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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 =:
-
(x+2)^3-1
-
((x+1)^3)/(x-1)^2
-
6/(x^2-4)
-
3*x^5-5*x^3+16
-
Expresiones idénticas
- (sqrt(uno +x^ dos)+|ln^3x|/ uno . seis +x^ cuatro)*sin7x
- ( raíz cuadrada de (1 más x al cuadrado ) más módulo de ln al cubo x| dividir por 1.6 más x en el grado 4) multiplicar por seno de 7x
- ( raíz cuadrada de (uno más x en el grado dos) más módulo de ln al cubo x| dividir por uno . seis más x en el grado cuatro) multiplicar por seno de 7x
- (√(1+x^2)+|ln^3x|/1.6+x^4)*sin7x
- (sqrt(1+x2)+|ln3x|/1.6+x4)*sin7x
- sqrt1+x2+|ln3x|/1.6+x4*sin7x
- (sqrt(1+x²)+|ln³x|/1.6+x⁴)*sin7x
- (sqrt(1+x en el grado 2)+|ln en el grado 3x|/1.6+x en el grado 4)*sin7x
- (sqrt(1+x^2)+|ln^3x|/1.6+x^4)sin7x
- (sqrt(1+x2)+|ln3x|/1.6+x4)sin7x
- sqrt1+x2+|ln3x|/1.6+x4sin7x
- sqrt1+x^2+|ln^3x|/1.6+x^4sin7x
- (sqrt(1+x^2)+|ln^3x| dividir por 1.6+x^4)*sin7x
-
Expresiones semejantes
- (sqrt(1+x^2)-|ln^3x|/1.6+x^4)*sin7x
- (sqrt(1-x^2)+|ln^3x|/1.6+x^4)*sin7x
- (sqrt(1+x^2)+|ln^3x|/1.6-x^4)*sin7x
-
Expresiones con funciones
- Módulo |
- ||x^2+1|+1|
- |1-x|+|x-x^2|
- |1/3x-2|-|x+4|
- |x|-4:x^2-4|x|
- |-x^2-2|x|+3|
Gráfico de la función y = (sqrt(1+x^2)+|ln^3x|/1.6+x^4)*sin7x
Solución
Ha introducido
[src]
/ ________ | 3 | \
| / 2 |log (x)| 4|
f(x) = |\/ 1 + x + --------- + x |*sin(7*x)
\ 8/5 /
$$f{\left(x \right)} = \left(x^{4} + \left(\sqrt{x^{2} + 1} + \frac{\left|{\log{\left(x \right)}^{3}}\right|}{\frac{8}{5}}\right)\right) \sin{\left(7 x \right)}$$
f = (x^4 + sqrt(x^2 + 1) + Abs(log(x)^3)/(8/5))*sin(7*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:
$$\left(x^{4} + \left(\sqrt{x^{2} + 1} + \frac{\left|{\log{\left(x \right)}^{3}}\right|}{\frac{8}{5}}\right)\right) \sin{\left(7 x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 65.9734457253857$$
$$x_{2} = -21.9911485751286$$
$$x_{3} = 21.9911485751286$$
$$x_{4} = 13.4639685153848$$
$$x_{5} = -89.7597901025655$$
$$x_{6} = -15.707963267949$$
$$x_{7} = -49.8166835069239$$
$$x_{8} = -23.7863443771799$$
$$x_{9} = 8.0783811092309$$
$$x_{10} = -31.8647254864108$$
$$x_{11} = -45.7774929523084$$
$$x_{12} = 2.24399475256414$$
$$x_{13} = -48.0214877048726$$
$$x_{14} = 12.1175716638463$$
$$x_{15} = 6.28318530717959$$
$$x_{16} = -85.7205995479501$$
$$x_{17} = 54.3046730120521$$
$$x_{18} = 28.2743338823081$$
$$x_{19} = -70.0126362800011$$
$$x_{20} = 81.2326100428218$$
$$x_{21} = 39.9431065956417$$
$$x_{22} = -92.0037848551297$$
$$x_{23} = 26.030339129744$$
$$x_{24} = -4.03919055461545$$
$$x_{25} = -13.9127674658977$$
$$x_{26} = 76.2958215871807$$
$$x_{27} = -52.060678259488$$
$$x_{28} = -97.8381712117964$$
$$x_{29} = 38.1479107935903$$
$$x_{30} = -17.9519580205131$$
$$x_{31} = 10.322375861795$$
$$x_{32} = 52.060678259488$$
$$x_{33} = 96.0429754097451$$
$$x_{34} = -96.9405733107708$$
$$x_{35} = 100.082165964361$$
$$x_{36} = 70.0126362800011$$
$$x_{37} = 72.2566310325652$$
$$x_{38} = 24.2351433276927$$
$$x_{39} = 20.1959527730772$$
$$x_{40} = 83.4766047953859$$
$$x_{41} = -79.8862131912833$$
$$x_{42} = -87.9645943005142$$
$$x_{43} = -37.6991118430775$$
$$x_{44} = -43.9822971502571$$
$$x_{45} = 68.2174404779498$$
$$x_{46} = -1.79519580205131$$
$$x_{47} = 48.0214877048726$$
$$x_{48} = -71.8078320820524$$
$$x_{49} = 90.2085890530783$$
$$x_{50} = 17.9519580205131$$
$$x_{51} = 63.7294509728215$$
$$x_{52} = 4.03919055461545$$
$$x_{53} = 98.2869701623092$$
$$x_{54} = -9.87357691128221$$
$$x_{55} = 30.0695296843594$$
$$x_{56} = -81.6814089933346$$
$$x_{57} = 30.9671275853851$$
$$x_{58} = -65.9734457253857$$
$$x_{59} = -39.9431065956417$$
$$x_{60} = 61.9342551707702$$
$$x_{61} = -11.2199737628207$$
$$x_{62} = -83.9254037458988$$
$$x_{63} = 43.9822971502571$$
$$x_{64} = 34.1087202389749$$
$$x_{65} = -61.9342551707702$$
$$x_{66} = 82.1302079438475$$
$$x_{67} = 32.3135244369236$$
$$x_{68} = -63.7294509728215$$
$$x_{69} = -27.8255349317953$$
$$x_{70} = -93.798980657181$$
$$x_{71} = 78.091017389232$$
$$x_{72} = 42.1871013482058$$
$$x_{73} = 50.2654824574367$$
$$x_{74} = 94.2477796076938$$
$$x_{75} = -53.8558740615393$$
$$x_{76} = -35.9039160410262$$
$$x_{77} = -5.83438635666676$$
$$x_{78} = -74.0518268346165$$
$$x_{79} = 86.1693984984629$$
$$x_{80} = 60.1390593687189$$
$$x_{81} = 92.0037848551297$$
$$x_{82} = -59.6902604182061$$
$$x_{83} = 16.1567622184618$$
$$x_{84} = -30.0695296843594$$
$$x_{85} = -75.8470226366679$$
$$x_{86} = -96.0429754097451$$
$$x_{87} = -41.738302397693$$
$$x_{88} = -26.030339129744$$
$$x_{89} = 87.9645943005142$$
$$x_{90} = 56.0998688141035$$
$$x_{91} = 46.2262919028212$$
$$x_{92} = -19.7471538225644$$
$$x_{93} = 74.0518268346165$$
$$x_{94} = -67.768641527437$$
$$x_{95} = 64.1782499233343$$
$$x_{96} = -57.8950646161548$$
o sea hay que resolver la ecuación:
$$\left(x^{4} + \left(\sqrt{x^{2} + 1} + \frac{\left|{\log{\left(x \right)}^{3}}\right|}{\frac{8}{5}}\right)\right) \sin{\left(7 x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 65.9734457253857$$
$$x_{2} = -21.9911485751286$$
$$x_{3} = 21.9911485751286$$
$$x_{4} = 13.4639685153848$$
$$x_{5} = -89.7597901025655$$
$$x_{6} = -15.707963267949$$
$$x_{7} = -49.8166835069239$$
$$x_{8} = -23.7863443771799$$
$$x_{9} = 8.0783811092309$$
$$x_{10} = -31.8647254864108$$
$$x_{11} = -45.7774929523084$$
$$x_{12} = 2.24399475256414$$
$$x_{13} = -48.0214877048726$$
$$x_{14} = 12.1175716638463$$
$$x_{15} = 6.28318530717959$$
$$x_{16} = -85.7205995479501$$
$$x_{17} = 54.3046730120521$$
$$x_{18} = 28.2743338823081$$
$$x_{19} = -70.0126362800011$$
$$x_{20} = 81.2326100428218$$
$$x_{21} = 39.9431065956417$$
$$x_{22} = -92.0037848551297$$
$$x_{23} = 26.030339129744$$
$$x_{24} = -4.03919055461545$$
$$x_{25} = -13.9127674658977$$
$$x_{26} = 76.2958215871807$$
$$x_{27} = -52.060678259488$$
$$x_{28} = -97.8381712117964$$
$$x_{29} = 38.1479107935903$$
$$x_{30} = -17.9519580205131$$
$$x_{31} = 10.322375861795$$
$$x_{32} = 52.060678259488$$
$$x_{33} = 96.0429754097451$$
$$x_{34} = -96.9405733107708$$
$$x_{35} = 100.082165964361$$
$$x_{36} = 70.0126362800011$$
$$x_{37} = 72.2566310325652$$
$$x_{38} = 24.2351433276927$$
$$x_{39} = 20.1959527730772$$
$$x_{40} = 83.4766047953859$$
$$x_{41} = -79.8862131912833$$
$$x_{42} = -87.9645943005142$$
$$x_{43} = -37.6991118430775$$
$$x_{44} = -43.9822971502571$$
$$x_{45} = 68.2174404779498$$
$$x_{46} = -1.79519580205131$$
$$x_{47} = 48.0214877048726$$
$$x_{48} = -71.8078320820524$$
$$x_{49} = 90.2085890530783$$
$$x_{50} = 17.9519580205131$$
$$x_{51} = 63.7294509728215$$
$$x_{52} = 4.03919055461545$$
$$x_{53} = 98.2869701623092$$
$$x_{54} = -9.87357691128221$$
$$x_{55} = 30.0695296843594$$
$$x_{56} = -81.6814089933346$$
$$x_{57} = 30.9671275853851$$
$$x_{58} = -65.9734457253857$$
$$x_{59} = -39.9431065956417$$
$$x_{60} = 61.9342551707702$$
$$x_{61} = -11.2199737628207$$
$$x_{62} = -83.9254037458988$$
$$x_{63} = 43.9822971502571$$
$$x_{64} = 34.1087202389749$$
$$x_{65} = -61.9342551707702$$
$$x_{66} = 82.1302079438475$$
$$x_{67} = 32.3135244369236$$
$$x_{68} = -63.7294509728215$$
$$x_{69} = -27.8255349317953$$
$$x_{70} = -93.798980657181$$
$$x_{71} = 78.091017389232$$
$$x_{72} = 42.1871013482058$$
$$x_{73} = 50.2654824574367$$
$$x_{74} = 94.2477796076938$$
$$x_{75} = -53.8558740615393$$
$$x_{76} = -35.9039160410262$$
$$x_{77} = -5.83438635666676$$
$$x_{78} = -74.0518268346165$$
$$x_{79} = 86.1693984984629$$
$$x_{80} = 60.1390593687189$$
$$x_{81} = 92.0037848551297$$
$$x_{82} = -59.6902604182061$$
$$x_{83} = 16.1567622184618$$
$$x_{84} = -30.0695296843594$$
$$x_{85} = -75.8470226366679$$
$$x_{86} = -96.0429754097451$$
$$x_{87} = -41.738302397693$$
$$x_{88} = -26.030339129744$$
$$x_{89} = 87.9645943005142$$
$$x_{90} = 56.0998688141035$$
$$x_{91} = 46.2262919028212$$
$$x_{92} = -19.7471538225644$$
$$x_{93} = 74.0518268346165$$
$$x_{94} = -67.768641527437$$
$$x_{95} = 64.1782499233343$$
$$x_{96} = -57.8950646161548$$
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
$$7 \left(x^{4} + \left(\sqrt{x^{2} + 1} + \frac{\left|{\log{\left(x \right)}^{3}}\right|}{\frac{8}{5}}\right)\right) \cos{\left(7 x \right)} + \left(4 x^{3} + \frac{x}{\sqrt{x^{2} + 1}} + \frac{5 \left(\left(\frac{3 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} \operatorname{sign}^{2}{\left(x \right)}}{x} - \frac{3 \arg^{2}{\left(x \right)} \operatorname{sign}^{2}{\left(x \right)}}{x}\right) \left(\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{3} - 3 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \arg^{2}{\left(x \right)}\right) + \frac{6 \left(3 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} \arg{\left(x \right)} - \arg^{3}{\left(x \right)}\right) \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \arg{\left(x \right)} \operatorname{sign}^{2}{\left(x \right)}}{x}\right) \operatorname{sign}{\left(\log{\left(x \right)}^{3} \right)}}{8 \log{\left(x \right)}^{3}}\right) \sin{\left(7 x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -77.8676662421385$$
$$x_{2} = 63.9551268127915$$
$$x_{3} = -57.2232926907379$$
$$x_{4} = 28.0528437814417$$
$$x_{5} = -89.9850967431251$$
$$x_{6} = 40.1695380936668$$
$$x_{7} = 14.1429330458931$$
$$x_{8} = -3.83372811967992$$
$$x_{9} = 4.28230647709453$$
$$x_{10} = 98.0634031242016$$
$$x_{11} = 48.247579033909$$
$$x_{12} = -75.1749100880996$$
$$x_{13} = -21.7704968763189$$
$$x_{14} = 67.9942415512575$$
$$x_{15} = -24.0141418825183$$
$$x_{16} = -2.0381095495968$$
$$x_{17} = -11.90001170804$$
$$x_{18} = -17.7321580056565$$
$$x_{19} = 36.1305746405087$$
$$x_{20} = 96.268222844754$$
$$x_{21} = -41.9646469647508$$
$$x_{22} = -72.0333647894745$$
$$x_{23} = 8.3125648497237$$
$$x_{24} = -28.0528436172354$$
$$x_{25} = -43.7597629919067$$
$$x_{26} = -98.0634031237967$$
$$x_{27} = -65.7502877661554$$
$$x_{28} = -61.7111784643883$$
$$x_{29} = -69.7894064717788$$
$$x_{30} = 88.1899194071673$$
$$x_{31} = -99.858583961357$$
$$x_{32} = -85.9459488203819$$
$$x_{33} = -51.8378534656629$$
$$x_{34} = -55.8769302072371$$
$$x_{35} = 10.1060354920338$$
$$x_{36} = -7.86424878510195$$
$$x_{37} = 6.52002221049098$$
$$x_{38} = -94.0242483284409$$
$$x_{39} = -35.681804043653$$
$$x_{40} = 76.0724951788205$$
$$x_{41} = -87.7411251903887$$
$$x_{42} = -39.7207620657404$$
$$x_{43} = 15.9374808881019$$
$$x_{44} = 78.3164591866865$$
$$x_{45} = -12.3485643108583$$
$$x_{46} = 85.9459488211459$$
$$x_{47} = -47.7987959533079$$
$$x_{48} = 22.2192207421213$$
$$x_{49} = 62.1599678690624$$
$$x_{50} = -63.9551268096275$$
$$x_{51} = 33.8867294355745$$
$$x_{52} = 19.9756380867426$$
$$x_{53} = 66.1990783032156$$
$$x_{54} = 41.964646988644$$
$$x_{55} = 52.2866389104854$$
$$x_{56} = -67.9942415489$$
$$x_{57} = 94.0242483289367$$
$$x_{58} = -46.0036667775284$$
$$x_{59} = 70.238197951115$$
$$x_{60} = 6.07212402548424$$
$$x_{61} = 56.7745050207678$$
$$x_{62} = -36.1305745915694$$
$$x_{63} = 26.2578467616661$$
$$x_{64} = 80.5604249071651$$
$$x_{65} = 92.2290694252539$$
$$x_{66} = -37.9256635006403$$
$$x_{67} = 44.2085430291782$$
$$x_{68} = 32.0916683196631$$
$$x_{69} = 54.0817828975494$$
$$x_{70} = 89.9850967437376$$
$$x_{71} = -29.8478645679819$$
$$x_{72} = 30.2966231528638$$
$$x_{73} = -95.8194278653148$$
$$x_{74} = -15.9374784510599$$
$$x_{75} = -50.0427141428815$$
$$x_{76} = 72.0333647912609$$
$$x_{77} = -33.8867293690557$$
$$x_{78} = -19.9756372558986$$
$$x_{79} = -73.8285330381195$$
$$x_{80} = 80.1116316329674$$
$$x_{81} = 50.0427141531543$$
$$x_{82} = 2.05503340050281$$
$$x_{83} = -81.9068051019944$$
$$x_{84} = -59.9160222905198$$
$$x_{85} = 100.307379254798$$
$$x_{86} = -83.7019795299298$$
$$x_{87} = 24.014142227597$$
$$x_{88} = -13.6943174821224$$
$$x_{89} = 58.1208685691598$$
$$x_{90} = 18.1808450293136$$
$$x_{91} = 84.1507732802386$$
$$x_{92} = -25.8091015779888$$
$$x_{93} = -8.76082556046972$$
$$x_{94} = 32.9891970465441$$
$$x_{95} = -78.765252197294$$
$$x_{96} = 46.003666792909$$
$$x_{97} = 74.2773253099006$$
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:
$$x_{1} = -89.9850967431251$$
$$x_{2} = 40.1695380936668$$
$$x_{3} = 14.1429330458931$$
$$x_{4} = -3.83372811967992$$
$$x_{5} = 4.28230647709453$$
$$x_{6} = 48.247579033909$$
$$x_{7} = -21.7704968763189$$
$$x_{8} = 67.9942415512575$$
$$x_{9} = -2.0381095495968$$
$$x_{10} = -11.90001170804$$
$$x_{11} = -72.0333647894745$$
$$x_{12} = -28.0528436172354$$
$$x_{13} = -98.0634031237967$$
$$x_{14} = -65.7502877661554$$
$$x_{15} = -99.858583961357$$
$$x_{16} = -55.8769302072371$$
$$x_{17} = 76.0724951788205$$
$$x_{18} = -39.7207620657404$$
$$x_{19} = 15.9374808881019$$
$$x_{20} = 85.9459488211459$$
$$x_{21} = -47.7987959533079$$
$$x_{22} = 22.2192207421213$$
$$x_{23} = -63.9551268096275$$
$$x_{24} = 33.8867294355745$$
$$x_{25} = 66.1990783032156$$
$$x_{26} = 41.964646988644$$
$$x_{27} = 94.0242483289367$$
$$x_{28} = -46.0036667775284$$
$$x_{29} = 6.07212402548424$$
$$x_{30} = -36.1305745915694$$
$$x_{31} = 80.5604249071651$$
$$x_{32} = 92.2290694252539$$
$$x_{33} = -37.9256635006403$$
$$x_{34} = 32.0916683196631$$
$$x_{35} = -29.8478645679819$$
$$x_{36} = 30.2966231528638$$
$$x_{37} = -19.9756372558986$$
$$x_{38} = -73.8285330381195$$
$$x_{39} = 50.0427141531543$$
$$x_{40} = -81.9068051019944$$
$$x_{41} = 100.307379254798$$
$$x_{42} = -83.7019795299298$$
$$x_{43} = 24.014142227597$$
$$x_{44} = -13.6943174821224$$
$$x_{45} = 58.1208685691598$$
$$x_{46} = 84.1507732802386$$
$$x_{47} = 32.9891970465441$$
$$x_{48} = 74.2773253099006$$
Puntos máximos de la función:
$$x_{48} = -77.8676662421385$$
$$x_{48} = 63.9551268127915$$
$$x_{48} = -57.2232926907379$$
$$x_{48} = 28.0528437814417$$
$$x_{48} = 98.0634031242016$$
$$x_{48} = -75.1749100880996$$
$$x_{48} = -24.0141418825183$$
$$x_{48} = -17.7321580056565$$
$$x_{48} = 36.1305746405087$$
$$x_{48} = 96.268222844754$$
$$x_{48} = -41.9646469647508$$
$$x_{48} = 8.3125648497237$$
$$x_{48} = -43.7597629919067$$
$$x_{48} = -61.7111784643883$$
$$x_{48} = -69.7894064717788$$
$$x_{48} = 88.1899194071673$$
$$x_{48} = -85.9459488203819$$
$$x_{48} = -51.8378534656629$$
$$x_{48} = 10.1060354920338$$
$$x_{48} = -7.86424878510195$$
$$x_{48} = 6.52002221049098$$
$$x_{48} = -94.0242483284409$$
$$x_{48} = -35.681804043653$$
$$x_{48} = -87.7411251903887$$
$$x_{48} = 78.3164591866865$$
$$x_{48} = -12.3485643108583$$
$$x_{48} = 62.1599678690624$$
$$x_{48} = 19.9756380867426$$
$$x_{48} = 52.2866389104854$$
$$x_{48} = -67.9942415489$$
$$x_{48} = 70.238197951115$$
$$x_{48} = 56.7745050207678$$
$$x_{48} = 26.2578467616661$$
$$x_{48} = 44.2085430291782$$
$$x_{48} = 54.0817828975494$$
$$x_{48} = 89.9850967437376$$
$$x_{48} = -95.8194278653148$$
$$x_{48} = -15.9374784510599$$
$$x_{48} = -50.0427141428815$$
$$x_{48} = 72.0333647912609$$
$$x_{48} = -33.8867293690557$$
$$x_{48} = 80.1116316329674$$
$$x_{48} = 2.05503340050281$$
$$x_{48} = -59.9160222905198$$
$$x_{48} = 18.1808450293136$$
$$x_{48} = -25.8091015779888$$
$$x_{48} = -8.76082556046972$$
$$x_{48} = -78.765252197294$$
$$x_{48} = 46.003666792909$$
Decrece en los intervalos
$$\left[100.307379254798, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.858583961357\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$7 \left(x^{4} + \left(\sqrt{x^{2} + 1} + \frac{\left|{\log{\left(x \right)}^{3}}\right|}{\frac{8}{5}}\right)\right) \cos{\left(7 x \right)} + \left(4 x^{3} + \frac{x}{\sqrt{x^{2} + 1}} + \frac{5 \left(\left(\frac{3 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} \operatorname{sign}^{2}{\left(x \right)}}{x} - \frac{3 \arg^{2}{\left(x \right)} \operatorname{sign}^{2}{\left(x \right)}}{x}\right) \left(\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{3} - 3 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \arg^{2}{\left(x \right)}\right) + \frac{6 \left(3 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} \arg{\left(x \right)} - \arg^{3}{\left(x \right)}\right) \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \arg{\left(x \right)} \operatorname{sign}^{2}{\left(x \right)}}{x}\right) \operatorname{sign}{\left(\log{\left(x \right)}^{3} \right)}}{8 \log{\left(x \right)}^{3}}\right) \sin{\left(7 x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -77.8676662421385$$
$$x_{2} = 63.9551268127915$$
$$x_{3} = -57.2232926907379$$
$$x_{4} = 28.0528437814417$$
$$x_{5} = -89.9850967431251$$
$$x_{6} = 40.1695380936668$$
$$x_{7} = 14.1429330458931$$
$$x_{8} = -3.83372811967992$$
$$x_{9} = 4.28230647709453$$
$$x_{10} = 98.0634031242016$$
$$x_{11} = 48.247579033909$$
$$x_{12} = -75.1749100880996$$
$$x_{13} = -21.7704968763189$$
$$x_{14} = 67.9942415512575$$
$$x_{15} = -24.0141418825183$$
$$x_{16} = -2.0381095495968$$
$$x_{17} = -11.90001170804$$
$$x_{18} = -17.7321580056565$$
$$x_{19} = 36.1305746405087$$
$$x_{20} = 96.268222844754$$
$$x_{21} = -41.9646469647508$$
$$x_{22} = -72.0333647894745$$
$$x_{23} = 8.3125648497237$$
$$x_{24} = -28.0528436172354$$
$$x_{25} = -43.7597629919067$$
$$x_{26} = -98.0634031237967$$
$$x_{27} = -65.7502877661554$$
$$x_{28} = -61.7111784643883$$
$$x_{29} = -69.7894064717788$$
$$x_{30} = 88.1899194071673$$
$$x_{31} = -99.858583961357$$
$$x_{32} = -85.9459488203819$$
$$x_{33} = -51.8378534656629$$
$$x_{34} = -55.8769302072371$$
$$x_{35} = 10.1060354920338$$
$$x_{36} = -7.86424878510195$$
$$x_{37} = 6.52002221049098$$
$$x_{38} = -94.0242483284409$$
$$x_{39} = -35.681804043653$$
$$x_{40} = 76.0724951788205$$
$$x_{41} = -87.7411251903887$$
$$x_{42} = -39.7207620657404$$
$$x_{43} = 15.9374808881019$$
$$x_{44} = 78.3164591866865$$
$$x_{45} = -12.3485643108583$$
$$x_{46} = 85.9459488211459$$
$$x_{47} = -47.7987959533079$$
$$x_{48} = 22.2192207421213$$
$$x_{49} = 62.1599678690624$$
$$x_{50} = -63.9551268096275$$
$$x_{51} = 33.8867294355745$$
$$x_{52} = 19.9756380867426$$
$$x_{53} = 66.1990783032156$$
$$x_{54} = 41.964646988644$$
$$x_{55} = 52.2866389104854$$
$$x_{56} = -67.9942415489$$
$$x_{57} = 94.0242483289367$$
$$x_{58} = -46.0036667775284$$
$$x_{59} = 70.238197951115$$
$$x_{60} = 6.07212402548424$$
$$x_{61} = 56.7745050207678$$
$$x_{62} = -36.1305745915694$$
$$x_{63} = 26.2578467616661$$
$$x_{64} = 80.5604249071651$$
$$x_{65} = 92.2290694252539$$
$$x_{66} = -37.9256635006403$$
$$x_{67} = 44.2085430291782$$
$$x_{68} = 32.0916683196631$$
$$x_{69} = 54.0817828975494$$
$$x_{70} = 89.9850967437376$$
$$x_{71} = -29.8478645679819$$
$$x_{72} = 30.2966231528638$$
$$x_{73} = -95.8194278653148$$
$$x_{74} = -15.9374784510599$$
$$x_{75} = -50.0427141428815$$
$$x_{76} = 72.0333647912609$$
$$x_{77} = -33.8867293690557$$
$$x_{78} = -19.9756372558986$$
$$x_{79} = -73.8285330381195$$
$$x_{80} = 80.1116316329674$$
$$x_{81} = 50.0427141531543$$
$$x_{82} = 2.05503340050281$$
$$x_{83} = -81.9068051019944$$
$$x_{84} = -59.9160222905198$$
$$x_{85} = 100.307379254798$$
$$x_{86} = -83.7019795299298$$
$$x_{87} = 24.014142227597$$
$$x_{88} = -13.6943174821224$$
$$x_{89} = 58.1208685691598$$
$$x_{90} = 18.1808450293136$$
$$x_{91} = 84.1507732802386$$
$$x_{92} = -25.8091015779888$$
$$x_{93} = -8.76082556046972$$
$$x_{94} = 32.9891970465441$$
$$x_{95} = -78.765252197294$$
$$x_{96} = 46.003666792909$$
$$x_{97} = 74.2773253099006$$
Signos de extremos en los puntos:
3/2
/ 2\
(-77.8676662421385, 36763585.5268388 + 0.62498317176148*\18.9661190612996 + pi / )(63.95512681279155, 16729653.6596967)
3/2
/ 2\
(-57.223292690737885, 10721907.1452723 + 0.624968840802812*\16.3778935813279 + pi / )(28.052843781441652, 619232.055552049)
3/2
/ 2\
(-89.98509674312508, -65565320.9203627 - 0.624987398637509*\20.2467967105683 + pi / )(40.16953809366681, -2603486.73077586)
(14.142933045893125, -40002.2145484911)
3/2
/ 2\
(-3.8337281196799187, -218.047774675892 - 0.619516797744193*\1.80589984226233 + pi / )(4.282306477094526, -339.672142574491)
(98.06340312420161, 92474334.1454528)
(48.24757903390897, -5418491.95816916)
3/2
/ 2\
(-75.1749100880996, 31935972.4835302 + 0.624981944669594*\18.6608235170257 + pi / ) 3/2
/ 2\
(-21.770496876318866, -224577.389613057 - 0.624784935589424*\9.48982341394347 + pi / )(67.99424155125755, -21373494.5410285)
3/2
/ 2\
(-24.014141882518345, 332488.63541371 + 0.624823200894358*\10.1037706979019 + pi / ) 3/2
/ 2\
(-2.0381095495967987, -19.3612816537827 - 0.61975855011924*\0.506976306572152 + pi / ) 3/2
/ 2\
(-11.900011708039967, -20042.4206748781 - 0.624283832973532*\6.13324732043328 + pi / ) 3/2
/ 2\
(-17.73215800565654, 98832.3784387754 + 0.624676091493372*\8.2678091530064 + pi / )(36.1305746405087, 1703969.16914455)
(96.26822284475404, 85886510.0124375)
3/2
/ 2\
(-41.9646469647508, 3100986.79316712 + 0.624942067856341*\13.9638799507585 + pi / ) 3/2
/ 2\
(-72.03336478947448, -26922928.9344104 - 0.624980335558768*\18.2938360030033 + pi / )(8.312564849723701, 4777.71614045141)
3/2
/ 2\
(-28.052843617235393, -619208.896438957 - 0.624870407075377*\11.1161561518318 + pi / ) 3/2
/ 2\
(-43.759762991906726, 3666637.75433932 + 0.624946722334816*\14.2786851014775 + pi / ) 3/2
/ 2\
(-98.0634031237967, -92474273.8804479 - 0.624989389218869*\21.0278579592488 + pi / ) 3/2
/ 2\
(-65.75028776615544, -18688556.5540318 - 0.624976398042261*\17.5214578259937 + pi / ) 3/2
/ 2\
(-61.71117846438833, 14502356.9550307 + 0.624973207642275*\16.9947184097317 + pi / ) 3/2
/ 2\
(-69.7894064717788, 23721641.5948766 + 0.62497905076467*\18.0241193525505 + pi / )(88.18991940716732, 60487787.9989068)
3/2
/ 2\
(-99.85858396135696, -99434006.4932241 - 0.624989767283134*\21.1945603277558 + pi / ) 3/2
/ 2\
(-85.94594882038191, 54562307.2869222 + 0.624986186419932*\19.8356093374753 + pi / ) 3/2
/ 2\
(-51.83785346566289, 7220458.44481716 + 0.624962031178537*\15.5876566217454 + pi / ) 3/2
/ 2\
(-55.876930207237116, -9747874.65713994 - 0.624967321320338*\16.185748775928 + pi / )(10.106035492033786, 10432.2043321101)
3/2
/ 2\
(-7.864248785101955, 3823.01106908559 + 0.623386536800193*\4.25319273038268 + pi / )(6.520022210490983, 1810.98886094715)
3/2
/ 2\
(-94.02424832844093, 78154138.9895604 + 0.624988458019672*\20.6438712280455 + pi / ) 3/2
/ 2\
(-35.68180404365301, 1620843.56384979 + 0.624919878212474*\12.7780573365712 + pi / )(76.07249517882046, -33488834.9259571)
3/2
/ 2\
(-87.74112519038873, 59265810.8926763 + 0.624986745864169*\20.0201723134303 + pi / ) 3/2
/ 2\
(-39.72076206574036, -2489042.35818085 - 0.624935339483643*\13.5561963355135 + pi / )(15.937480888101911, -64505.4874078578)
(78.31645918668646, 37618555.8131065)
3/2
/ 2\
(-12.34856431085826, 23239.9137278575 + 0.624334485087884*\6.31788235726889 + pi / )(85.94594882114595, -54562362.4995908)
3/2
/ 2\
(-47.79879595330789, -5219642.68832783 - 0.624955344378601*\14.9536924796723 + pi / )(22.219220742121333, -243693.861017575)
(62.15996786906243, 14928901.9011532)
3/2
/ 2\
(-63.95512680962746, -16729608.7259049 - 0.624975054590488*\17.2904749990191 + pi / )(33.88672943557448, -1318490.64389417)
(19.9756380867426, 159193.52560612)
(66.19907830321556, -19204107.6056984)
(41.96464698864401, -3101019.40302608)
(52.28663891048536, 7473813.50068794)
3/2
/ 2\
(-67.99424154890005, 21373447.5922946 + 0.624977930052226*\17.8035306089285 + pi / )(94.02424832893675, -78154197.611301)
3/2
/ 2\
(-46.00366677752841, -4478584.35070138 - 0.62495179200354*\14.6591053065128 + pi / )(70.23819795111503, 24337792.3576222)
(6.07212402548424, -1363.3053437222)
(56.77450502076781, 10389521.0020657)
3/2
/ 2\
(-36.13057459156939, -1703940.32414775 - 0.624921855492092*\12.8675694228479 + pi / )(26.25784676166614, 475310.677303957)
(80.56042490716513, -42118941.0165585)
(92.2290694252539, -72354221.0436473)
3/2
/ 2\
(-37.925663500640304, -2068671.00577381 - 0.624929075546038*\13.2177911009323 + pi / )(44.208543029178166, 3819420.07211078)
(32.09166831966311, -1060532.96844024)
(54.08178289754936, 8554300.91262707)
(89.9850967437376, 65565377.8588263)
3/2
/ 2\
(-29.847864567981937, -793578.410397534 - 0.624885516243542*\11.5335855363743 + pi / )(30.29662315286384, -842418.865069975)
3/2
/ 2\
(-95.81942786531481, 84296018.9361107 + 0.624988886432783*\20.8160910783043 + pi / ) 3/2
/ 2\
(-15.93747845105994, 64492.2312696156 + 0.624599307548149*\7.6655527880633 + pi / ) 3/2
/ 2\
(-50.042714142881515, 6271025.69697361 + 0.624959258700001*\15.3106058191934 + pi / )(72.03336479126092, 26922977.8360618)
3/2
/ 2\
(-33.88672936905573, 1318463.31859664 + 0.624911168478151*\12.4116943962743 + pi / ) 3/2
/ 2\
(-19.975637255898576, -159176.749878053 - 0.624744624591282*\8.96711046574779 + pi / ) 3/2
/ 2\
(-73.8285330381195, -29708794.3715874 - 0.624981280171441*\18.5050124834127 + pi / )(80.11163163296737, 41188185.5400107)
(50.0427141531543, -6271063.1373555)
(2.0550334005028112, 19.7310260503707)
3/2
/ 2\
(-81.90680510199444, -45005974.0519576 - 0.624984790490158*\19.4091534441061 + pi / ) 3/2
/ 2\
(-59.9160222905198, 12887069.307287 + 0.624971578289993*\16.752190205149 + pi / )(100.30737925479832, -101233716.376168)
3/2
/ 2\
(-83.70197952992979, -49083275.4510067 - 0.624985435868061*\19.6006543729787 + pi / )(24.014142227597034, -332508.702410933)
3/2
/ 2\
(-13.694317482122427, -35152.3495764905 - 0.624458070211339*\6.84858936835261 + pi / )(58.12086856915984, -11410671.4766279)
(18.18084502931363, 109238.297241613)
(84.15077328023864, -50144537.7557322)
3/2
/ 2\
(-25.80910157798876, 443619.308653025 + 0.624846915546222*\10.5672273552026 + pi / ) 3/2
/ 2\
(-8.760825560469717, 5887.34463249375 + 0.623691406786257*\4.71015930293889 + pi / )(32.9891970465441, -1184250.93821092)
3/2
/ 2\
(-78.765252197294, 38488247.0717001 + 0.624983553095703*\19.0660771822696 + pi / )(46.00366679290904, 4478619.42651188)
(74.27732530990062, -30437848.2722757)
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:
$$x_{1} = -89.9850967431251$$
$$x_{2} = 40.1695380936668$$
$$x_{3} = 14.1429330458931$$
$$x_{4} = -3.83372811967992$$
$$x_{5} = 4.28230647709453$$
$$x_{6} = 48.247579033909$$
$$x_{7} = -21.7704968763189$$
$$x_{8} = 67.9942415512575$$
$$x_{9} = -2.0381095495968$$
$$x_{10} = -11.90001170804$$
$$x_{11} = -72.0333647894745$$
$$x_{12} = -28.0528436172354$$
$$x_{13} = -98.0634031237967$$
$$x_{14} = -65.7502877661554$$
$$x_{15} = -99.858583961357$$
$$x_{16} = -55.8769302072371$$
$$x_{17} = 76.0724951788205$$
$$x_{18} = -39.7207620657404$$
$$x_{19} = 15.9374808881019$$
$$x_{20} = 85.9459488211459$$
$$x_{21} = -47.7987959533079$$
$$x_{22} = 22.2192207421213$$
$$x_{23} = -63.9551268096275$$
$$x_{24} = 33.8867294355745$$
$$x_{25} = 66.1990783032156$$
$$x_{26} = 41.964646988644$$
$$x_{27} = 94.0242483289367$$
$$x_{28} = -46.0036667775284$$
$$x_{29} = 6.07212402548424$$
$$x_{30} = -36.1305745915694$$
$$x_{31} = 80.5604249071651$$
$$x_{32} = 92.2290694252539$$
$$x_{33} = -37.9256635006403$$
$$x_{34} = 32.0916683196631$$
$$x_{35} = -29.8478645679819$$
$$x_{36} = 30.2966231528638$$
$$x_{37} = -19.9756372558986$$
$$x_{38} = -73.8285330381195$$
$$x_{39} = 50.0427141531543$$
$$x_{40} = -81.9068051019944$$
$$x_{41} = 100.307379254798$$
$$x_{42} = -83.7019795299298$$
$$x_{43} = 24.014142227597$$
$$x_{44} = -13.6943174821224$$
$$x_{45} = 58.1208685691598$$
$$x_{46} = 84.1507732802386$$
$$x_{47} = 32.9891970465441$$
$$x_{48} = 74.2773253099006$$
Puntos máximos de la función:
$$x_{48} = -77.8676662421385$$
$$x_{48} = 63.9551268127915$$
$$x_{48} = -57.2232926907379$$
$$x_{48} = 28.0528437814417$$
$$x_{48} = 98.0634031242016$$
$$x_{48} = -75.1749100880996$$
$$x_{48} = -24.0141418825183$$
$$x_{48} = -17.7321580056565$$
$$x_{48} = 36.1305746405087$$
$$x_{48} = 96.268222844754$$
$$x_{48} = -41.9646469647508$$
$$x_{48} = 8.3125648497237$$
$$x_{48} = -43.7597629919067$$
$$x_{48} = -61.7111784643883$$
$$x_{48} = -69.7894064717788$$
$$x_{48} = 88.1899194071673$$
$$x_{48} = -85.9459488203819$$
$$x_{48} = -51.8378534656629$$
$$x_{48} = 10.1060354920338$$
$$x_{48} = -7.86424878510195$$
$$x_{48} = 6.52002221049098$$
$$x_{48} = -94.0242483284409$$
$$x_{48} = -35.681804043653$$
$$x_{48} = -87.7411251903887$$
$$x_{48} = 78.3164591866865$$
$$x_{48} = -12.3485643108583$$
$$x_{48} = 62.1599678690624$$
$$x_{48} = 19.9756380867426$$
$$x_{48} = 52.2866389104854$$
$$x_{48} = -67.9942415489$$
$$x_{48} = 70.238197951115$$
$$x_{48} = 56.7745050207678$$
$$x_{48} = 26.2578467616661$$
$$x_{48} = 44.2085430291782$$
$$x_{48} = 54.0817828975494$$
$$x_{48} = 89.9850967437376$$
$$x_{48} = -95.8194278653148$$
$$x_{48} = -15.9374784510599$$
$$x_{48} = -50.0427141428815$$
$$x_{48} = 72.0333647912609$$
$$x_{48} = -33.8867293690557$$
$$x_{48} = 80.1116316329674$$
$$x_{48} = 2.05503340050281$$
$$x_{48} = -59.9160222905198$$
$$x_{48} = 18.1808450293136$$
$$x_{48} = -25.8091015779888$$
$$x_{48} = -8.76082556046972$$
$$x_{48} = -78.765252197294$$
$$x_{48} = 46.003666792909$$
Decrece en los intervalos
$$\left[100.307379254798, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.858583961357\right]$$
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
$$\frac{14 \left(32 x^{3} + \frac{8 x}{\sqrt{x^{2} + 1}} + \frac{15 \left(\left(\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - 3 \arg^{2}{\left(x \right)}\right) \left(\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - \arg^{2}{\left(x \right)}\right) + 2 \left(3 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - \arg^{2}{\left(x \right)}\right) \arg^{2}{\left(x \right)}\right) \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}^{2}{\left(x \right)} \operatorname{sign}{\left(\log{\left(x \right)}^{3} \right)}}{x \log{\left(x \right)}^{3}}\right) \cos{\left(7 x \right)} - 49 \left(8 x^{4} + 8 \sqrt{x^{2} + 1} + 5 \left|{\log{\left(x \right)}^{3}}\right|\right) \sin{\left(7 x \right)} + \left(96 x^{2} - \frac{8 x^{2}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{8}{\sqrt{x^{2} + 1}} + \frac{15 \left(\left(\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - 3 \arg^{2}{\left(x \right)}\right) \left(\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - \arg^{2}{\left(x \right)}\right) + 2 \left(3 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - \arg^{2}{\left(x \right)}\right) \arg^{2}{\left(x \right)}\right) \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}^{2}{\left(x \right)} \frac{d}{d x} \operatorname{sign}{\left(\log{\left(x \right)}^{3} \right)}}{x \log{\left(x \right)}^{3}} - \frac{15 \left(\left(\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - 3 \arg^{2}{\left(x \right)}\right) \left(- 4 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} \delta\left(x\right) + 4 \arg^{2}{\left(x \right)} \delta\left(x\right) + \frac{2 \left(\frac{2 x \delta\left(x\right)}{\operatorname{sign}{\left(x \right)}} - 1\right) \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}{\left(x \right)}}{x} + \frac{\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} \operatorname{sign}{\left(x \right)}}{x} - \frac{\arg^{2}{\left(x \right)} \operatorname{sign}{\left(x \right)}}{x}\right) \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} - 8 \left(3 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - \arg^{2}{\left(x \right)}\right) \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \arg^{2}{\left(x \right)} \delta\left(x\right) + \frac{3 \left(\frac{2 x \delta\left(x\right)}{\operatorname{sign}{\left(x \right)}} - 1\right) \left(\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - \arg^{2}{\left(x \right)}\right)^{2} \operatorname{sign}{\left(x \right)}}{x} + \frac{2 \left(\frac{2 x \delta\left(x\right)}{\operatorname{sign}{\left(x \right)}} - 1\right) \left(3 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - \arg^{2}{\left(x \right)}\right) \arg^{2}{\left(x \right)} \operatorname{sign}{\left(x \right)}}{x} + \frac{12 \left(\frac{2 x \delta\left(x\right)}{\operatorname{sign}{\left(x \right)}} - 1\right) \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} \arg^{2}{\left(x \right)} \operatorname{sign}{\left(x \right)}}{x} + \frac{2 \left(3 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - \arg^{2}{\left(x \right)}\right) \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \arg^{2}{\left(x \right)} \operatorname{sign}{\left(x \right)}}{x}\right) \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\log{\left(x \right)}^{3} \right)}}{x \log{\left(x \right)}^{3}} - \frac{45 \left(\left(\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - 3 \arg^{2}{\left(x \right)}\right) \left(\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - \arg^{2}{\left(x \right)}\right) + 2 \left(3 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - \arg^{2}{\left(x \right)}\right) \arg^{2}{\left(x \right)}\right) \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}^{2}{\left(x \right)} \operatorname{sign}{\left(\log{\left(x \right)}^{3} \right)}}{x^{2} \log{\left(x \right)}^{4}}\right) \sin{\left(7 x \right)}}{8} = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{14 \left(32 x^{3} + \frac{8 x}{\sqrt{x^{2} + 1}} + \frac{15 \left(\left(\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - 3 \arg^{2}{\left(x \right)}\right) \left(\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - \arg^{2}{\left(x \right)}\right) + 2 \left(3 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - \arg^{2}{\left(x \right)}\right) \arg^{2}{\left(x \right)}\right) \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}^{2}{\left(x \right)} \operatorname{sign}{\left(\log{\left(x \right)}^{3} \right)}}{x \log{\left(x \right)}^{3}}\right) \cos{\left(7 x \right)} - 49 \left(8 x^{4} + 8 \sqrt{x^{2} + 1} + 5 \left|{\log{\left(x \right)}^{3}}\right|\right) \sin{\left(7 x \right)} + \left(96 x^{2} - \frac{8 x^{2}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{8}{\sqrt{x^{2} + 1}} + \frac{15 \left(\left(\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - 3 \arg^{2}{\left(x \right)}\right) \left(\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - \arg^{2}{\left(x \right)}\right) + 2 \left(3 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - \arg^{2}{\left(x \right)}\right) \arg^{2}{\left(x \right)}\right) \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}^{2}{\left(x \right)} \frac{d}{d x} \operatorname{sign}{\left(\log{\left(x \right)}^{3} \right)}}{x \log{\left(x \right)}^{3}} - \frac{15 \left(\left(\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - 3 \arg^{2}{\left(x \right)}\right) \left(- 4 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} \delta\left(x\right) + 4 \arg^{2}{\left(x \right)} \delta\left(x\right) + \frac{2 \left(\frac{2 x \delta\left(x\right)}{\operatorname{sign}{\left(x \right)}} - 1\right) \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}{\left(x \right)}}{x} + \frac{\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} \operatorname{sign}{\left(x \right)}}{x} - \frac{\arg^{2}{\left(x \right)} \operatorname{sign}{\left(x \right)}}{x}\right) \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} - 8 \left(3 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - \arg^{2}{\left(x \right)}\right) \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \arg^{2}{\left(x \right)} \delta\left(x\right) + \frac{3 \left(\frac{2 x \delta\left(x\right)}{\operatorname{sign}{\left(x \right)}} - 1\right) \left(\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - \arg^{2}{\left(x \right)}\right)^{2} \operatorname{sign}{\left(x \right)}}{x} + \frac{2 \left(\frac{2 x \delta\left(x\right)}{\operatorname{sign}{\left(x \right)}} - 1\right) \left(3 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - \arg^{2}{\left(x \right)}\right) \arg^{2}{\left(x \right)} \operatorname{sign}{\left(x \right)}}{x} + \frac{12 \left(\frac{2 x \delta\left(x\right)}{\operatorname{sign}{\left(x \right)}} - 1\right) \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} \arg^{2}{\left(x \right)} \operatorname{sign}{\left(x \right)}}{x} + \frac{2 \left(3 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - \arg^{2}{\left(x \right)}\right) \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \arg^{2}{\left(x \right)} \operatorname{sign}{\left(x \right)}}{x}\right) \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\log{\left(x \right)}^{3} \right)}}{x \log{\left(x \right)}^{3}} - \frac{45 \left(\left(\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - 3 \arg^{2}{\left(x \right)}\right) \left(\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - \arg^{2}{\left(x \right)}\right) + 2 \left(3 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}^{2} - \arg^{2}{\left(x \right)}\right) \arg^{2}{\left(x \right)}\right) \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}^{2}{\left(x \right)} \operatorname{sign}{\left(\log{\left(x \right)}^{3} \right)}}{x^{2} \log{\left(x \right)}^{4}}\right) \sin{\left(7 x \right)}}{8} = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\left(x^{4} + \left(\sqrt{x^{2} + 1} + \frac{\left|{\log{\left(x \right)}^{3}}\right|}{\frac{8}{5}}\right)\right) \sin{\left(7 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\left(x^{4} + \left(\sqrt{x^{2} + 1} + \frac{\left|{\log{\left(x \right)}^{3}}\right|}{\frac{8}{5}}\right)\right) \sin{\left(7 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty}\left(\left(x^{4} + \left(\sqrt{x^{2} + 1} + \frac{\left|{\log{\left(x \right)}^{3}}\right|}{\frac{8}{5}}\right)\right) \sin{\left(7 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\left(x^{4} + \left(\sqrt{x^{2} + 1} + \frac{\left|{\log{\left(x \right)}^{3}}\right|}{\frac{8}{5}}\right)\right) \sin{\left(7 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
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 (sqrt(1 + x^2) + Abs(log(x)^3)/(8/5) + x^4)*sin(7*x), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(x^{4} + \left(\sqrt{x^{2} + 1} + \frac{\left|{\log{\left(x \right)}^{3}}\right|}{\frac{8}{5}}\right)\right) \sin{\left(7 x \right)}}{x}\right) = \left\langle -1, 1\right\rangle \lim_{x \to 0^-}\left(\frac{x \sqrt{x^{2} + 1}}{\left|{x}\right|} + \frac{5 x \left|{\log{\left(\frac{1}{x} \right)}^{3}}\right|}{8} + \frac{1}{x^{3}}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle -1, 1\right\rangle x \lim_{x \to 0^-}\left(\frac{x \sqrt{x^{2} + 1}}{\left|{x}\right|} + \frac{5 x \left|{\log{\left(\frac{1}{x} \right)}^{3}}\right|}{8} + \frac{1}{x^{3}}\right)$$
$$\lim_{x \to \infty}\left(\frac{\left(x^{4} + \left(\sqrt{x^{2} + 1} + \frac{\left|{\log{\left(x \right)}^{3}}\right|}{\frac{8}{5}}\right)\right) \sin{\left(7 x \right)}}{x}\right) = \left\langle -1, 1\right\rangle \lim_{x \to 0^+}\left(\frac{x \sqrt{x^{2} + 1}}{\left|{x}\right|} + \frac{5 x \left|{\log{\left(\frac{1}{x} \right)}^{3}}\right|}{8} + \frac{1}{x^{3}}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle -1, 1\right\rangle x \lim_{x \to 0^+}\left(\frac{x \sqrt{x^{2} + 1}}{\left|{x}\right|} + \frac{5 x \left|{\log{\left(\frac{1}{x} \right)}^{3}}\right|}{8} + \frac{1}{x^{3}}\right)$$
$$\lim_{x \to -\infty}\left(\frac{\left(x^{4} + \left(\sqrt{x^{2} + 1} + \frac{\left|{\log{\left(x \right)}^{3}}\right|}{\frac{8}{5}}\right)\right) \sin{\left(7 x \right)}}{x}\right) = \left\langle -1, 1\right\rangle \lim_{x \to 0^-}\left(\frac{x \sqrt{x^{2} + 1}}{\left|{x}\right|} + \frac{5 x \left|{\log{\left(\frac{1}{x} \right)}^{3}}\right|}{8} + \frac{1}{x^{3}}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle -1, 1\right\rangle x \lim_{x \to 0^-}\left(\frac{x \sqrt{x^{2} + 1}}{\left|{x}\right|} + \frac{5 x \left|{\log{\left(\frac{1}{x} \right)}^{3}}\right|}{8} + \frac{1}{x^{3}}\right)$$
$$\lim_{x \to \infty}\left(\frac{\left(x^{4} + \left(\sqrt{x^{2} + 1} + \frac{\left|{\log{\left(x \right)}^{3}}\right|}{\frac{8}{5}}\right)\right) \sin{\left(7 x \right)}}{x}\right) = \left\langle -1, 1\right\rangle \lim_{x \to 0^+}\left(\frac{x \sqrt{x^{2} + 1}}{\left|{x}\right|} + \frac{5 x \left|{\log{\left(\frac{1}{x} \right)}^{3}}\right|}{8} + \frac{1}{x^{3}}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle -1, 1\right\rangle x \lim_{x \to 0^+}\left(\frac{x \sqrt{x^{2} + 1}}{\left|{x}\right|} + \frac{5 x \left|{\log{\left(\frac{1}{x} \right)}^{3}}\right|}{8} + \frac{1}{x^{3}}\right)$$
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:
$$\left(x^{4} + \left(\sqrt{x^{2} + 1} + \frac{\left|{\log{\left(x \right)}^{3}}\right|}{\frac{8}{5}}\right)\right) \sin{\left(7 x \right)} = - \left(x^{4} + \sqrt{x^{2} + 1} + \frac{5 \left|{\log{\left(- x \right)}^{3}}\right|}{8}\right) \sin{\left(7 x \right)}$$
- No
$$\left(x^{4} + \left(\sqrt{x^{2} + 1} + \frac{\left|{\log{\left(x \right)}^{3}}\right|}{\frac{8}{5}}\right)\right) \sin{\left(7 x \right)} = \left(x^{4} + \sqrt{x^{2} + 1} + \frac{5 \left|{\log{\left(- x \right)}^{3}}\right|}{8}\right) \sin{\left(7 x \right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\left(x^{4} + \left(\sqrt{x^{2} + 1} + \frac{\left|{\log{\left(x \right)}^{3}}\right|}{\frac{8}{5}}\right)\right) \sin{\left(7 x \right)} = - \left(x^{4} + \sqrt{x^{2} + 1} + \frac{5 \left|{\log{\left(- x \right)}^{3}}\right|}{8}\right) \sin{\left(7 x \right)}$$
- No
$$\left(x^{4} + \left(\sqrt{x^{2} + 1} + \frac{\left|{\log{\left(x \right)}^{3}}\right|}{\frac{8}{5}}\right)\right) \sin{\left(7 x \right)} = \left(x^{4} + \sqrt{x^{2} + 1} + \frac{5 \left|{\log{\left(- x \right)}^{3}}\right|}{8}\right) \sin{\left(7 x \right)}$$
- No
es decir, función
no es
par ni impar