} = 72.2566310277192$$
$$x_{41} = 78.5398161880055$$
$$x_{42} = -94.2477794530841$$
$$x_{43} = 81.6814091763464$$
$$x_{44} = -47.1238901634649$$
$$x_{45} = -84.8230014108525$$
$$x_{46} = -69.1150386812732$$
$$x_{47} = -50.26548229554$$
$$x_{48} = -15.7079632965444$$
$$x_{49} = -81.6814090380142$$
$$x_{50} = -28.2743337169939$$
$$x_{51} = -78.5398160961768$$
$$x_{52} = -34.5575189435564$$
$$x_{53} = 9.42477821969831$$
$$x_{54} = -18.849556125498$$
$$x_{55} = 25.1327410259853$$
$$x_{56} = 28.274333865208$$
$$x_{57} = 31.4159267885251$$
$$x_{58} = 15.7079634423521$$
$$x_{59} = 87.9645943357646$$
$$x_{60} = 75.398223939555$$
$$x_{61} = -62.8318532599408$$
$$x_{62} = 18.849555467321$$
$$x_{63} = 43.9822971694319$$
$$x_{64} = 69.1150381619555$$
$$x_{65} = 97.3893727189685$$
$$x_{66} = -72.2566308742628$$
$$x_{67} = 6.28318528435388$$
$$x_{68} = 94.2477796093525$$
$$x_{69} = -100.530964672801$$
$$x_{70} = -31.4159267055451$$
$$x_{71} = 12.5663704539928$$
$$x_{72} = -25.132741657845$$
$$x_{73} = 40.8407039438721$$
$$x_{74} = -3.1415929030096$$
$$x_{75} = 97.3893725153965$$
$$x_{76} = -40.8407046921536$$
$$x_{77} = -6.28318513957697$$
$$x_{78} = 91.1061871597628$$
$$x_{79} = -37.6991118771621$$
$$x_{80} = 50.2654824463487$$
o sea hay que resolver la ecuación:
$$\log{\left(x \right)} \sin^{2}{\left(x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 1$$
$$x_{2} = \pi$$
Solución numérica
$$x_{1} = 100.530964766714$$
$$x_{2} = 56.5486676093942$$
$$x_{3} = -12.5663703688317$$
$$x_{4} = 62.8318528334018$$
$$x_{5} = 65.9734457529035$$
$$x_{6} = -87.9645943587859$$
$$x_{7} = -62.8318528390238$$
$$x_{8} = -75.398223862187$$
$$x_{9} = 75.3982242072123$$
$$x_{10} = 69.1150385904649$$
$$x_{11} = -21.9911485864536$$
$$x_{12} = -53.4070752838512$$
$$x_{13} = -56.5486675197118$$
$$x_{14} = -91.1061872054414$$
$$x_{15} = 37.699112019774$$
$$x_{16} = 53.4070753637887$$
$$x_{17} = 59.6902605980227$$
$$x_{18} = 40.8407042574134$$
$$x_{19} = -91.1061872017085$$
$$x_{20} = 47.1238895923028$$
$$x_{21} = -84.8230018275441$$
$$x_{22} = 47.1238900212775$$
$$x_{23} = -43.9822971745836$$
$$x_{24} = 91.1061867326316$$
$$x_{25} = -40.8407042678506$$
$$x_{26} = -97.3893724404962$$
$$x_{27} = -47.1238900499755$$
$$x_{28} = -25.1327414743941$$
$$x_{29} = -18.8495556986729$$
$$x_{30} = -59.6902604576489$$
$$x_{31} = 21.9911485851996$$
$$x_{32} = 18.8495556834211$$
$$x_{33} = 34.5575190309907$$
$$x_{34} = -69.1150386258394$$
$$x_{35} = 25.1327414538853$$
$$x_{36} = 3.14159296679403$$
$$x_{37} = -9.42477812775273$$
$$x_{38} = -65.9734457650256$$
$$x_{39} = 84.8230014098108$$
$$x_{40} = 72.2566310277192$$
$$x_{41} = 78.5398161880055$$
$$x_{42} = -94.2477794530841$$
$$x_{43} = 81.6814091763464$$
$$x_{44} = -47.1238901634649$$
$$x_{45} = -84.8230014108525$$
$$x_{46} = -69.1150386812732$$
$$x_{47} = -50.26548229554$$
$$x_{48} = -15.7079632965444$$
$$x_{49} = -81.6814090380142$$
$$x_{50} = -28.2743337169939$$
$$x_{51} = -78.5398160961768$$
$$x_{52} = -34.5575189435564$$
$$x_{53} = 9.42477821969831$$
$$x_{54} = -18.849556125498$$
$$x_{55} = 25.1327410259853$$
$$x_{56} = 28.274333865208$$
$$x_{57} = 31.4159267885251$$
$$x_{58} = 15.7079634423521$$
$$x_{59} = 87.9645943357646$$
$$x_{60} = 75.398223939555$$
$$x_{61} = -62.8318532599408$$
$$x_{62} = 18.849555467321$$
$$x_{63} = 43.9822971694319$$
$$x_{64} = 69.1150381619555$$
$$x_{65} = 97.3893727189685$$
$$x_{66} = -72.2566308742628$$
$$x_{67} = 6.28318528435388$$
$$x_{68} = 94.2477796093525$$
$$x_{69} = -100.530964672801$$
$$x_{70} = -31.4159267055451$$
$$x_{71} = 12.5663704539928$$
$$x_{72} = -25.132741657845$$
$$x_{73} = 40.8407039438721$$
$$x_{74} = -3.1415929030096$$
$$x_{75} = 97.3893725153965$$
$$x_{76} = -40.8407046921536$$
$$x_{77} = -6.28318513957697$$
$$x_{78} = 91.1061871597628$$
$$x_{79} = -37.6991118771621$$
$$x_{80} = 50.2654824463487$$
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
$$2 \log{\left(x \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\sin^{2}{\left(x \right)}}{x} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 95.8197196421744$$
$$x_{2} = 80.1120364968984$$
$$x_{3} = 70.6874957987652$$
$$x_{4} = -6.28318530717959$$
$$x_{5} = -31.4159265358979$$
$$x_{6} = 67.5459991590417$$
$$x_{7} = 21.9911485751286$$
$$x_{8} = 43.9822971502571$$
$$x_{9} = 15.707963267949$$
$$x_{10} = 42.4146464836393$$
$$x_{11} = 7.88468215226503$$
$$x_{12} = -53.4070751110265$$
$$x_{13} = 92.6781744605177$$
$$x_{14} = 58.1215816459086$$
$$x_{15} = -9.42477796076938$$
$$x_{16} = 78.5398163397448$$
$$x_{17} = -81.6814089933346$$
$$x_{18} = -94.2477796076938$$
$$x_{19} = 36.1321731425561$$
$$x_{20} = 26.7092361429331$$
$$x_{21} = 94.2477796076938$$
$$x_{22} = 12.5663706143592$$
$$x_{23} = 81.6814089933346$$
$$x_{24} = 29.8500622839131$$
$$x_{25} = -50.2654824574367$$
$$x_{26} = 23.5686584612553$$
$$x_{27} = -15.707963267949$$
$$x_{28} = 64.4045132832651$$
$$x_{29} = -59.6902604182061$$
$$x_{30} = 270.176968208722$$
$$x_{31} = 51.8387217793455$$
$$x_{32} = 6.28318530717959$$
$$x_{33} = 48.697328558041$$
$$x_{34} = 73.829001694965$$
$$x_{35} = -87.9645943005142$$
$$x_{36} = 34.5575191894877$$
$$x_{37} = 65.9734457253857$$
$$x_{38} = -40.8407044966673$$
$$x_{39} = -37.6991118430775$$
$$x_{40} = 72.2566310325652$$
$$x_{41} = -43.9822971502571$$
$$x_{42} = 20.4284648400094$$
$$x_{43} = 14.1505011586627$$
$$x_{44} = -100.530964914873$$
$$x_{45} = 45.5559674357001$$
$$x_{46} = -75.398223686155$$
$$x_{47} = 1.94119311490964$$
$$x_{48} = -97.3893722612836$$
$$x_{49} = 87.9645943005142$$
$$x_{50} = 59.6902604182061$$
$$x_{51} = 100.530964914873$$
$$x_{52} = 28.2743338823081$$
$$x_{53} = -72.2566310325652$$
$$x_{54} = -21.9911485751286$$
$$x_{55} = 56.5486677646163$$
$$x_{56} = 89.5366330613754$$
$$x_{57} = -28.2743338823081$$
$$x_{58} = 50.2654824574367$$
$$x_{59} = -65.9734457253857$$
$$x_{60} = 37.6991118430775$$
$$x_{61} = 86.3950958997003$$
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:
$$x_{1} = -6.28318530717959$$
$$x_{2} = -31.4159265358979$$
$$x_{3} = 21.9911485751286$$
$$x_{4} = 43.9822971502571$$
$$x_{5} = 15.707963267949$$
$$x_{6} = -53.4070751110265$$
$$x_{7} = -9.42477796076938$$
$$x_{8} = 78.5398163397448$$
$$x_{9} = -81.6814089933346$$
$$x_{10} = -94.2477796076938$$
$$x_{11} = 94.2477796076938$$
$$x_{12} = 12.5663706143592$$
$$x_{13} = 81.6814089933346$$
$$x_{14} = -50.2654824574367$$
$$x_{15} = -15.707963267949$$
$$x_{16} = -59.6902604182061$$
$$x_{17} = 270.176968208722$$
$$x_{18} = 6.28318530717959$$
$$x_{19} = -87.9645943005142$$
$$x_{20} = 34.5575191894877$$
$$x_{21} = 65.9734457253857$$
$$x_{22} = -40.8407044966673$$
$$x_{23} = -37.6991118430775$$
$$x_{24} = 72.2566310325652$$
$$x_{25} = -43.9822971502571$$
$$x_{26} = -100.530964914873$$
$$x_{27} = -75.398223686155$$
$$x_{28} = -97.3893722612836$$
$$x_{29} = 87.9645943005142$$
$$x_{30} = 59.6902604182061$$
$$x_{31} = 100.530964914873$$
$$x_{32} = 28.2743338823081$$
$$x_{33} = -72.2566310325652$$
$$x_{34} = -21.9911485751286$$
$$x_{35} = 56.5486677646163$$
$$x_{36} = -28.2743338823081$$
$$x_{37} = 50.2654824574367$$
$$x_{38} = -65.9734457253857$$
$$x_{39} = 37.6991118430775$$
Puntos máximos de la función:
$$x_{39} = 95.8197196421744$$
$$x_{39} = 80.1120364968984$$
$$x_{39} = 70.6874957987652$$
$$x_{39} = 67.5459991590417$$
$$x_{39} = 42.4146464836393$$
$$x_{39} = 7.88468215226503$$
$$x_{39} = 92.6781744605177$$
$$x_{39} = 58.1215816459086$$
$$x_{39} = 36.1321731425561$$
$$x_{39} = 26.7092361429331$$
$$x_{39} = 29.8500622839131$$
$$x_{39} = 23.5686584612553$$
$$x_{39} = 64.4045132832651$$
$$x_{39} = 51.8387217793455$$
$$x_{39} = 48.697328558041$$
$$x_{39} = 73.829001694965$$
$$x_{39} = 20.4284648400094$$
$$x_{39} = 14.1505011586627$$
$$x_{39} = 45.5559674357001$$
$$x_{39} = 1.94119311490964$$
$$x_{39} = 89.5366330613754$$
$$x_{39} = 86.3950958997003$$
Decrece en los intervalos
$$\left[270.176968208722, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100.530964914873\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$2 \log{\left(x \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\sin^{2}{\left(x \right)}}{x} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 95.8197196421744$$
$$x_{2} = 80.1120364968984$$
$$x_{3} = 70.6874957987652$$
$$x_{4} = -6.28318530717959$$
$$x_{5} = -31.4159265358979$$
$$x_{6} = 67.5459991590417$$
$$x_{7} = 21.9911485751286$$
$$x_{8} = 43.9822971502571$$
$$x_{9} = 15.707963267949$$
$$x_{10} = 42.4146464836393$$
$$x_{11} = 7.88468215226503$$
$$x_{12} = -53.4070751110265$$
$$x_{13} = 92.6781744605177$$
$$x_{14} = 58.1215816459086$$
$$x_{15} = -9.42477796076938$$
$$x_{16} = 78.5398163397448$$
$$x_{17} = -81.6814089933346$$
$$x_{18} = -94.2477796076938$$
$$x_{19} = 36.1321731425561$$
$$x_{20} = 26.7092361429331$$
$$x_{21} = 94.2477796076938$$
$$x_{22} = 12.5663706143592$$
$$x_{23} = 81.6814089933346$$
$$x_{24} = 29.8500622839131$$
$$x_{25} = -50.2654824574367$$
$$x_{26} = 23.5686584612553$$
$$x_{27} = -15.707963267949$$
$$x_{28} = 64.4045132832651$$
$$x_{29} = -59.6902604182061$$
$$x_{30} = 270.176968208722$$
$$x_{31} = 51.8387217793455$$
$$x_{32} = 6.28318530717959$$
$$x_{33} = 48.697328558041$$
$$x_{34} = 73.829001694965$$
$$x_{35} = -87.9645943005142$$
$$x_{36} = 34.5575191894877$$
$$x_{37} = 65.9734457253857$$
$$x_{38} = -40.8407044966673$$
$$x_{39} = -37.6991118430775$$
$$x_{40} = 72.2566310325652$$
$$x_{41} = -43.9822971502571$$
$$x_{42} = 20.4284648400094$$
$$x_{43} = 14.1505011586627$$
$$x_{44} = -100.530964914873$$
$$x_{45} = 45.5559674357001$$
$$x_{46} = -75.398223686155$$
$$x_{47} = 1.94119311490964$$
$$x_{48} = -97.3893722612836$$
$$x_{49} = 87.9645943005142$$
$$x_{50} = 59.6902604182061$$
$$x_{51} = 100.530964914873$$
$$x_{52} = 28.2743338823081$$
$$x_{53} = -72.2566310325652$$
$$x_{54} = -21.9911485751286$$
$$x_{55} = 56.5486677646163$$
$$x_{56} = 89.5366330613754$$
$$x_{57} = -28.2743338823081$$
$$x_{58} = 50.2654824574367$$
$$x_{59} = -65.9734457253857$$
$$x_{60} = 37.6991118430775$$
$$x_{61} = 86.3950958997003$$
Signos de extremos en los puntos:
(95.81971964217442, 4.56246253755753)
(80.11203649689836, 4.38341722467833)
(70.68749579876517, 4.25825694491077)
(-6.283185307179586, 1.10254964387092e-31 + 5.99903913064743e-32*pi*I)
(-31.41592653589793, 5.17014436343721e-30 + 1.49975978266186e-30*pi*I)
(67.54599915904168, 4.21279582809083)
(21.991148575128552, 2.27125663724702e-30)
(43.982297150257104, 1.11225529081318e-29)
(15.707963267948966, 1.03264752464199e-30)
(42.414646483639295, 3.74745665652845)
(7.884682152265031, 2.06297628658956)
(-53.40707511102649, 8.60546142301337e-30 + 2.16329417632678e-30*pi*I)
(92.67817446051765, 4.52912657571395)
(58.12158164590861, 4.06251883524572)
(-9.42477796076938, 3.0280269348815e-31 + 1.34978380439567e-31*pi*I)
(78.53981633974483, 1.05239675280611e-30)
(-81.68140899333463, 6.77020503227825e-29 + 1.53769519406264e-29*pi*I)
(-94.2477796076938, 5.35287607041647e-29 + 1.17751027577409e-29*pi*I)
(36.132173142556084, 3.5871303095993)
(26.709236142933094, 3.28490275263666)
(94.2477796076938, 5.35287607041647e-29)
(12.566370614359172, 6.07348539927449e-31)
(81.68140899333463, 6.77020503227825e-29)
(29.85006228391305, 3.39610431359338)
(-50.26548245743669, 1.50400944749698e-29 + 3.83938504361436e-30*pi*I)
(23.568658461255307, 3.15977537682115)
(-15.707963267948966, 1.03264752464199e-30 + 3.74939945665464e-31*pi*I)
(64.40451328326512, 4.16516924258626)
(-59.69026041820607, 6.14517616924322e-30 + 1.5027934458313e-30*pi*I)
(270.1769682087222, 1.7417470442363e-27)
(51.838721779345526, 3.94811383137987)
(6.283185307179586, 1.10254964387092e-31)
(48.69732855804098, 3.88559704250618)
(73.82900169496496, 4.30174096928494)
(-87.96459430051421, 5.26403170691023e-29 + 1.1758116696069e-29*pi*I)
(34.55751918948773, 1.72337415243203e-29)
(65.97344572538566, 4.03120648719441e-30)
(-40.840704496667314, 1.42608898598829e-29 + 3.84423798515659e-30*pi*I)
(-37.69911184307752, 7.83875937863388e-30 + 2.15965408703307e-30*pi*I)
(72.25663103256524, 1.73645580663874e-28)
(-43.982297150257104, 1.11225529081318e-29 + 2.93952917401724e-30*pi*I)
(20.428464840009365, 3.01673071130889)
(14.150501158662737, 2.64927893980534)
(-100.53096491487338, 7.08054135721405e-29 + 1.53575401744574e-29*pi*I)
(45.55596743570007, 3.81891008060303)
(-75.39822368615503, 3.73428700801825e-29 + 8.6386163481323e-30*pi*I)
(1.9411931149096389, 0.576387980778498)
(-97.3893722612836, 2.15581661527067e-28 + 4.70834203716472e-29*pi*I)
(87.96459430051421, 5.26403170691023e-29)
(59.69026041820607, 6.14517616924322e-30)
(100.53096491487338, 7.08054135721405e-29)
(28.274333882308138, 4.0598244084922e-30)
(-72.25663103256524, 1.73645580663874e-28 + 4.05692731349557e-29*pi*I)
(-21.991148575128552, 2.27125663724702e-30 + 7.3488229350431e-31*pi*I)
(56.548667764616276, 1.96074534521452e-29)
(89.53663306137544, 4.49464091136003)
(-28.274333882308138, 4.0598244084922e-30 + 1.2148054239561e-30*pi*I)
(50.26548245743669, 1.50400944749698e-29)
(-65.97344572538566, 4.03120648719441e-30 + 9.62273497948765e-31*pi*I)
(37.69911184307752, 7.83875937863388e-30)
(86.39509589970032, 4.45892340221529)
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:
$$x_{1} = -6.28318530717959$$
$$x_{2} = -31.4159265358979$$
$$x_{3} = 21.9911485751286$$
$$x_{4} = 43.9822971502571$$
$$x_{5} = 15.707963267949$$
$$x_{6} = -53.4070751110265$$
$$x_{7} = -9.42477796076938$$
$$x_{8} = 78.5398163397448$$
$$x_{9} = -81.6814089933346$$
$$x_{10} = -94.2477796076938$$
$$x_{11} = 94.2477796076938$$
$$x_{12} = 12.5663706143592$$
$$x_{13} = 81.6814089933346$$
$$x_{14} = -50.2654824574367$$
$$x_{15} = -15.707963267949$$
$$x_{16} = -59.6902604182061$$
$$x_{17} = 270.176968208722$$
$$x_{18} = 6.28318530717959$$
$$x_{19} = -87.9645943005142$$
$$x_{20} = 34.5575191894877$$
$$x_{21} = 65.9734457253857$$
$$x_{22} = -40.8407044966673$$
$$x_{23} = -37.6991118430775$$
$$x_{24} = 72.2566310325652$$
$$x_{25} = -43.9822971502571$$
$$x_{26} = -100.530964914873$$
$$x_{27} = -75.398223686155$$
$$x_{28} = -97.3893722612836$$
$$x_{29} = 87.9645943005142$$
$$x_{30} = 59.6902604182061$$
$$x_{31} = 100.530964914873$$
$$x_{32} = 28.2743338823081$$
$$x_{33} = -72.2566310325652$$
$$x_{34} = -21.9911485751286$$
$$x_{35} = 56.5486677646163$$
$$x_{36} = -28.2743338823081$$
$$x_{37} = 50.2654824574367$$
$$x_{38} = -65.9734457253857$$
$$x_{39} = 37.6991118430775$$
Puntos máximos de la función:
$$x_{39} = 95.8197196421744$$
$$x_{39} = 80.1120364968984$$
$$x_{39} = 70.6874957987652$$
$$x_{39} = 67.5459991590417$$
$$x_{39} = 42.4146464836393$$
$$x_{39} = 7.88468215226503$$
$$x_{39} = 92.6781744605177$$
$$x_{39} = 58.1215816459086$$
$$x_{39} = 36.1321731425561$$
$$x_{39} = 26.7092361429331$$
$$x_{39} = 29.8500622839131$$
$$x_{39} = 23.5686584612553$$
$$x_{39} = 64.4045132832651$$
$$x_{39} = 51.8387217793455$$
$$x_{39} = 48.697328558041$$
$$x_{39} = 73.829001694965$$
$$x_{39} = 20.4284648400094$$
$$x_{39} = 14.1505011586627$$
$$x_{39} = 45.5559674357001$$
$$x_{39} = 1.94119311490964$$
$$x_{39} = 89.5366330613754$$
$$x_{39} = 86.3950958997003$$
Decrece en los intervalos
$$\left[270.176968208722, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100.530964914873\right]$$
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
$$- 2 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(x \right)} + \frac{4 \sin{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{\sin^{2}{\left(x \right)}}{x^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 24.3538383764564$$
$$x_{2} = 96.6051094036356$$
$$x_{3} = 69.9021146690577$$
$$x_{4} = 68.3313786837675$$
$$x_{5} = 46.3413193708802$$
$$x_{6} = 49.4826871182256$$
$$x_{7} = 16.5039986039154$$
$$x_{8} = 32.2057609130582$$
$$x_{9} = 2.56372347785954$$
$$x_{10} = 74.6143846512062$$
$$x_{11} = 109.171322870945$$
$$x_{12} = 63.6191362325953$$
$$x_{13} = 77.7558999415355$$
$$x_{14} = 5.55222612151163$$
$$x_{15} = 85.6097084865142$$
$$x_{16} = 41.6293042263401$$
$$x_{17} = 47.911970825717$$
$$x_{18} = 30.6353359960064$$
$$x_{19} = 4.00943109513867$$
$$x_{20} = 25.9240063903879$$
$$x_{21} = 82.4681770421537$$
$$x_{22} = 40.0587094975312$$
$$x_{23} = 88.7512448059522$$
$$x_{24} = 99.7466585671434$$
$$x_{25} = 76.1851314475436$$
$$x_{26} = 33.7763575468426$$
$$x_{27} = 55.7655092736559$$
$$x_{28} = 66.7606199103709$$
$$x_{29} = 98.1758779372257$$
$$x_{30} = 90.3220214249147$$
$$x_{31} = 10.2306466686845$$
$$x_{32} = 18.0738438609999$$
$$x_{33} = 52.6240856735649$$
$$x_{34} = 93.4635635620593$$
$$x_{35} = 91.8927854554234$$
$$x_{36} = 19.643390684436$$
$$x_{37} = 8.66679695828245$$
$$x_{38} = 38.4880455188002$$
$$x_{39} = 60.4776655555505$$
$$x_{40} = 84.0389501054096$$
$$x_{41} = 11.7984879147268$$
$$x_{42} = 54.1947733288452$$
$$x_{43} = 27.4944723337379$$
$$x_{44} = 62.048414862195$$
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
$$\left[109.171322870945, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, 2.56372347785954\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- 2 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(x \right)} + \frac{4 \sin{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{\sin^{2}{\left(x \right)}}{x^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 24.3538383764564$$
$$x_{2} = 96.6051094036356$$
$$x_{3} = 69.9021146690577$$
$$x_{4} = 68.3313786837675$$
$$x_{5} = 46.3413193708802$$
$$x_{6} = 49.4826871182256$$
$$x_{7} = 16.5039986039154$$
$$x_{8} = 32.2057609130582$$
$$x_{9} = 2.56372347785954$$
$$x_{10} = 74.6143846512062$$
$$x_{11} = 109.171322870945$$
$$x_{12} = 63.6191362325953$$
$$x_{13} = 77.7558999415355$$
$$x_{14} = 5.55222612151163$$
$$x_{15} = 85.6097084865142$$
$$x_{16} = 41.6293042263401$$
$$x_{17} = 47.911970825717$$
$$x_{18} = 30.6353359960064$$
$$x_{19} = 4.00943109513867$$
$$x_{20} = 25.9240063903879$$
$$x_{21} = 82.4681770421537$$
$$x_{22} = 40.0587094975312$$
$$x_{23} = 88.7512448059522$$
$$x_{24} = 99.7466585671434$$
$$x_{25} = 76.1851314475436$$
$$x_{26} = 33.7763575468426$$
$$x_{27} = 55.7655092736559$$
$$x_{28} = 66.7606199103709$$
$$x_{29} = 98.1758779372257$$
$$x_{30} = 90.3220214249147$$
$$x_{31} = 10.2306466686845$$
$$x_{32} = 18.0738438609999$$
$$x_{33} = 52.6240856735649$$
$$x_{34} = 93.4635635620593$$
$$x_{35} = 91.8927854554234$$
$$x_{36} = 19.643390684436$$
$$x_{37} = 8.66679695828245$$
$$x_{38} = 38.4880455188002$$
$$x_{39} = 60.4776655555505$$
$$x_{40} = 84.0389501054096$$
$$x_{41} = 11.7984879147268$$
$$x_{42} = 54.1947733288452$$
$$x_{43} = 27.4944723337379$$
$$x_{44} = 62.048414862195$$
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
$$\left[109.171322870945, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, 2.56372347785954\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\log{\left(x \right)} \sin^{2}{\left(x \right)}\right) = \left\langle 0, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle 0, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\log{\left(x \right)} \sin^{2}{\left(x \right)}\right) = \left\langle 0, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle 0, \infty\right\rangle$$
$$\lim_{x \to -\infty}\left(\log{\left(x \right)} \sin^{2}{\left(x \right)}\right) = \left\langle 0, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle 0, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\log{\left(x \right)} \sin^{2}{\left(x \right)}\right) = \left\langle 0, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle 0, \infty\right\rangle$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función log(x)*sin(x)^2, dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)} \sin^{2}{\left(x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)} \sin^{2}{\left(x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)} \sin^{2}{\left(x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)} \sin^{2}{\left(x \right)}}{x}\right) = 0$$
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:
$$\log{\left(x \right)} \sin^{2}{\left(x \right)} = \log{\left(- x \right)} \sin^{2}{\left(x \right)}$$
- No
$$\log{\left(x \right)} \sin^{2}{\left(x \right)} = - \log{\left(- x \right)} \sin^{2}{\left(x \right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\log{\left(x \right)} \sin^{2}{\left(x \right)} = \log{\left(- x \right)} \sin^{2}{\left(x \right)}$$
- No
$$\log{\left(x \right)} \sin^{2}{\left(x \right)} = - \log{\left(- x \right)} \sin^{2}{\left(x \right)}$$
- No
es decir, función
no es
par ni impar