Sr Examen
Otras calculadoras
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- 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 =:
-
y=(x+1)^3
-
y=3x^2-x^3
-
3-x^2
-
1/((x-1)^2)
-
Expresiones idénticas
- sin(tres *x)/(x^ tres - dos *x^ dos +x)
- seno de (3 multiplicar por x) dividir por (x al cubo menos 2 multiplicar por x al cuadrado más x)
- seno de (tres multiplicar por x) dividir por (x en el grado tres menos dos multiplicar por x en el grado dos más x)
- sin(3*x)/(x3-2*x2+x)
- sin3*x/x3-2*x2+x
- sin(3*x)/(x³-2*x²+x)
- sin(3*x)/(x en el grado 3-2*x en el grado 2+x)
- sin(3x)/(x^3-2x^2+x)
- sin(3x)/(x3-2x2+x)
- sin3x/x3-2x2+x
- sin3x/x^3-2x^2+x
- sin(3*x) dividir por (x^3-2*x^2+x)
-
Expresiones semejantes
- sin(3*x)/(x^3-2*x^2-x)
- sin(3*x)/(x^3+2*x^2+x)
-
Expresiones con funciones
- Seno sin
- sin(x)*2*x
- sin(asin(x))
- sin(2*x)+5
- sin(2x)/(sinx)
- sin^2(x+0.5*sin^2(x))
Gráfico de la función y = sin(3*x)/(x^3-2*x^2+x)
Solución
Ha introducido
[src]
sin(3*x)
f(x) = -------------
3 2
x - 2*x + x
$$f{\left(x \right)} = \frac{\sin{\left(3 x \right)}}{x + \left(x^{3} - 2 x^{2}\right)}$$
f = sin(3*x)/(x + x^3 - 2*x^2)
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 0$$
$$x_{2} = 1$$
$$x_{1} = 0$$
$$x_{2} = 1$$
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:
$$\frac{\sin{\left(3 x \right)}}{x + \left(x^{3} - 2 x^{2}\right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \frac{\pi}{3}$$
Solución numérica
$$x_{1} = 65.9734457253857$$
$$x_{2} = -68.0678408277789$$
$$x_{3} = -79.5870138909414$$
$$x_{4} = -21.9911485751286$$
$$x_{5} = 21.9911485751286$$
$$x_{6} = 63.8790506229925$$
$$x_{7} = -85.870199198121$$
$$x_{8} = -15.707963267949$$
$$x_{9} = -4.18879020478639$$
$$x_{10} = -108.908545324446$$
$$x_{11} = 74.3510261349584$$
$$x_{12} = 61.7846555205993$$
$$x_{13} = -35.6047167406843$$
$$x_{14} = 6.28318530717959$$
$$x_{15} = 2.0943951023932$$
$$x_{16} = -32.4631240870945$$
$$x_{17} = 30.3687289847013$$
$$x_{18} = -63.8790506229925$$
$$x_{19} = 39.7935069454707$$
$$x_{20} = -70.162235930172$$
$$x_{21} = 70.162235930172$$
$$x_{22} = 76.4454212373516$$
$$x_{23} = -39.7935069454707$$
$$x_{24} = -77.4926187885482$$
$$x_{25} = 28.2743338823081$$
$$x_{26} = -94.2477796076938$$
$$x_{27} = -96.342174710087$$
$$x_{28} = 85.870199198121$$
$$x_{29} = 98.4365698124802$$
$$x_{30} = -24.0855436775217$$
$$x_{31} = -90.0589894029074$$
$$x_{32} = -41.8879020478639$$
$$x_{33} = 64.9262481741891$$
$$x_{34} = 90.0589894029074$$
$$x_{35} = 56.5486677646163$$
$$x_{36} = -2.0943951023932$$
$$x_{37} = 41.8879020478639$$
$$x_{38} = 72.2566310325652$$
$$x_{39} = 26.1799387799149$$
$$x_{40} = -50.2654824574367$$
$$x_{41} = 78.5398163397448$$
$$x_{42} = -87.9645943005142$$
$$x_{43} = 37.6991118430775$$
$$x_{44} = -61.7846555205993$$
$$x_{45} = -6.28318530717959$$
$$x_{46} = 68.0678408277789$$
$$x_{47} = -37.6991118430775$$
$$x_{48} = -43.9822971502571$$
$$x_{49} = -13.6135681655558$$
$$x_{50} = -55.5014702134197$$
$$x_{51} = -26.1799387799149$$
$$x_{52} = 8.37758040957278$$
$$x_{53} = -72.2566310325652$$
$$x_{54} = 52.3598775598299$$
$$x_{55} = -81.6814089933346$$
$$x_{56} = -46.0766922526503$$
$$x_{57} = -33.5103216382911$$
$$x_{58} = -65.9734457253857$$
$$x_{59} = 156.032435128293$$
$$x_{60} = -60.7374579694027$$
$$x_{61} = -99.4837673636768$$
$$x_{62} = -229.336263712055$$
$$x_{63} = -28.2743338823081$$
$$x_{64} = -56.5486677646163$$
$$x_{65} = 96.342174710087$$
$$x_{66} = -11.5191730631626$$
$$x_{67} = 43.9822971502571$$
$$x_{68} = 80.634211442138$$
$$x_{69} = -83.7758040957278$$
$$x_{70} = 100.530964914873$$
$$x_{71} = 19.8967534727354$$
$$x_{72} = 50.2654824574367$$
$$x_{73} = 94.2477796076938$$
$$x_{74} = -10.471975511966$$
$$x_{75} = -36.6519142918809$$
$$x_{76} = 4.18879020478639$$
$$x_{77} = 32.4631240870945$$
$$x_{78} = -59.6902604182061$$
$$x_{79} = 83.7758040957278$$
$$x_{80} = 12.5663706143592$$
$$x_{81} = -19.8967534727354$$
$$x_{82} = 54.4542726622231$$
$$x_{83} = -20.943951023932$$
$$x_{84} = 34.5575191894877$$
$$x_{85} = 92.1533845053006$$
$$x_{86} = -48.1710873550435$$
$$x_{87} = 46.0766922526503$$
$$x_{88} = 24.0855436775217$$
$$x_{89} = 15.707963267949$$
$$x_{90} = 10.471975511966$$
$$x_{91} = 87.9645943005142$$
$$x_{92} = 17.8023583703422$$
$$x_{93} = -17.8023583703422$$
$$x_{94} = -92.1533845053006$$
o sea hay que resolver la ecuación:
$$\frac{\sin{\left(3 x \right)}}{x + \left(x^{3} - 2 x^{2}\right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \frac{\pi}{3}$$
Solución numérica
$$x_{1} = 65.9734457253857$$
$$x_{2} = -68.0678408277789$$
$$x_{3} = -79.5870138909414$$
$$x_{4} = -21.9911485751286$$
$$x_{5} = 21.9911485751286$$
$$x_{6} = 63.8790506229925$$
$$x_{7} = -85.870199198121$$
$$x_{8} = -15.707963267949$$
$$x_{9} = -4.18879020478639$$
$$x_{10} = -108.908545324446$$
$$x_{11} = 74.3510261349584$$
$$x_{12} = 61.7846555205993$$
$$x_{13} = -35.6047167406843$$
$$x_{14} = 6.28318530717959$$
$$x_{15} = 2.0943951023932$$
$$x_{16} = -32.4631240870945$$
$$x_{17} = 30.3687289847013$$
$$x_{18} = -63.8790506229925$$
$$x_{19} = 39.7935069454707$$
$$x_{20} = -70.162235930172$$
$$x_{21} = 70.162235930172$$
$$x_{22} = 76.4454212373516$$
$$x_{23} = -39.7935069454707$$
$$x_{24} = -77.4926187885482$$
$$x_{25} = 28.2743338823081$$
$$x_{26} = -94.2477796076938$$
$$x_{27} = -96.342174710087$$
$$x_{28} = 85.870199198121$$
$$x_{29} = 98.4365698124802$$
$$x_{30} = -24.0855436775217$$
$$x_{31} = -90.0589894029074$$
$$x_{32} = -41.8879020478639$$
$$x_{33} = 64.9262481741891$$
$$x_{34} = 90.0589894029074$$
$$x_{35} = 56.5486677646163$$
$$x_{36} = -2.0943951023932$$
$$x_{37} = 41.8879020478639$$
$$x_{38} = 72.2566310325652$$
$$x_{39} = 26.1799387799149$$
$$x_{40} = -50.2654824574367$$
$$x_{41} = 78.5398163397448$$
$$x_{42} = -87.9645943005142$$
$$x_{43} = 37.6991118430775$$
$$x_{44} = -61.7846555205993$$
$$x_{45} = -6.28318530717959$$
$$x_{46} = 68.0678408277789$$
$$x_{47} = -37.6991118430775$$
$$x_{48} = -43.9822971502571$$
$$x_{49} = -13.6135681655558$$
$$x_{50} = -55.5014702134197$$
$$x_{51} = -26.1799387799149$$
$$x_{52} = 8.37758040957278$$
$$x_{53} = -72.2566310325652$$
$$x_{54} = 52.3598775598299$$
$$x_{55} = -81.6814089933346$$
$$x_{56} = -46.0766922526503$$
$$x_{57} = -33.5103216382911$$
$$x_{58} = -65.9734457253857$$
$$x_{59} = 156.032435128293$$
$$x_{60} = -60.7374579694027$$
$$x_{61} = -99.4837673636768$$
$$x_{62} = -229.336263712055$$
$$x_{63} = -28.2743338823081$$
$$x_{64} = -56.5486677646163$$
$$x_{65} = 96.342174710087$$
$$x_{66} = -11.5191730631626$$
$$x_{67} = 43.9822971502571$$
$$x_{68} = 80.634211442138$$
$$x_{69} = -83.7758040957278$$
$$x_{70} = 100.530964914873$$
$$x_{71} = 19.8967534727354$$
$$x_{72} = 50.2654824574367$$
$$x_{73} = 94.2477796076938$$
$$x_{74} = -10.471975511966$$
$$x_{75} = -36.6519142918809$$
$$x_{76} = 4.18879020478639$$
$$x_{77} = 32.4631240870945$$
$$x_{78} = -59.6902604182061$$
$$x_{79} = 83.7758040957278$$
$$x_{80} = 12.5663706143592$$
$$x_{81} = -19.8967534727354$$
$$x_{82} = 54.4542726622231$$
$$x_{83} = -20.943951023932$$
$$x_{84} = 34.5575191894877$$
$$x_{85} = 92.1533845053006$$
$$x_{86} = -48.1710873550435$$
$$x_{87} = 46.0766922526503$$
$$x_{88} = 24.0855436775217$$
$$x_{89} = 15.707963267949$$
$$x_{90} = 10.471975511966$$
$$x_{91} = 87.9645943005142$$
$$x_{92} = 17.8023583703422$$
$$x_{93} = -17.8023583703422$$
$$x_{94} = -92.1533845053006$$
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
$$\frac{3 \cos{\left(3 x \right)}}{x + \left(x^{3} - 2 x^{2}\right)} + \frac{\left(- 3 x^{2} + 4 x - 1\right) \sin{\left(3 x \right)}}{\left(x + \left(x^{3} - 2 x^{2}\right)\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -93.7206493278661$$
$$x_{2} = 91.6261211428138$$
$$x_{3} = 71.7283416001381$$
$$x_{4} = 36.1189140439607$$
$$x_{5} = -56.0191888282368$$
$$x_{6} = -14.1146268428529$$
$$x_{7} = 14.112391865157$$
$$x_{8} = -27.739001936835$$
$$x_{9} = 9.91237393696233$$
$$x_{10} = 31.9288640440463$$
$$x_{11} = -69.6338957143956$$
$$x_{12} = -75.9174700320155$$
$$x_{13} = -95.8151210919736$$
$$x_{14} = -100.004055050435$$
$$x_{15} = 12.0134216092849$$
$$x_{16} = -80.1064859524893$$
$$x_{17} = 73.822870994613$$
$$x_{18} = 16.2101270633009$$
$$x_{19} = 48.6877450734286$$
$$x_{20} = -44.4985160055418$$
$$x_{21} = -62.3029608856172$$
$$x_{22} = 29.8337033085781$$
$$x_{23} = 58.1136618387278$$
$$x_{24} = -78.0119809966491$$
$$x_{25} = 66.4919807241567$$
$$x_{26} = 78.0119079636624$$
$$x_{27} = -84.2954796154091$$
$$x_{28} = 51.829763951856$$
$$x_{29} = 64.3974192058359$$
$$x_{30} = -89.5316951187848$$
$$x_{31} = 38.2138336079304$$
$$x_{32} = 27.7384240631162$$
$$x_{33} = -49.7352704648011$$
$$x_{34} = 22.4994834778756$$
$$x_{35} = -71.7284279901462$$
$$x_{36} = 6.75218934974203$$
$$x_{37} = 25.6429964736596$$
$$x_{38} = -58.1137934529666$$
$$x_{39} = -73.8229525516993$$
$$x_{40} = -67.5393550137299$$
$$x_{41} = 40.3086977417864$$
$$x_{42} = -40.3089713354788$$
$$x_{43} = -7.81473552792952$$
$$x_{44} = 20.4034682989027$$
$$x_{45} = -60.2083836591177$$
$$x_{46} = -31.9293001445384$$
$$x_{47} = 49.7350907645845$$
$$x_{48} = 60.2082610439035$$
$$x_{49} = 100.004010608094$$
$$x_{50} = -47.6405886286574$$
$$x_{51} = -53.9245681251302$$
$$x_{52} = 2.43774972842973$$
$$x_{53} = -45.5458807986161$$
$$x_{54} = -16.2118204243123$$
$$x_{55} = -51.8299294189034$$
$$x_{56} = -36.119254808942$$
$$x_{57} = 44.4982915131037$$
$$x_{58} = 86.3899095504266$$
$$x_{59} = -106.28743487485$$
$$x_{60} = -23.5481817781288$$
$$x_{61} = 56.019047186555$$
$$x_{62} = 18.3070681200226$$
$$x_{63} = 4.62893757661424$$
$$x_{64} = 84.2954170650793$$
$$x_{65} = -43.4511432716931$$
$$x_{66} = 80.1064166888903$$
$$x_{67} = -25.6436726944804$$
$$x_{68} = -20.4045366997778$$
$$x_{69} = 7.80741137056402$$
$$x_{70} = -9.91691055112144$$
$$x_{71} = -4.64990392099382$$
$$x_{72} = 69.6338040487669$$
$$x_{73} = 34.0239286046872$$
$$x_{74} = -480.139383943735$$
$$x_{75} = 75.9173929133357$$
$$x_{76} = -5.70741802829714$$
$$x_{77} = 46.5930334212042$$
$$x_{78} = 95.8150726785981$$
$$x_{79} = -3.58722715778822$$
$$x_{80} = -38.214138027046$$
$$x_{81} = 93.7205987263395$$
$$x_{82} = 89.531639671256$$
$$x_{83} = 97.9095432342321$$
$$x_{84} = -91.6261740842685$$
$$x_{85} = 42.4035147930643$$
$$x_{86} = -12.0165074489364$$
$$x_{87} = 53.9244152646776$$
$$x_{88} = -18.3083954532018$$
$$x_{89} = -82.2009853551742$$
$$x_{90} = -34.0243126378108$$
$$x_{91} = -29.8342028350082$$
$$x_{92} = 67.5392575740105$$
$$x_{93} = 82.2009195764864$$
$$x_{94} = 88.4843973905275$$
$$x_{95} = 62.3028463769924$$
$$x_{96} = -97.9095895983848$$
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:
$$x_{1} = -93.7206493278661$$
$$x_{2} = 91.6261211428138$$
$$x_{3} = -56.0191888282368$$
$$x_{4} = -14.1146268428529$$
$$x_{5} = 14.112391865157$$
$$x_{6} = 9.91237393696233$$
$$x_{7} = -95.8151210919736$$
$$x_{8} = -100.004055050435$$
$$x_{9} = 12.0134216092849$$
$$x_{10} = 16.2101270633009$$
$$x_{11} = -62.3029608856172$$
$$x_{12} = 58.1136618387278$$
$$x_{13} = 66.4919807241567$$
$$x_{14} = 51.829763951856$$
$$x_{15} = 64.3974192058359$$
$$x_{16} = -89.5316951187848$$
$$x_{17} = -49.7352704648011$$
$$x_{18} = 22.4994834778756$$
$$x_{19} = -58.1137934529666$$
$$x_{20} = -7.81473552792952$$
$$x_{21} = 20.4034682989027$$
$$x_{22} = -60.2083836591177$$
$$x_{23} = 49.7350907645845$$
$$x_{24} = 60.2082610439035$$
$$x_{25} = 100.004010608094$$
$$x_{26} = -47.6405886286574$$
$$x_{27} = -53.9245681251302$$
$$x_{28} = -45.5458807986161$$
$$x_{29} = -16.2118204243123$$
$$x_{30} = -51.8299294189034$$
$$x_{31} = -106.28743487485$$
$$x_{32} = 56.019047186555$$
$$x_{33} = 18.3070681200226$$
$$x_{34} = -43.4511432716931$$
$$x_{35} = -20.4045366997778$$
$$x_{36} = 7.80741137056402$$
$$x_{37} = -9.91691055112144$$
$$x_{38} = -5.70741802829714$$
$$x_{39} = 95.8150726785981$$
$$x_{40} = -3.58722715778822$$
$$x_{41} = 93.7205987263395$$
$$x_{42} = 89.531639671256$$
$$x_{43} = 97.9095432342321$$
$$x_{44} = -91.6261740842685$$
$$x_{45} = -12.0165074489364$$
$$x_{46} = 53.9244152646776$$
$$x_{47} = -18.3083954532018$$
$$x_{48} = 62.3028463769924$$
$$x_{49} = -97.9095895983848$$
Puntos máximos de la función:
$$x_{49} = 71.7283416001381$$
$$x_{49} = 36.1189140439607$$
$$x_{49} = -27.739001936835$$
$$x_{49} = 31.9288640440463$$
$$x_{49} = -69.6338957143956$$
$$x_{49} = -75.9174700320155$$
$$x_{49} = -80.1064859524893$$
$$x_{49} = 73.822870994613$$
$$x_{49} = 48.6877450734286$$
$$x_{49} = -44.4985160055418$$
$$x_{49} = 29.8337033085781$$
$$x_{49} = -78.0119809966491$$
$$x_{49} = 78.0119079636624$$
$$x_{49} = -84.2954796154091$$
$$x_{49} = 38.2138336079304$$
$$x_{49} = 27.7384240631162$$
$$x_{49} = -71.7284279901462$$
$$x_{49} = 6.75218934974203$$
$$x_{49} = 25.6429964736596$$
$$x_{49} = -73.8229525516993$$
$$x_{49} = -67.5393550137299$$
$$x_{49} = 40.3086977417864$$
$$x_{49} = -40.3089713354788$$
$$x_{49} = -31.9293001445384$$
$$x_{49} = 2.43774972842973$$
$$x_{49} = -36.119254808942$$
$$x_{49} = 44.4982915131037$$
$$x_{49} = 86.3899095504266$$
$$x_{49} = -23.5481817781288$$
$$x_{49} = 4.62893757661424$$
$$x_{49} = 84.2954170650793$$
$$x_{49} = 80.1064166888903$$
$$x_{49} = -25.6436726944804$$
$$x_{49} = -4.64990392099382$$
$$x_{49} = 69.6338040487669$$
$$x_{49} = 34.0239286046872$$
$$x_{49} = -480.139383943735$$
$$x_{49} = 75.9173929133357$$
$$x_{49} = 46.5930334212042$$
$$x_{49} = -38.214138027046$$
$$x_{49} = 42.4035147930643$$
$$x_{49} = -82.2009853551742$$
$$x_{49} = -34.0243126378108$$
$$x_{49} = -29.8342028350082$$
$$x_{49} = 67.5392575740105$$
$$x_{49} = 82.2009195764864$$
$$x_{49} = 88.4843973905275$$
Decrece en los intervalos
$$\left[100.004010608094, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -106.28743487485\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{3 \cos{\left(3 x \right)}}{x + \left(x^{3} - 2 x^{2}\right)} + \frac{\left(- 3 x^{2} + 4 x - 1\right) \sin{\left(3 x \right)}}{\left(x + \left(x^{3} - 2 x^{2}\right)\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -93.7206493278661$$
$$x_{2} = 91.6261211428138$$
$$x_{3} = 71.7283416001381$$
$$x_{4} = 36.1189140439607$$
$$x_{5} = -56.0191888282368$$
$$x_{6} = -14.1146268428529$$
$$x_{7} = 14.112391865157$$
$$x_{8} = -27.739001936835$$
$$x_{9} = 9.91237393696233$$
$$x_{10} = 31.9288640440463$$
$$x_{11} = -69.6338957143956$$
$$x_{12} = -75.9174700320155$$
$$x_{13} = -95.8151210919736$$
$$x_{14} = -100.004055050435$$
$$x_{15} = 12.0134216092849$$
$$x_{16} = -80.1064859524893$$
$$x_{17} = 73.822870994613$$
$$x_{18} = 16.2101270633009$$
$$x_{19} = 48.6877450734286$$
$$x_{20} = -44.4985160055418$$
$$x_{21} = -62.3029608856172$$
$$x_{22} = 29.8337033085781$$
$$x_{23} = 58.1136618387278$$
$$x_{24} = -78.0119809966491$$
$$x_{25} = 66.4919807241567$$
$$x_{26} = 78.0119079636624$$
$$x_{27} = -84.2954796154091$$
$$x_{28} = 51.829763951856$$
$$x_{29} = 64.3974192058359$$
$$x_{30} = -89.5316951187848$$
$$x_{31} = 38.2138336079304$$
$$x_{32} = 27.7384240631162$$
$$x_{33} = -49.7352704648011$$
$$x_{34} = 22.4994834778756$$
$$x_{35} = -71.7284279901462$$
$$x_{36} = 6.75218934974203$$
$$x_{37} = 25.6429964736596$$
$$x_{38} = -58.1137934529666$$
$$x_{39} = -73.8229525516993$$
$$x_{40} = -67.5393550137299$$
$$x_{41} = 40.3086977417864$$
$$x_{42} = -40.3089713354788$$
$$x_{43} = -7.81473552792952$$
$$x_{44} = 20.4034682989027$$
$$x_{45} = -60.2083836591177$$
$$x_{46} = -31.9293001445384$$
$$x_{47} = 49.7350907645845$$
$$x_{48} = 60.2082610439035$$
$$x_{49} = 100.004010608094$$
$$x_{50} = -47.6405886286574$$
$$x_{51} = -53.9245681251302$$
$$x_{52} = 2.43774972842973$$
$$x_{53} = -45.5458807986161$$
$$x_{54} = -16.2118204243123$$
$$x_{55} = -51.8299294189034$$
$$x_{56} = -36.119254808942$$
$$x_{57} = 44.4982915131037$$
$$x_{58} = 86.3899095504266$$
$$x_{59} = -106.28743487485$$
$$x_{60} = -23.5481817781288$$
$$x_{61} = 56.019047186555$$
$$x_{62} = 18.3070681200226$$
$$x_{63} = 4.62893757661424$$
$$x_{64} = 84.2954170650793$$
$$x_{65} = -43.4511432716931$$
$$x_{66} = 80.1064166888903$$
$$x_{67} = -25.6436726944804$$
$$x_{68} = -20.4045366997778$$
$$x_{69} = 7.80741137056402$$
$$x_{70} = -9.91691055112144$$
$$x_{71} = -4.64990392099382$$
$$x_{72} = 69.6338040487669$$
$$x_{73} = 34.0239286046872$$
$$x_{74} = -480.139383943735$$
$$x_{75} = 75.9173929133357$$
$$x_{76} = -5.70741802829714$$
$$x_{77} = 46.5930334212042$$
$$x_{78} = 95.8150726785981$$
$$x_{79} = -3.58722715778822$$
$$x_{80} = -38.214138027046$$
$$x_{81} = 93.7205987263395$$
$$x_{82} = 89.531639671256$$
$$x_{83} = 97.9095432342321$$
$$x_{84} = -91.6261740842685$$
$$x_{85} = 42.4035147930643$$
$$x_{86} = -12.0165074489364$$
$$x_{87} = 53.9244152646776$$
$$x_{88} = -18.3083954532018$$
$$x_{89} = -82.2009853551742$$
$$x_{90} = -34.0243126378108$$
$$x_{91} = -29.8342028350082$$
$$x_{92} = 67.5392575740105$$
$$x_{93} = 82.2009195764864$$
$$x_{94} = 88.4843973905275$$
$$x_{95} = 62.3028463769924$$
$$x_{96} = -97.9095895983848$$
Signos de extremos en los puntos:
(-93.7206493278661, -1.18918933082121e-6)
(91.62612114281382, -1.32876343331136e-6)
(71.7283416001381, 2.78662968373046e-6)
(36.11891404396066, 2.2439356500701e-5)
(-56.01918882823683, -5.48976227053796e-6)
(-14.114626842852926, -0.000309415507111686)
(14.11239186515704, -0.000410993405876693)
(-27.739001936834992, 4.36211090958447e-5)
(9.91237393696233, -0.00126269186646904)
(31.928864044046257, 3.2724002844681e-5)
(-69.63389571439565, 2.87812101425112e-6)
(-75.91747003201546, 2.22623634276153e-6)
(-95.81512109197357, -1.11341286248088e-6)
(-100.00405505043516, -9.80129235053595e-7)
(12.013421609284917, -0.000683601560410484)
(-80.10648595248932, 1.89752656977739e-6)
(73.82287099461301, 2.55407062149765e-6)
(16.210127063300924, -0.000266102739816768)
(48.687745073428594, 9.02968503039291e-6)
(-44.49851600554178, 1.08530939137805e-5)
(-62.30296088561718, -4.00487073408351e-6)
(29.83370330857811, 4.02936628921896e-5)
(58.11366183872781, -5.27443504749581e-6)
(-78.01198099664914, 2.05313598241677e-6)
(66.49198072415669, -3.50593907355159e-6)
(78.0119079636624, 2.16116057338361e-6)
(-84.29547961540915, 1.63047296189196e-6)
(51.829763951856044, -7.46623457469376e-6)
(64.39741920583589, -3.86309080504557e-6)
(-89.53169511878481, -1.36268413010618e-6)
(38.21383360793043, 1.88893297102999e-5)
(27.738424063116163, 5.03906934241378e-5)
(-49.73527046480113, -7.80962317126772e-6)
(22.499483477875625, -9.60541006969072e-5)
(-71.72842799014623, 2.63547471890058e-6)
(6.752189349742026, 0.00441608859164183)
(25.6429964736596, 6.41646954649086e-5)
(-58.11379345296657, -4.92357171749194e-6)
(-73.82295255169933, 2.4193559632e-6)
(-67.53935501372992, 3.15149329580133e-6)
(40.30869774178637, 1.60503957763343e-5)
(-40.30897133547875, 1.4533834040501e-5)
(-7.814735527929522, -0.00163549793047796)
(20.40346829890268, -0.000130010945929864)
(-60.20838365911769, -4.43264155037943e-6)
(-31.929300144538356, 2.8869631117966e-5)
(49.73509076458449, -8.46376001302792e-6)
(60.20826104390349, -4.73716000647905e-6)
(100.004010608094, -1.02012878249773e-6)
(-47.640588628657355, -8.87017496753908e-6)
(-53.9245681251302, -6.14620669030315e-6)
(2.437749728429733, 0.170134842985545)
(-45.54588079861613, -1.01318024097916e-5)
(-16.211820424312325, -0.000207851180312834)
(-51.82992941890342, -6.91163076908239e-6)
(-36.11925480894203, 2.00864069706778e-5)
(44.49829151310371, 1.18740600662628e-5)
(86.38990955042655, 1.58742905091439e-6)
(-106.2874348748496, -8.17337391644411e-7)
(-23.548181778128818, 7.04098888513651e-5)
(56.019047186554964, -5.89613312119428e-6)
(18.307068120022624, -0.000182069218810388)
(4.628937576614239, 0.0158929638416328)
(84.29541706507928, 1.7097112669521e-6)
(-43.451143271693084, -1.16445137935266e-5)
(80.10641668889033, 1.99468747330621e-6)
(-25.6436726944804, 5.48931214334835e-5)
(-20.404536699777786, -0.000106849520074739)
(7.807411370564021, -0.00273701113995656)
(-9.916910551121438, -0.000842337628119065)
(-4.649903920993817, 0.00661909098505801)
(69.63380404876689, 3.04830266333865e-6)
(34.02392860468716, 2.69378497651625e-5)
(-480.1393839437345, 8.9968382840227e-9)
(75.91739291333568, 2.34668639900381e-6)
(-5.707418028297144, -0.00384688244042313)
(46.5930334212042, 1.03223621133104e-5)
(95.81507267859806, -1.16088023385071e-6)
(-3.587227157788216, -0.01288698984462)
(-38.214138027046005, 1.70116699384642e-5)
(93.72059872633949, -1.24104466953864e-6)
(89.53163967125596, -1.42494777988513e-6)
(97.90954323423207, -1.08747409142919e-6)
(-91.6261740842685, -1.27200114660059e-6)
(42.40351479306426, 1.37530515355943e-5)
(-12.016507448936391, -0.000489647195323849)
(53.92441526467758, -6.61951030492112e-6)
(-18.308395453201843, -0.000146303369078272)
(-82.20098535517423, 1.75725349473657e-6)
(-34.024312637810766, 2.39491977732805e-5)
(-29.83420283500819, 3.52360087443922e-5)
(67.53925757401053, 3.34379203504136e-6)
(82.20091957648638, 1.84488272763417e-6)
(88.48439739052748, 1.47653564635636e-6)
(62.30284637699239, -4.27045032073535e-6)
(-97.90958959838484, -1.04394020485464e-6)
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:
$$x_{1} = -93.7206493278661$$
$$x_{2} = 91.6261211428138$$
$$x_{3} = -56.0191888282368$$
$$x_{4} = -14.1146268428529$$
$$x_{5} = 14.112391865157$$
$$x_{6} = 9.91237393696233$$
$$x_{7} = -95.8151210919736$$
$$x_{8} = -100.004055050435$$
$$x_{9} = 12.0134216092849$$
$$x_{10} = 16.2101270633009$$
$$x_{11} = -62.3029608856172$$
$$x_{12} = 58.1136618387278$$
$$x_{13} = 66.4919807241567$$
$$x_{14} = 51.829763951856$$
$$x_{15} = 64.3974192058359$$
$$x_{16} = -89.5316951187848$$
$$x_{17} = -49.7352704648011$$
$$x_{18} = 22.4994834778756$$
$$x_{19} = -58.1137934529666$$
$$x_{20} = -7.81473552792952$$
$$x_{21} = 20.4034682989027$$
$$x_{22} = -60.2083836591177$$
$$x_{23} = 49.7350907645845$$
$$x_{24} = 60.2082610439035$$
$$x_{25} = 100.004010608094$$
$$x_{26} = -47.6405886286574$$
$$x_{27} = -53.9245681251302$$
$$x_{28} = -45.5458807986161$$
$$x_{29} = -16.2118204243123$$
$$x_{30} = -51.8299294189034$$
$$x_{31} = -106.28743487485$$
$$x_{32} = 56.019047186555$$
$$x_{33} = 18.3070681200226$$
$$x_{34} = -43.4511432716931$$
$$x_{35} = -20.4045366997778$$
$$x_{36} = 7.80741137056402$$
$$x_{37} = -9.91691055112144$$
$$x_{38} = -5.70741802829714$$
$$x_{39} = 95.8150726785981$$
$$x_{40} = -3.58722715778822$$
$$x_{41} = 93.7205987263395$$
$$x_{42} = 89.531639671256$$
$$x_{43} = 97.9095432342321$$
$$x_{44} = -91.6261740842685$$
$$x_{45} = -12.0165074489364$$
$$x_{46} = 53.9244152646776$$
$$x_{47} = -18.3083954532018$$
$$x_{48} = 62.3028463769924$$
$$x_{49} = -97.9095895983848$$
Puntos máximos de la función:
$$x_{49} = 71.7283416001381$$
$$x_{49} = 36.1189140439607$$
$$x_{49} = -27.739001936835$$
$$x_{49} = 31.9288640440463$$
$$x_{49} = -69.6338957143956$$
$$x_{49} = -75.9174700320155$$
$$x_{49} = -80.1064859524893$$
$$x_{49} = 73.822870994613$$
$$x_{49} = 48.6877450734286$$
$$x_{49} = -44.4985160055418$$
$$x_{49} = 29.8337033085781$$
$$x_{49} = -78.0119809966491$$
$$x_{49} = 78.0119079636624$$
$$x_{49} = -84.2954796154091$$
$$x_{49} = 38.2138336079304$$
$$x_{49} = 27.7384240631162$$
$$x_{49} = -71.7284279901462$$
$$x_{49} = 6.75218934974203$$
$$x_{49} = 25.6429964736596$$
$$x_{49} = -73.8229525516993$$
$$x_{49} = -67.5393550137299$$
$$x_{49} = 40.3086977417864$$
$$x_{49} = -40.3089713354788$$
$$x_{49} = -31.9293001445384$$
$$x_{49} = 2.43774972842973$$
$$x_{49} = -36.119254808942$$
$$x_{49} = 44.4982915131037$$
$$x_{49} = 86.3899095504266$$
$$x_{49} = -23.5481817781288$$
$$x_{49} = 4.62893757661424$$
$$x_{49} = 84.2954170650793$$
$$x_{49} = 80.1064166888903$$
$$x_{49} = -25.6436726944804$$
$$x_{49} = -4.64990392099382$$
$$x_{49} = 69.6338040487669$$
$$x_{49} = 34.0239286046872$$
$$x_{49} = -480.139383943735$$
$$x_{49} = 75.9173929133357$$
$$x_{49} = 46.5930334212042$$
$$x_{49} = -38.214138027046$$
$$x_{49} = 42.4035147930643$$
$$x_{49} = -82.2009853551742$$
$$x_{49} = -34.0243126378108$$
$$x_{49} = -29.8342028350082$$
$$x_{49} = 67.5392575740105$$
$$x_{49} = 82.2009195764864$$
$$x_{49} = 88.4843973905275$$
Decrece en los intervalos
$$\left[100.004010608094, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -106.28743487485\right]$$
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
$$\frac{- 9 \sin{\left(3 x \right)} + \frac{2 \left(- 3 x + 2 + \frac{\left(3 x^{2} - 4 x + 1\right)^{2}}{x \left(x^{2} - 2 x + 1\right)}\right) \sin{\left(3 x \right)}}{x \left(x^{2} - 2 x + 1\right)} - \frac{6 \left(3 x^{2} - 4 x + 1\right) \cos{\left(3 x \right)}}{x \left(x^{2} - 2 x + 1\right)}}{x \left(x^{2} - 2 x + 1\right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 52.3469766744115$$
$$x_{2} = 90.0515308119679$$
$$x_{3} = -6.18541703387252$$
$$x_{4} = 46.0620048855565$$
$$x_{5} = -57.5844198543657$$
$$x_{6} = 17.7633354486767$$
$$x_{7} = -73.2948145151447$$
$$x_{8} = -1.81620741244233$$
$$x_{9} = -8.30304557067179$$
$$x_{10} = -28.2512739527987$$
$$x_{11} = -63.8687198192252$$
$$x_{12} = 36.6333754800966$$
$$x_{13} = -90.0516404481773$$
$$x_{14} = -21.9616740436999$$
$$x_{15} = -4.04586252825108$$
$$x_{16} = 21.959824540472$$
$$x_{17} = -41.8722281771942$$
$$x_{18} = -77.4840879557206$$
$$x_{19} = -94.2407550446189$$
$$x_{20} = 74.3419770542282$$
$$x_{21} = 6.16103997134633$$
$$x_{22} = -61.7739781026929$$
$$x_{23} = 54.4418744290766$$
$$x_{24} = -52.3473012574026$$
$$x_{25} = -79.5787057512241$$
$$x_{26} = -50.2523886528575$$
$$x_{27} = 56.5367344726679$$
$$x_{28} = 10.4033525982465$$
$$x_{29} = -39.7770208638373$$
$$x_{30} = -74.3421379369112$$
$$x_{31} = -37.6817247195243$$
$$x_{32} = -92.1462014073263$$
$$x_{33} = -85.8624944368759$$
$$x_{34} = 32.4421389791125$$
$$x_{35} = -68.0581398447829$$
$$x_{36} = 26.1537729450639$$
$$x_{37} = 70.152641228493$$
$$x_{38} = 68.0579478697132$$
$$x_{39} = 63.8685018224872$$
$$x_{40} = -15.667113483403$$
$$x_{41} = 94.2406549405045$$
$$x_{42} = -26.1550755592994$$
$$x_{43} = 62.8211264915579$$
$$x_{44} = -72.2474874765178$$
$$x_{45} = 92.1460966996919$$
$$x_{46} = 50.252036424943$$
$$x_{47} = -99.4771101203796$$
$$x_{48} = -33.4908004372481$$
$$x_{49} = -65.9634397351124$$
$$x_{50} = -56.5370127067789$$
$$x_{51} = -24.0585707553518$$
$$x_{52} = 41.8717207035144$$
$$x_{53} = 28.2501578713658$$
$$x_{54} = -46.06242416064$$
$$x_{55} = 39.7764584620477$$
$$x_{56} = 3.98460271944375$$
$$x_{57} = -48.1574316145709$$
$$x_{58} = 30.3462612661142$$
$$x_{59} = 96.3352060367939$$
$$x_{60} = -11.4641317642476$$
$$x_{61} = 58.631560896683$$
$$x_{62} = -19.8642647824478$$
$$x_{63} = -59.6792123139416$$
$$x_{64} = -97.3825727987749$$
$$x_{65} = 19.8620022848173$$
$$x_{66} = 78.5312541548672$$
$$x_{67} = 83.767781483373$$
$$x_{68} = 8.28981335485049$$
$$x_{69} = 85.8623738391078$$
$$x_{70} = -35.5863243176632$$
$$x_{71} = 65.9632353691337$$
$$x_{72} = 80.6258735688115$$
$$x_{73} = 61.7737450641939$$
$$x_{74} = 48.1570480517877$$
$$x_{75} = -93.1934791105607$$
$$x_{76} = 24.0570305203629$$
$$x_{77} = -153.933727793189$$
$$x_{78} = -83.767908189014$$
$$x_{79} = -55.489597737261$$
$$x_{80} = 100.52428859453$$
$$x_{81} = 43.9668989641642$$
$$x_{82} = -17.7661670055876$$
$$x_{83} = -43.9673591844553$$
$$x_{84} = -87.9570716478884$$
$$x_{85} = 34.5378329934001$$
$$x_{86} = 95.2879313520537$$
$$x_{87} = 72.2473171266951$$
$$x_{88} = -32.4429848520372$$
$$x_{89} = 2.82344924606991$$
$$x_{90} = 51.2995121914725$$
$$x_{91} = 76.4366223361522$$
$$x_{92} = 87.956956726706$$
$$x_{93} = 12.5099940720095$$
$$x_{94} = 16.7135809936809$$
$$x_{95} = 98.4297504478792$$
$$x_{96} = -70.1528219065421$$
$$x_{97} = -81.6733122144386$$
$$x_{98} = 15.6634665799347$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$x_{2} = 1$$
$$\lim_{x \to 0^-}\left(\frac{- 9 \sin{\left(3 x \right)} + \frac{2 \left(- 3 x + 2 + \frac{\left(3 x^{2} - 4 x + 1\right)^{2}}{x \left(x^{2} - 2 x + 1\right)}\right) \sin{\left(3 x \right)}}{x \left(x^{2} - 2 x + 1\right)} - \frac{6 \left(3 x^{2} - 4 x + 1\right) \cos{\left(3 x \right)}}{x \left(x^{2} - 2 x + 1\right)}}{x \left(x^{2} - 2 x + 1\right)}\right) = 9$$
$$\lim_{x \to 0^+}\left(\frac{- 9 \sin{\left(3 x \right)} + \frac{2 \left(- 3 x + 2 + \frac{\left(3 x^{2} - 4 x + 1\right)^{2}}{x \left(x^{2} - 2 x + 1\right)}\right) \sin{\left(3 x \right)}}{x \left(x^{2} - 2 x + 1\right)} - \frac{6 \left(3 x^{2} - 4 x + 1\right) \cos{\left(3 x \right)}}{x \left(x^{2} - 2 x + 1\right)}}{x \left(x^{2} - 2 x + 1\right)}\right) = 9$$
- los límites son iguales, es decir omitimos el punto correspondiente
$$\lim_{x \to 1^-}\left(\frac{- 9 \sin{\left(3 x \right)} + \frac{2 \left(- 3 x + 2 + \frac{\left(3 x^{2} - 4 x + 1\right)^{2}}{x \left(x^{2} - 2 x + 1\right)}\right) \sin{\left(3 x \right)}}{x \left(x^{2} - 2 x + 1\right)} - \frac{6 \left(3 x^{2} - 4 x + 1\right) \cos{\left(3 x \right)}}{x \left(x^{2} - 2 x + 1\right)}}{x \left(x^{2} - 2 x + 1\right)}\right) = \infty$$
$$\lim_{x \to 1^+}\left(\frac{- 9 \sin{\left(3 x \right)} + \frac{2 \left(- 3 x + 2 + \frac{\left(3 x^{2} - 4 x + 1\right)^{2}}{x \left(x^{2} - 2 x + 1\right)}\right) \sin{\left(3 x \right)}}{x \left(x^{2} - 2 x + 1\right)} - \frac{6 \left(3 x^{2} - 4 x + 1\right) \cos{\left(3 x \right)}}{x \left(x^{2} - 2 x + 1\right)}}{x \left(x^{2} - 2 x + 1\right)}\right) = \infty$$
- los límites son iguales, es decir omitimos el punto correspondiente
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
$$\left[95.2879313520537, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -94.2407550446189\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{- 9 \sin{\left(3 x \right)} + \frac{2 \left(- 3 x + 2 + \frac{\left(3 x^{2} - 4 x + 1\right)^{2}}{x \left(x^{2} - 2 x + 1\right)}\right) \sin{\left(3 x \right)}}{x \left(x^{2} - 2 x + 1\right)} - \frac{6 \left(3 x^{2} - 4 x + 1\right) \cos{\left(3 x \right)}}{x \left(x^{2} - 2 x + 1\right)}}{x \left(x^{2} - 2 x + 1\right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 52.3469766744115$$
$$x_{2} = 90.0515308119679$$
$$x_{3} = -6.18541703387252$$
$$x_{4} = 46.0620048855565$$
$$x_{5} = -57.5844198543657$$
$$x_{6} = 17.7633354486767$$
$$x_{7} = -73.2948145151447$$
$$x_{8} = -1.81620741244233$$
$$x_{9} = -8.30304557067179$$
$$x_{10} = -28.2512739527987$$
$$x_{11} = -63.8687198192252$$
$$x_{12} = 36.6333754800966$$
$$x_{13} = -90.0516404481773$$
$$x_{14} = -21.9616740436999$$
$$x_{15} = -4.04586252825108$$
$$x_{16} = 21.959824540472$$
$$x_{17} = -41.8722281771942$$
$$x_{18} = -77.4840879557206$$
$$x_{19} = -94.2407550446189$$
$$x_{20} = 74.3419770542282$$
$$x_{21} = 6.16103997134633$$
$$x_{22} = -61.7739781026929$$
$$x_{23} = 54.4418744290766$$
$$x_{24} = -52.3473012574026$$
$$x_{25} = -79.5787057512241$$
$$x_{26} = -50.2523886528575$$
$$x_{27} = 56.5367344726679$$
$$x_{28} = 10.4033525982465$$
$$x_{29} = -39.7770208638373$$
$$x_{30} = -74.3421379369112$$
$$x_{31} = -37.6817247195243$$
$$x_{32} = -92.1462014073263$$
$$x_{33} = -85.8624944368759$$
$$x_{34} = 32.4421389791125$$
$$x_{35} = -68.0581398447829$$
$$x_{36} = 26.1537729450639$$
$$x_{37} = 70.152641228493$$
$$x_{38} = 68.0579478697132$$
$$x_{39} = 63.8685018224872$$
$$x_{40} = -15.667113483403$$
$$x_{41} = 94.2406549405045$$
$$x_{42} = -26.1550755592994$$
$$x_{43} = 62.8211264915579$$
$$x_{44} = -72.2474874765178$$
$$x_{45} = 92.1460966996919$$
$$x_{46} = 50.252036424943$$
$$x_{47} = -99.4771101203796$$
$$x_{48} = -33.4908004372481$$
$$x_{49} = -65.9634397351124$$
$$x_{50} = -56.5370127067789$$
$$x_{51} = -24.0585707553518$$
$$x_{52} = 41.8717207035144$$
$$x_{53} = 28.2501578713658$$
$$x_{54} = -46.06242416064$$
$$x_{55} = 39.7764584620477$$
$$x_{56} = 3.98460271944375$$
$$x_{57} = -48.1574316145709$$
$$x_{58} = 30.3462612661142$$
$$x_{59} = 96.3352060367939$$
$$x_{60} = -11.4641317642476$$
$$x_{61} = 58.631560896683$$
$$x_{62} = -19.8642647824478$$
$$x_{63} = -59.6792123139416$$
$$x_{64} = -97.3825727987749$$
$$x_{65} = 19.8620022848173$$
$$x_{66} = 78.5312541548672$$
$$x_{67} = 83.767781483373$$
$$x_{68} = 8.28981335485049$$
$$x_{69} = 85.8623738391078$$
$$x_{70} = -35.5863243176632$$
$$x_{71} = 65.9632353691337$$
$$x_{72} = 80.6258735688115$$
$$x_{73} = 61.7737450641939$$
$$x_{74} = 48.1570480517877$$
$$x_{75} = -93.1934791105607$$
$$x_{76} = 24.0570305203629$$
$$x_{77} = -153.933727793189$$
$$x_{78} = -83.767908189014$$
$$x_{79} = -55.489597737261$$
$$x_{80} = 100.52428859453$$
$$x_{81} = 43.9668989641642$$
$$x_{82} = -17.7661670055876$$
$$x_{83} = -43.9673591844553$$
$$x_{84} = -87.9570716478884$$
$$x_{85} = 34.5378329934001$$
$$x_{86} = 95.2879313520537$$
$$x_{87} = 72.2473171266951$$
$$x_{88} = -32.4429848520372$$
$$x_{89} = 2.82344924606991$$
$$x_{90} = 51.2995121914725$$
$$x_{91} = 76.4366223361522$$
$$x_{92} = 87.956956726706$$
$$x_{93} = 12.5099940720095$$
$$x_{94} = 16.7135809936809$$
$$x_{95} = 98.4297504478792$$
$$x_{96} = -70.1528219065421$$
$$x_{97} = -81.6733122144386$$
$$x_{98} = 15.6634665799347$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$x_{2} = 1$$
$$\lim_{x \to 0^-}\left(\frac{- 9 \sin{\left(3 x \right)} + \frac{2 \left(- 3 x + 2 + \frac{\left(3 x^{2} - 4 x + 1\right)^{2}}{x \left(x^{2} - 2 x + 1\right)}\right) \sin{\left(3 x \right)}}{x \left(x^{2} - 2 x + 1\right)} - \frac{6 \left(3 x^{2} - 4 x + 1\right) \cos{\left(3 x \right)}}{x \left(x^{2} - 2 x + 1\right)}}{x \left(x^{2} - 2 x + 1\right)}\right) = 9$$
$$\lim_{x \to 0^+}\left(\frac{- 9 \sin{\left(3 x \right)} + \frac{2 \left(- 3 x + 2 + \frac{\left(3 x^{2} - 4 x + 1\right)^{2}}{x \left(x^{2} - 2 x + 1\right)}\right) \sin{\left(3 x \right)}}{x \left(x^{2} - 2 x + 1\right)} - \frac{6 \left(3 x^{2} - 4 x + 1\right) \cos{\left(3 x \right)}}{x \left(x^{2} - 2 x + 1\right)}}{x \left(x^{2} - 2 x + 1\right)}\right) = 9$$
- los límites son iguales, es decir omitimos el punto correspondiente
$$\lim_{x \to 1^-}\left(\frac{- 9 \sin{\left(3 x \right)} + \frac{2 \left(- 3 x + 2 + \frac{\left(3 x^{2} - 4 x + 1\right)^{2}}{x \left(x^{2} - 2 x + 1\right)}\right) \sin{\left(3 x \right)}}{x \left(x^{2} - 2 x + 1\right)} - \frac{6 \left(3 x^{2} - 4 x + 1\right) \cos{\left(3 x \right)}}{x \left(x^{2} - 2 x + 1\right)}}{x \left(x^{2} - 2 x + 1\right)}\right) = \infty$$
$$\lim_{x \to 1^+}\left(\frac{- 9 \sin{\left(3 x \right)} + \frac{2 \left(- 3 x + 2 + \frac{\left(3 x^{2} - 4 x + 1\right)^{2}}{x \left(x^{2} - 2 x + 1\right)}\right) \sin{\left(3 x \right)}}{x \left(x^{2} - 2 x + 1\right)} - \frac{6 \left(3 x^{2} - 4 x + 1\right) \cos{\left(3 x \right)}}{x \left(x^{2} - 2 x + 1\right)}}{x \left(x^{2} - 2 x + 1\right)}\right) = \infty$$
- los límites son iguales, es decir omitimos el punto correspondiente
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
$$\left[95.2879313520537, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -94.2407550446189\right]$$
Asíntotas verticales
Hay:
$$x_{1} = 0$$
$$x_{2} = 1$$
$$x_{1} = 0$$
$$x_{2} = 1$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(3 x \right)}}{x + \left(x^{3} - 2 x^{2}\right)}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(3 x \right)}}{x + \left(x^{3} - 2 x^{2}\right)}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(3 x \right)}}{x + \left(x^{3} - 2 x^{2}\right)}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(3 x \right)}}{x + \left(x^{3} - 2 x^{2}\right)}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(3*x)/(x^3 - 2*x^2 + x), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(3 x \right)}}{x \left(x + \left(x^{3} - 2 x^{2}\right)\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(3 x \right)}}{x \left(x + \left(x^{3} - 2 x^{2}\right)\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(3 x \right)}}{x \left(x + \left(x^{3} - 2 x^{2}\right)\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(3 x \right)}}{x \left(x + \left(x^{3} - 2 x^{2}\right)\right)}\right) = 0$$
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:
$$\frac{\sin{\left(3 x \right)}}{x + \left(x^{3} - 2 x^{2}\right)} = - \frac{\sin{\left(3 x \right)}}{- x^{3} - 2 x^{2} - x}$$
- No
$$\frac{\sin{\left(3 x \right)}}{x + \left(x^{3} - 2 x^{2}\right)} = \frac{\sin{\left(3 x \right)}}{- x^{3} - 2 x^{2} - x}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\sin{\left(3 x \right)}}{x + \left(x^{3} - 2 x^{2}\right)} = - \frac{\sin{\left(3 x \right)}}{- x^{3} - 2 x^{2} - x}$$
- No
$$\frac{\sin{\left(3 x \right)}}{x + \left(x^{3} - 2 x^{2}\right)} = \frac{\sin{\left(3 x \right)}}{- x^{3} - 2 x^{2} - x}$$
- No
es decir, función
no es
par ni impar