Sr Examen
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*e^(2-x)
-
x+27/x^3
-
(x^2-8)/(x-3)
-
-x^2+4*x+2
-
Expresiones idénticas
- (x*sin(tres *x))/ cinco
- (x multiplicar por seno de (3 multiplicar por x)) dividir por 5
- (x multiplicar por seno de (tres multiplicar por x)) dividir por cinco
- (xsin(3x))/5
- xsin3x/5
- (x*sin(3*x)) dividir por 5
-
Expresiones semejantes
- (42/29+3*cos(3*x)*exp(-7*x)/58+7*exp(-7*x)*sin(3*x)/58)*exp(7*x)
- -cos(2*x)-sin(2*x)-12*cos(3*x)/25-sin(3*x)/5-2*x*sin(3*x)/5
-
Expresiones con funciones
- Seno sin
- sin(x-pi/4)*(-2)
- sin(pi*x)/asin(x)
- sin(pi×n/3)
- sin(x)*e^((1/2)*cot(x)^(2))
- sin(697*t)+sin(1209*t)
Gráfico de la función y = (x*sin(3*x))/5
Solución
Ha introducido
[src]
x*sin(3*x)
f(x) = ----------
5
$$f{\left(x \right)} = \frac{x \sin{\left(3 x \right)}}{5}$$
f = (x*sin(3*x))/5
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:
$$\frac{x \sin{\left(3 x \right)}}{5} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = - \frac{2 \pi}{3}$$
$$x_{3} = - \frac{\pi}{3}$$
$$x_{4} = \frac{\pi}{3}$$
$$x_{5} = \frac{2 \pi}{3}$$
$$x_{6} = \pi$$
Solución numérica
$$x_{1} = -55.5014702134197$$
$$x_{2} = 90.0589894029074$$
$$x_{3} = -1.0471975511966$$
$$x_{4} = -33.5103216382911$$
$$x_{5} = -72.2566310325652$$
$$x_{6} = 409.45424251787$$
$$x_{7} = -90.0589894029074$$
$$x_{8} = 74.3510261349584$$
$$x_{9} = -59.6902604182061$$
$$x_{10} = 17.8023583703422$$
$$x_{11} = -6.28318530717959$$
$$x_{12} = -85.870199198121$$
$$x_{13} = 56.5486677646163$$
$$x_{14} = 26.1799387799149$$
$$x_{15} = 72.2566310325652$$
$$x_{16} = 37.6991118430775$$
$$x_{17} = -83.7758040957278$$
$$x_{18} = 92.1533845053006$$
$$x_{19} = 100.530964914873$$
$$x_{20} = 48.1710873550435$$
$$x_{21} = 70.162235930172$$
$$x_{22} = 94.2477796076938$$
$$x_{23} = 54.4542726622231$$
$$x_{24} = -24.0855436775217$$
$$x_{25} = -95.2949771588904$$
$$x_{26} = -79.5870138909414$$
$$x_{27} = -87.9645943005142$$
$$x_{28} = -86.9173967493176$$
$$x_{29} = 46.0766922526503$$
$$x_{30} = 59.6902604182061$$
$$x_{31} = 0$$
$$x_{32} = 30.3687289847013$$
$$x_{33} = 50.2654824574367$$
$$x_{34} = -53.4070751110265$$
$$x_{35} = -92.1533845053006$$
$$x_{36} = -21.9911485751286$$
$$x_{37} = 41.8879020478639$$
$$x_{38} = 32.4631240870945$$
$$x_{39} = 75.398223686155$$
$$x_{40} = -41.8879020478639$$
$$x_{41} = 63.8790506229925$$
$$x_{42} = 87.9645943005142$$
$$x_{43} = -19.8967534727354$$
$$x_{44} = 4.18879020478639$$
$$x_{45} = -11.5191730631626$$
$$x_{46} = 39.7935069454707$$
$$x_{47} = -43.9822971502571$$
$$x_{48} = -37.6991118430775$$
$$x_{49} = -65.9734457253857$$
$$x_{50} = 10.471975511966$$
$$x_{51} = -99.4837673636768$$
$$x_{52} = 58.6430628670095$$
$$x_{53} = 76.4454212373516$$
$$x_{54} = 3.14159265358979$$
$$x_{55} = 6.28318530717959$$
$$x_{56} = -39.7935069454707$$
$$x_{57} = 83.7758040957278$$
$$x_{58} = -17.8023583703422$$
$$x_{59} = 28.2743338823081$$
$$x_{60} = -70.162235930172$$
$$x_{61} = -68.0678408277789$$
$$x_{62} = -94.2477796076938$$
$$x_{63} = -13.6135681655558$$
$$x_{64} = 43.9822971502571$$
$$x_{65} = -4.18879020478639$$
$$x_{66} = 65.9734457253857$$
$$x_{67} = 13.6135681655558$$
$$x_{68} = 96.342174710087$$
$$x_{69} = 52.3598775598299$$
$$x_{70} = -30.3687289847013$$
$$x_{71} = -15.707963267949$$
$$x_{72} = -48.1710873550435$$
$$x_{73} = -50.2654824574367$$
$$x_{74} = -46.0766922526503$$
$$x_{75} = -35.6047167406843$$
$$x_{76} = -8.37758040957278$$
$$x_{77} = 1.0471975511966$$
$$x_{78} = 15.707963267949$$
$$x_{79} = -57.5958653158129$$
$$x_{80} = -77.4926187885482$$
$$x_{81} = 24.0855436775217$$
$$x_{82} = -61.7846555205993$$
$$x_{83} = 21.9911485751286$$
$$x_{84} = -26.1799387799149$$
$$x_{85} = -2.0943951023932$$
$$x_{86} = -28.2743338823081$$
$$x_{87} = 78.5398163397448$$
$$x_{88} = 98.4365698124802$$
$$x_{89} = -63.8790506229925$$
$$x_{90} = 2.0943951023932$$
$$x_{91} = 68.0678408277789$$
$$x_{92} = 19.8967534727354$$
$$x_{93} = 245.044226980004$$
$$x_{94} = -81.6814089933346$$
$$x_{95} = 8.37758040957278$$
o sea hay que resolver la ecuación:
$$\frac{x \sin{\left(3 x \right)}}{5} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = - \frac{2 \pi}{3}$$
$$x_{3} = - \frac{\pi}{3}$$
$$x_{4} = \frac{\pi}{3}$$
$$x_{5} = \frac{2 \pi}{3}$$
$$x_{6} = \pi$$
Solución numérica
$$x_{1} = -55.5014702134197$$
$$x_{2} = 90.0589894029074$$
$$x_{3} = -1.0471975511966$$
$$x_{4} = -33.5103216382911$$
$$x_{5} = -72.2566310325652$$
$$x_{6} = 409.45424251787$$
$$x_{7} = -90.0589894029074$$
$$x_{8} = 74.3510261349584$$
$$x_{9} = -59.6902604182061$$
$$x_{10} = 17.8023583703422$$
$$x_{11} = -6.28318530717959$$
$$x_{12} = -85.870199198121$$
$$x_{13} = 56.5486677646163$$
$$x_{14} = 26.1799387799149$$
$$x_{15} = 72.2566310325652$$
$$x_{16} = 37.6991118430775$$
$$x_{17} = -83.7758040957278$$
$$x_{18} = 92.1533845053006$$
$$x_{19} = 100.530964914873$$
$$x_{20} = 48.1710873550435$$
$$x_{21} = 70.162235930172$$
$$x_{22} = 94.2477796076938$$
$$x_{23} = 54.4542726622231$$
$$x_{24} = -24.0855436775217$$
$$x_{25} = -95.2949771588904$$
$$x_{26} = -79.5870138909414$$
$$x_{27} = -87.9645943005142$$
$$x_{28} = -86.9173967493176$$
$$x_{29} = 46.0766922526503$$
$$x_{30} = 59.6902604182061$$
$$x_{31} = 0$$
$$x_{32} = 30.3687289847013$$
$$x_{33} = 50.2654824574367$$
$$x_{34} = -53.4070751110265$$
$$x_{35} = -92.1533845053006$$
$$x_{36} = -21.9911485751286$$
$$x_{37} = 41.8879020478639$$
$$x_{38} = 32.4631240870945$$
$$x_{39} = 75.398223686155$$
$$x_{40} = -41.8879020478639$$
$$x_{41} = 63.8790506229925$$
$$x_{42} = 87.9645943005142$$
$$x_{43} = -19.8967534727354$$
$$x_{44} = 4.18879020478639$$
$$x_{45} = -11.5191730631626$$
$$x_{46} = 39.7935069454707$$
$$x_{47} = -43.9822971502571$$
$$x_{48} = -37.6991118430775$$
$$x_{49} = -65.9734457253857$$
$$x_{50} = 10.471975511966$$
$$x_{51} = -99.4837673636768$$
$$x_{52} = 58.6430628670095$$
$$x_{53} = 76.4454212373516$$
$$x_{54} = 3.14159265358979$$
$$x_{55} = 6.28318530717959$$
$$x_{56} = -39.7935069454707$$
$$x_{57} = 83.7758040957278$$
$$x_{58} = -17.8023583703422$$
$$x_{59} = 28.2743338823081$$
$$x_{60} = -70.162235930172$$
$$x_{61} = -68.0678408277789$$
$$x_{62} = -94.2477796076938$$
$$x_{63} = -13.6135681655558$$
$$x_{64} = 43.9822971502571$$
$$x_{65} = -4.18879020478639$$
$$x_{66} = 65.9734457253857$$
$$x_{67} = 13.6135681655558$$
$$x_{68} = 96.342174710087$$
$$x_{69} = 52.3598775598299$$
$$x_{70} = -30.3687289847013$$
$$x_{71} = -15.707963267949$$
$$x_{72} = -48.1710873550435$$
$$x_{73} = -50.2654824574367$$
$$x_{74} = -46.0766922526503$$
$$x_{75} = -35.6047167406843$$
$$x_{76} = -8.37758040957278$$
$$x_{77} = 1.0471975511966$$
$$x_{78} = 15.707963267949$$
$$x_{79} = -57.5958653158129$$
$$x_{80} = -77.4926187885482$$
$$x_{81} = 24.0855436775217$$
$$x_{82} = -61.7846555205993$$
$$x_{83} = 21.9911485751286$$
$$x_{84} = -26.1799387799149$$
$$x_{85} = -2.0943951023932$$
$$x_{86} = -28.2743338823081$$
$$x_{87} = 78.5398163397448$$
$$x_{88} = 98.4365698124802$$
$$x_{89} = -63.8790506229925$$
$$x_{90} = 2.0943951023932$$
$$x_{91} = 68.0678408277789$$
$$x_{92} = 19.8967534727354$$
$$x_{93} = 245.044226980004$$
$$x_{94} = -81.6814089933346$$
$$x_{95} = 8.37758040957278$$
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
$$\frac{3 x \cos{\left(3 x \right)}}{5} + \frac{\sin{\left(3 x \right)}}{5} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -82.2063593736386$$
$$x_{2} = 16.2384035725192$$
$$x_{3} = -4.7358122417304$$
$$x_{4} = -51.8384221669793$$
$$x_{5} = 56.0270521345864$$
$$x_{6} = 84.3007208972085$$
$$x_{7} = -89.5366315785916$$
$$x_{8} = 67.5458870110976$$
$$x_{9} = 49.7441173016936$$
$$x_{10} = 26.7076976049501$$
$$x_{11} = -80.1119996057056$$
$$x_{12} = 80.1119996057056$$
$$x_{13} = 31.9430036030065$$
$$x_{14} = 73.8289323297373$$
$$x_{15} = 66.4987153630436$$
$$x_{16} = 22.5196809462695$$
$$x_{17} = -87.4422661984441$$
$$x_{18} = -14.1450206271366$$
$$x_{19} = 0.676252612703478$$
$$x_{20} = -23.5666593462033$$
$$x_{21} = -75.9232859178705$$
$$x_{22} = 89.5366315785916$$
$$x_{23} = 0$$
$$x_{24} = -58.1213757786594$$
$$x_{25} = 27.7547382346962$$
$$x_{26} = -36.1313906251304$$
$$x_{27} = 58.1213757786594$$
$$x_{28} = -5.77879264132779$$
$$x_{29} = 62.3100374768166$$
$$x_{30} = -43.4612548800528$$
$$x_{31} = 88.4894487126566$$
$$x_{32} = -1.63772681314496$$
$$x_{33} = 92.6781821675128$$
$$x_{34} = 9.95952883536913$$
$$x_{35} = -49.7441173016936$$
$$x_{36} = -29.8488525127497$$
$$x_{37} = -38.225617263561$$
$$x_{38} = 12.0519888065122$$
$$x_{39} = 78.0176417347899$$
$$x_{40} = 60.2157043931142$$
$$x_{41} = 18.3320175191655$$
$$x_{42} = 3.69517946883234$$
$$x_{43} = 100.008477152089$$
$$x_{44} = -69.6402326441114$$
$$x_{45} = -3.69517946883234$$
$$x_{46} = -45.5555324591521$$
$$x_{47} = -9.95952883536913$$
$$x_{48} = 86.3950840487432$$
$$x_{49} = 53.9327340398572$$
$$x_{50} = -73.8289323297373$$
$$x_{51} = -60.2157043931142$$
$$x_{52} = 14.1450206271366$$
$$x_{53} = 75.9232859178705$$
$$x_{54} = -41.3669891991005$$
$$x_{55} = -16.2384035725192$$
$$x_{56} = 7.8680949243268$$
$$x_{57} = 29.8488525127497$$
$$x_{58} = 64.4043745938079$$
$$x_{59} = -97.9141058139495$$
$$x_{60} = 97.9141058139495$$
$$x_{61} = 23.5666593462033$$
$$x_{62} = -53.9327340398572$$
$$x_{63} = -7.8680949243268$$
$$x_{64} = -25.6606697768062$$
$$x_{65} = -61.2628704049539$$
$$x_{66} = 34.037184713218$$
$$x_{67} = 5.77879264132779$$
$$x_{68} = -31.9430036030065$$
$$x_{69} = -67.5458870110976$$
$$x_{70} = -12.0519888065122$$
$$x_{71} = -95.8197355146347$$
$$x_{72} = -100.008477152089$$
$$x_{73} = 42.4141204421759$$
$$x_{74} = -34.037184713218$$
$$x_{75} = 36.1313906251304$$
$$x_{76} = -84.3007208972085$$
$$x_{77} = 51.8384221669793$$
$$x_{78} = -47.6498203678514$$
$$x_{79} = -27.7547382346962$$
$$x_{80} = 38.225617263561$$
$$x_{81} = -91.6309983173967$$
$$x_{82} = 54.9798923539233$$
$$x_{83} = -21.4727239072797$$
$$x_{84} = -65.4515445438056$$
$$x_{85} = -93.7253663237826$$
$$x_{86} = 40.3198613996847$$
$$x_{87} = -71.7345811655909$$
$$x_{88} = -106.291596785411$$
$$x_{89} = 95.8197355146347$$
$$x_{90} = 44.5083922872857$$
$$x_{91} = -56.0270521345864$$
$$x_{92} = 82.2063593736386$$
$$x_{93} = 93.7253663237826$$
$$x_{94} = 71.7345811655909$$
$$x_{95} = -78.0176417347899$$
$$x_{96} = 20.4257915111899$$
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:
$$x_{1} = 16.2384035725192$$
$$x_{2} = -51.8384221669793$$
$$x_{3} = 56.0270521345864$$
$$x_{4} = -89.5366315785916$$
$$x_{5} = 49.7441173016936$$
$$x_{6} = 26.7076976049501$$
$$x_{7} = 66.4987153630436$$
$$x_{8} = 22.5196809462695$$
$$x_{9} = -87.4422661984441$$
$$x_{10} = -14.1450206271366$$
$$x_{11} = 89.5366315785916$$
$$x_{12} = 0$$
$$x_{13} = -58.1213757786594$$
$$x_{14} = 58.1213757786594$$
$$x_{15} = -5.77879264132779$$
$$x_{16} = 62.3100374768166$$
$$x_{17} = -43.4612548800528$$
$$x_{18} = -1.63772681314496$$
$$x_{19} = 9.95952883536913$$
$$x_{20} = -49.7441173016936$$
$$x_{21} = 12.0519888065122$$
$$x_{22} = 60.2157043931142$$
$$x_{23} = 18.3320175191655$$
$$x_{24} = 3.69517946883234$$
$$x_{25} = 100.008477152089$$
$$x_{26} = -3.69517946883234$$
$$x_{27} = -45.5555324591521$$
$$x_{28} = -9.95952883536913$$
$$x_{29} = 53.9327340398572$$
$$x_{30} = -60.2157043931142$$
$$x_{31} = 14.1450206271366$$
$$x_{32} = -41.3669891991005$$
$$x_{33} = -16.2384035725192$$
$$x_{34} = 7.8680949243268$$
$$x_{35} = 64.4043745938079$$
$$x_{36} = -97.9141058139495$$
$$x_{37} = 97.9141058139495$$
$$x_{38} = -53.9327340398572$$
$$x_{39} = -7.8680949243268$$
$$x_{40} = 5.77879264132779$$
$$x_{41} = -12.0519888065122$$
$$x_{42} = -95.8197355146347$$
$$x_{43} = -100.008477152089$$
$$x_{44} = 51.8384221669793$$
$$x_{45} = -47.6498203678514$$
$$x_{46} = -91.6309983173967$$
$$x_{47} = -93.7253663237826$$
$$x_{48} = -106.291596785411$$
$$x_{49} = 95.8197355146347$$
$$x_{50} = -56.0270521345864$$
$$x_{51} = 93.7253663237826$$
$$x_{52} = 20.4257915111899$$
Puntos máximos de la función:
$$x_{52} = -82.2063593736386$$
$$x_{52} = -4.7358122417304$$
$$x_{52} = 84.3007208972085$$
$$x_{52} = 67.5458870110976$$
$$x_{52} = -80.1119996057056$$
$$x_{52} = 80.1119996057056$$
$$x_{52} = 31.9430036030065$$
$$x_{52} = 73.8289323297373$$
$$x_{52} = 0.676252612703478$$
$$x_{52} = -23.5666593462033$$
$$x_{52} = -75.9232859178705$$
$$x_{52} = 27.7547382346962$$
$$x_{52} = -36.1313906251304$$
$$x_{52} = 88.4894487126566$$
$$x_{52} = 92.6781821675128$$
$$x_{52} = -29.8488525127497$$
$$x_{52} = -38.225617263561$$
$$x_{52} = 78.0176417347899$$
$$x_{52} = -69.6402326441114$$
$$x_{52} = 86.3950840487432$$
$$x_{52} = -73.8289323297373$$
$$x_{52} = 75.9232859178705$$
$$x_{52} = 29.8488525127497$$
$$x_{52} = 23.5666593462033$$
$$x_{52} = -25.6606697768062$$
$$x_{52} = -61.2628704049539$$
$$x_{52} = 34.037184713218$$
$$x_{52} = -31.9430036030065$$
$$x_{52} = -67.5458870110976$$
$$x_{52} = 42.4141204421759$$
$$x_{52} = -34.037184713218$$
$$x_{52} = 36.1313906251304$$
$$x_{52} = -84.3007208972085$$
$$x_{52} = -27.7547382346962$$
$$x_{52} = 38.225617263561$$
$$x_{52} = 54.9798923539233$$
$$x_{52} = -21.4727239072797$$
$$x_{52} = -65.4515445438056$$
$$x_{52} = 40.3198613996847$$
$$x_{52} = -71.7345811655909$$
$$x_{52} = 44.5083922872857$$
$$x_{52} = 82.2063593736386$$
$$x_{52} = 71.7345811655909$$
$$x_{52} = -78.0176417347899$$
Decrece en los intervalos
$$\left[100.008477152089, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -106.291596785411\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{3 x \cos{\left(3 x \right)}}{5} + \frac{\sin{\left(3 x \right)}}{5} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -82.2063593736386$$
$$x_{2} = 16.2384035725192$$
$$x_{3} = -4.7358122417304$$
$$x_{4} = -51.8384221669793$$
$$x_{5} = 56.0270521345864$$
$$x_{6} = 84.3007208972085$$
$$x_{7} = -89.5366315785916$$
$$x_{8} = 67.5458870110976$$
$$x_{9} = 49.7441173016936$$
$$x_{10} = 26.7076976049501$$
$$x_{11} = -80.1119996057056$$
$$x_{12} = 80.1119996057056$$
$$x_{13} = 31.9430036030065$$
$$x_{14} = 73.8289323297373$$
$$x_{15} = 66.4987153630436$$
$$x_{16} = 22.5196809462695$$
$$x_{17} = -87.4422661984441$$
$$x_{18} = -14.1450206271366$$
$$x_{19} = 0.676252612703478$$
$$x_{20} = -23.5666593462033$$
$$x_{21} = -75.9232859178705$$
$$x_{22} = 89.5366315785916$$
$$x_{23} = 0$$
$$x_{24} = -58.1213757786594$$
$$x_{25} = 27.7547382346962$$
$$x_{26} = -36.1313906251304$$
$$x_{27} = 58.1213757786594$$
$$x_{28} = -5.77879264132779$$
$$x_{29} = 62.3100374768166$$
$$x_{30} = -43.4612548800528$$
$$x_{31} = 88.4894487126566$$
$$x_{32} = -1.63772681314496$$
$$x_{33} = 92.6781821675128$$
$$x_{34} = 9.95952883536913$$
$$x_{35} = -49.7441173016936$$
$$x_{36} = -29.8488525127497$$
$$x_{37} = -38.225617263561$$
$$x_{38} = 12.0519888065122$$
$$x_{39} = 78.0176417347899$$
$$x_{40} = 60.2157043931142$$
$$x_{41} = 18.3320175191655$$
$$x_{42} = 3.69517946883234$$
$$x_{43} = 100.008477152089$$
$$x_{44} = -69.6402326441114$$
$$x_{45} = -3.69517946883234$$
$$x_{46} = -45.5555324591521$$
$$x_{47} = -9.95952883536913$$
$$x_{48} = 86.3950840487432$$
$$x_{49} = 53.9327340398572$$
$$x_{50} = -73.8289323297373$$
$$x_{51} = -60.2157043931142$$
$$x_{52} = 14.1450206271366$$
$$x_{53} = 75.9232859178705$$
$$x_{54} = -41.3669891991005$$
$$x_{55} = -16.2384035725192$$
$$x_{56} = 7.8680949243268$$
$$x_{57} = 29.8488525127497$$
$$x_{58} = 64.4043745938079$$
$$x_{59} = -97.9141058139495$$
$$x_{60} = 97.9141058139495$$
$$x_{61} = 23.5666593462033$$
$$x_{62} = -53.9327340398572$$
$$x_{63} = -7.8680949243268$$
$$x_{64} = -25.6606697768062$$
$$x_{65} = -61.2628704049539$$
$$x_{66} = 34.037184713218$$
$$x_{67} = 5.77879264132779$$
$$x_{68} = -31.9430036030065$$
$$x_{69} = -67.5458870110976$$
$$x_{70} = -12.0519888065122$$
$$x_{71} = -95.8197355146347$$
$$x_{72} = -100.008477152089$$
$$x_{73} = 42.4141204421759$$
$$x_{74} = -34.037184713218$$
$$x_{75} = 36.1313906251304$$
$$x_{76} = -84.3007208972085$$
$$x_{77} = 51.8384221669793$$
$$x_{78} = -47.6498203678514$$
$$x_{79} = -27.7547382346962$$
$$x_{80} = 38.225617263561$$
$$x_{81} = -91.6309983173967$$
$$x_{82} = 54.9798923539233$$
$$x_{83} = -21.4727239072797$$
$$x_{84} = -65.4515445438056$$
$$x_{85} = -93.7253663237826$$
$$x_{86} = 40.3198613996847$$
$$x_{87} = -71.7345811655909$$
$$x_{88} = -106.291596785411$$
$$x_{89} = 95.8197355146347$$
$$x_{90} = 44.5083922872857$$
$$x_{91} = -56.0270521345864$$
$$x_{92} = 82.2063593736386$$
$$x_{93} = 93.7253663237826$$
$$x_{94} = 71.7345811655909$$
$$x_{95} = -78.0176417347899$$
$$x_{96} = 20.4257915111899$$
Signos de extremos en los puntos:
(-82.20635937363859, 16.4411367151831)
(16.238403572519243, -3.2469966816912)
(-4.735812241730396, 0.944824940918285)
(-51.83842216697935, -10.3674700988137)
(56.027052134586405, -11.2052121152852)
(84.3007208972085, 16.8600123777121)
(-89.53663157859164, -17.9072022213067)
(67.54588701109756, 13.509012907997)
(49.74411730169364, -9.94860010253207)
(26.707697604950084, -5.34112354304395)
(-80.1119996057056, 16.022261228225)
(80.1119996057056, 16.022261228225)
(31.943003603006492, 6.38825290723105)
(73.82893232973727, 14.765635970188)
(66.49871536304362, -13.299575988152)
(22.519680946269467, -4.5034428747315)
(-87.44226619844414, -17.4883261731054)
(-14.145020627136628, -2.82821893848394)
(0.676252612703478, 0.12131371607731)
(-23.566659346203334, 4.71286046410621)
(-75.92328591787046, 15.1845108391377)
(89.53663157859164, -17.9072022213067)
(0, 0)
(-58.121375778659406, -11.6240839896269)
(27.754738234696216, 5.55054735819688)
(-36.131390625130436, 7.22597062504596)
(58.121375778659406, -11.6240839896269)
(-5.778792641327787, -1.15384057385723)
(62.31003747681657, -12.4618291794277)
(-43.46125488005277, -8.69199533173688)
(88.4894487126566, 17.6977641796193)
(-1.6377268131449612, -0.320964659314151)
(92.6781821675128, 18.5355165454874)
(9.959528835369131, -1.99079107727912)
(-49.74411730169364, -9.94860010253207)
(-29.848852512749733, 5.96939829152568)
(-38.225617263561, 7.64483279743256)
(12.05198880651224, -2.40947635814947)
(78.01764173478989, 15.6033859309769)
(60.21570439311422, -12.0429563610478)
(18.332017519165465, -3.66579754997984)
(3.695179468832341, -0.736047201062)
(100.00847715208945, -20.0015843296507)
(-69.64023264411136, 13.9278869813916)
(-3.695179468832341, -0.736047201062)
(-45.555532459152055, -9.11086259906113)
(-9.959528835369131, -1.99079107727912)
(86.39508404874319, 17.2788882030439)
(53.93273403985717, -10.7863407959271)
(-73.82893232973727, 14.765635970188)
(-60.21570439311422, -12.0429563610478)
(14.145020627136628, -2.82821893848394)
(75.92328591787046, 15.1845108391377)
(-41.36698919910045, -8.2731292544029)
(-16.238403572519243, -3.2469966816912)
(7.868094924326803, -1.57220870997556)
(29.848852512749733, 5.96939829152568)
(64.40437459380786, -12.8807024011654)
(-97.91410581394948, -19.5827076856311)
(97.91410581394948, -19.5827076856311)
(23.566659346203334, 4.71286046410621)
(-53.93273403985717, -10.7863407959271)
(-7.868094924326803, -1.57220870997556)
(-25.66066977680624, 5.13170100855091)
(-61.262870404953894, 12.2523927172327)
(34.037184713217975, 6.80711052575443)
(5.778792641327787, -1.15384057385723)
(-31.943003603006492, 6.38825290723105)
(-67.54588701109756, 13.509012907997)
(-12.05198880651224, -2.40947635814947)
(-95.81973551463474, -19.163831145497)
(-100.00847715208945, -20.0015843296507)
(42.41412044217588, 8.48256213330513)
(-34.037184713217975, 6.80711052575443)
(36.131390625130436, 7.22597062504596)
(-84.3007208972085, 16.8600123777121)
(51.83842216697935, -10.3674700988137)
(-47.64982036785143, -9.52973089948409)
(-27.754738234696216, 5.55054735819688)
(38.225617263561, 7.64483279743256)
(-91.63099831739673, -18.3260784053785)
(54.97989235392331, 10.9957763822696)
(-21.47272390727972, 4.29402742262502)
(-65.45154454380558, 13.0901391511939)
(-93.7253663237826, -18.7449547162126)
(40.319861399684655, 8.0636967199226)
(-71.73458116559091, 14.3467613436493)
(-106.29159678541116, -21.2582148236149)
(95.81973551463474, -19.163831145497)
(44.50839228728569, 8.90142882714793)
(-56.027052134586405, -11.2052121152852)
(82.20635937363859, 16.4411367151831)
(93.7253663237826, -18.7449547162126)
(71.73458116559091, 14.3467613436493)
(-78.01764173478989, 15.6033859309769)
(20.425791511189885, -4.08461443629844)
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:
$$x_{1} = 16.2384035725192$$
$$x_{2} = -51.8384221669793$$
$$x_{3} = 56.0270521345864$$
$$x_{4} = -89.5366315785916$$
$$x_{5} = 49.7441173016936$$
$$x_{6} = 26.7076976049501$$
$$x_{7} = 66.4987153630436$$
$$x_{8} = 22.5196809462695$$
$$x_{9} = -87.4422661984441$$
$$x_{10} = -14.1450206271366$$
$$x_{11} = 89.5366315785916$$
$$x_{12} = 0$$
$$x_{13} = -58.1213757786594$$
$$x_{14} = 58.1213757786594$$
$$x_{15} = -5.77879264132779$$
$$x_{16} = 62.3100374768166$$
$$x_{17} = -43.4612548800528$$
$$x_{18} = -1.63772681314496$$
$$x_{19} = 9.95952883536913$$
$$x_{20} = -49.7441173016936$$
$$x_{21} = 12.0519888065122$$
$$x_{22} = 60.2157043931142$$
$$x_{23} = 18.3320175191655$$
$$x_{24} = 3.69517946883234$$
$$x_{25} = 100.008477152089$$
$$x_{26} = -3.69517946883234$$
$$x_{27} = -45.5555324591521$$
$$x_{28} = -9.95952883536913$$
$$x_{29} = 53.9327340398572$$
$$x_{30} = -60.2157043931142$$
$$x_{31} = 14.1450206271366$$
$$x_{32} = -41.3669891991005$$
$$x_{33} = -16.2384035725192$$
$$x_{34} = 7.8680949243268$$
$$x_{35} = 64.4043745938079$$
$$x_{36} = -97.9141058139495$$
$$x_{37} = 97.9141058139495$$
$$x_{38} = -53.9327340398572$$
$$x_{39} = -7.8680949243268$$
$$x_{40} = 5.77879264132779$$
$$x_{41} = -12.0519888065122$$
$$x_{42} = -95.8197355146347$$
$$x_{43} = -100.008477152089$$
$$x_{44} = 51.8384221669793$$
$$x_{45} = -47.6498203678514$$
$$x_{46} = -91.6309983173967$$
$$x_{47} = -93.7253663237826$$
$$x_{48} = -106.291596785411$$
$$x_{49} = 95.8197355146347$$
$$x_{50} = -56.0270521345864$$
$$x_{51} = 93.7253663237826$$
$$x_{52} = 20.4257915111899$$
Puntos máximos de la función:
$$x_{52} = -82.2063593736386$$
$$x_{52} = -4.7358122417304$$
$$x_{52} = 84.3007208972085$$
$$x_{52} = 67.5458870110976$$
$$x_{52} = -80.1119996057056$$
$$x_{52} = 80.1119996057056$$
$$x_{52} = 31.9430036030065$$
$$x_{52} = 73.8289323297373$$
$$x_{52} = 0.676252612703478$$
$$x_{52} = -23.5666593462033$$
$$x_{52} = -75.9232859178705$$
$$x_{52} = 27.7547382346962$$
$$x_{52} = -36.1313906251304$$
$$x_{52} = 88.4894487126566$$
$$x_{52} = 92.6781821675128$$
$$x_{52} = -29.8488525127497$$
$$x_{52} = -38.225617263561$$
$$x_{52} = 78.0176417347899$$
$$x_{52} = -69.6402326441114$$
$$x_{52} = 86.3950840487432$$
$$x_{52} = -73.8289323297373$$
$$x_{52} = 75.9232859178705$$
$$x_{52} = 29.8488525127497$$
$$x_{52} = 23.5666593462033$$
$$x_{52} = -25.6606697768062$$
$$x_{52} = -61.2628704049539$$
$$x_{52} = 34.037184713218$$
$$x_{52} = -31.9430036030065$$
$$x_{52} = -67.5458870110976$$
$$x_{52} = 42.4141204421759$$
$$x_{52} = -34.037184713218$$
$$x_{52} = 36.1313906251304$$
$$x_{52} = -84.3007208972085$$
$$x_{52} = -27.7547382346962$$
$$x_{52} = 38.225617263561$$
$$x_{52} = 54.9798923539233$$
$$x_{52} = -21.4727239072797$$
$$x_{52} = -65.4515445438056$$
$$x_{52} = 40.3198613996847$$
$$x_{52} = -71.7345811655909$$
$$x_{52} = 44.5083922872857$$
$$x_{52} = 82.2063593736386$$
$$x_{52} = 71.7345811655909$$
$$x_{52} = -78.0176417347899$$
Decrece en los intervalos
$$\left[100.008477152089, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -106.291596785411\right]$$
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
$$\frac{3 \left(- 3 x \sin{\left(3 x \right)} + 2 \cos{\left(3 x \right)}\right)}{5} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 83.7784565381466$$
$$x_{2} = 61.7882518935787$$
$$x_{3} = 68.0711052836269$$
$$x_{4} = 52.3641211184374$$
$$x_{5} = -13.6298592553469$$
$$x_{6} = -59.6939829539286$$
$$x_{7} = -61.7882518935787$$
$$x_{8} = -90.0614568087362$$
$$x_{9} = 26.1884224615332$$
$$x_{10} = -55.5054736300684$$
$$x_{11} = 72.2597062722654$$
$$x_{12} = 65.9768137973912$$
$$x_{13} = -28.2821897477697$$
$$x_{14} = -81.6841294396421$$
$$x_{15} = 56.5525970612491$$
$$x_{16} = 46.0815142886463$$
$$x_{17} = 59.6939829539286$$
$$x_{18} = -15.7220892009256$$
$$x_{19} = -17.8148265565878$$
$$x_{20} = 12.5840132115367$$
$$x_{21} = -72.2597062722654$$
$$x_{22} = -2.19277791090745$$
$$x_{23} = -33.5169509084747$$
$$x_{24} = -24.0947641678942$$
$$x_{25} = 1.2145323891418$$
$$x_{26} = -11.5384110184102$$
$$x_{27} = 54.4583530486096$$
$$x_{28} = 41.8932060932537$$
$$x_{29} = -57.5997231867839$$
$$x_{30} = -95.2973090043489$$
$$x_{31} = 22.0012459236092$$
$$x_{32} = -0.358957995437268$$
$$x_{33} = 8.40396768807019$$
$$x_{34} = -70.1654029544622$$
$$x_{35} = -46.0815142886463$$
$$x_{36} = -3.20985344776581$$
$$x_{37} = -26.1884224615332$$
$$x_{38} = 96.3444812114328$$
$$x_{39} = 6.31822725550968$$
$$x_{40} = -22.0012459236092$$
$$x_{41} = 48.1756998057147$$
$$x_{42} = -77.4954862683366$$
$$x_{43} = -37.7050049352483$$
$$x_{44} = -53.4112354844189$$
$$x_{45} = 39.7990900239077$$
$$x_{46} = 4.24076625725554$$
$$x_{47} = 100.533175319193$$
$$x_{48} = 43.9873487211332$$
$$x_{49} = 28.2821897477697$$
$$x_{50} = 15.7220892009256$$
$$x_{51} = -85.8727869534015$$
$$x_{52} = -50.2699027802066$$
$$x_{53} = 94.2501373603978$$
$$x_{54} = -79.5898059196778$$
$$x_{55} = 98.4388272431212$$
$$x_{56} = 85.8727869534015$$
$$x_{57} = -1.2145323891418$$
$$x_{58} = -41.8932060932537$$
$$x_{59} = 32.4699670568908$$
$$x_{60} = 105.769053656653$$
$$x_{61} = -9.44825895659546$$
$$x_{62} = 10.493124973438$$
$$x_{63} = -19.9079118108102$$
$$x_{64} = -87.967120448543$$
$$x_{65} = -94.2501373603978$$
$$x_{66} = -65.9768137973912$$
$$x_{67} = 90.0614568087362$$
$$x_{68} = 30.3760435170464$$
$$x_{69} = 70.1654029544622$$
$$x_{70} = 37.7050049352483$$
$$x_{71} = -39.7990900239077$$
$$x_{72} = 74.3540147599617$$
$$x_{73} = 19.9079118108102$$
$$x_{74} = 87.967120448543$$
$$x_{75} = 17.8148265565878$$
$$x_{76} = -43.9873487211332$$
$$x_{77} = -35.6109562885689$$
$$x_{78} = -48.1756998057147$$
$$x_{79} = 2.19277791090745$$
$$x_{80} = 34.5639476991297$$
$$x_{81} = -63.8825291038655$$
$$x_{82} = 78.5426455910908$$
$$x_{83} = 50.2699027802066$$
$$x_{84} = 63.8825291038655$$
$$x_{85} = 0.358957995437268$$
$$x_{86} = -99.4860010336778$$
$$x_{87} = -68.0711052836269$$
$$x_{88} = 92.1557958386718$$
$$x_{89} = 24.0947641678942$$
$$x_{90} = -4.24076625725554$$
$$x_{91} = -30.3760435170464$$
$$x_{92} = -92.1557958386718$$
$$x_{93} = 76.4483279927407$$
$$x_{94} = -83.7784565381466$$
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
$$\left[105.769053656653, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -94.2501373603978\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{3 \left(- 3 x \sin{\left(3 x \right)} + 2 \cos{\left(3 x \right)}\right)}{5} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 83.7784565381466$$
$$x_{2} = 61.7882518935787$$
$$x_{3} = 68.0711052836269$$
$$x_{4} = 52.3641211184374$$
$$x_{5} = -13.6298592553469$$
$$x_{6} = -59.6939829539286$$
$$x_{7} = -61.7882518935787$$
$$x_{8} = -90.0614568087362$$
$$x_{9} = 26.1884224615332$$
$$x_{10} = -55.5054736300684$$
$$x_{11} = 72.2597062722654$$
$$x_{12} = 65.9768137973912$$
$$x_{13} = -28.2821897477697$$
$$x_{14} = -81.6841294396421$$
$$x_{15} = 56.5525970612491$$
$$x_{16} = 46.0815142886463$$
$$x_{17} = 59.6939829539286$$
$$x_{18} = -15.7220892009256$$
$$x_{19} = -17.8148265565878$$
$$x_{20} = 12.5840132115367$$
$$x_{21} = -72.2597062722654$$
$$x_{22} = -2.19277791090745$$
$$x_{23} = -33.5169509084747$$
$$x_{24} = -24.0947641678942$$
$$x_{25} = 1.2145323891418$$
$$x_{26} = -11.5384110184102$$
$$x_{27} = 54.4583530486096$$
$$x_{28} = 41.8932060932537$$
$$x_{29} = -57.5997231867839$$
$$x_{30} = -95.2973090043489$$
$$x_{31} = 22.0012459236092$$
$$x_{32} = -0.358957995437268$$
$$x_{33} = 8.40396768807019$$
$$x_{34} = -70.1654029544622$$
$$x_{35} = -46.0815142886463$$
$$x_{36} = -3.20985344776581$$
$$x_{37} = -26.1884224615332$$
$$x_{38} = 96.3444812114328$$
$$x_{39} = 6.31822725550968$$
$$x_{40} = -22.0012459236092$$
$$x_{41} = 48.1756998057147$$
$$x_{42} = -77.4954862683366$$
$$x_{43} = -37.7050049352483$$
$$x_{44} = -53.4112354844189$$
$$x_{45} = 39.7990900239077$$
$$x_{46} = 4.24076625725554$$
$$x_{47} = 100.533175319193$$
$$x_{48} = 43.9873487211332$$
$$x_{49} = 28.2821897477697$$
$$x_{50} = 15.7220892009256$$
$$x_{51} = -85.8727869534015$$
$$x_{52} = -50.2699027802066$$
$$x_{53} = 94.2501373603978$$
$$x_{54} = -79.5898059196778$$
$$x_{55} = 98.4388272431212$$
$$x_{56} = 85.8727869534015$$
$$x_{57} = -1.2145323891418$$
$$x_{58} = -41.8932060932537$$
$$x_{59} = 32.4699670568908$$
$$x_{60} = 105.769053656653$$
$$x_{61} = -9.44825895659546$$
$$x_{62} = 10.493124973438$$
$$x_{63} = -19.9079118108102$$
$$x_{64} = -87.967120448543$$
$$x_{65} = -94.2501373603978$$
$$x_{66} = -65.9768137973912$$
$$x_{67} = 90.0614568087362$$
$$x_{68} = 30.3760435170464$$
$$x_{69} = 70.1654029544622$$
$$x_{70} = 37.7050049352483$$
$$x_{71} = -39.7990900239077$$
$$x_{72} = 74.3540147599617$$
$$x_{73} = 19.9079118108102$$
$$x_{74} = 87.967120448543$$
$$x_{75} = 17.8148265565878$$
$$x_{76} = -43.9873487211332$$
$$x_{77} = -35.6109562885689$$
$$x_{78} = -48.1756998057147$$
$$x_{79} = 2.19277791090745$$
$$x_{80} = 34.5639476991297$$
$$x_{81} = -63.8825291038655$$
$$x_{82} = 78.5426455910908$$
$$x_{83} = 50.2699027802066$$
$$x_{84} = 63.8825291038655$$
$$x_{85} = 0.358957995437268$$
$$x_{86} = -99.4860010336778$$
$$x_{87} = -68.0711052836269$$
$$x_{88} = 92.1557958386718$$
$$x_{89} = 24.0947641678942$$
$$x_{90} = -4.24076625725554$$
$$x_{91} = -30.3760435170464$$
$$x_{92} = -92.1557958386718$$
$$x_{93} = 76.4483279927407$$
$$x_{94} = -83.7784565381466$$
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
$$\left[105.769053656653, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -94.2501373603978\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{x \sin{\left(3 x \right)}}{5}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\frac{x \sin{\left(3 x \right)}}{5}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty}\left(\frac{x \sin{\left(3 x \right)}}{5}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\frac{x \sin{\left(3 x \right)}}{5}\right) = \left\langle -\infty, \infty\right\rangle$$
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 (x*sin(3*x))/5, dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(3 x \right)}}{5}\right) = \left\langle - \frac{1}{5}, \frac{1}{5}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle - \frac{1}{5}, \frac{1}{5}\right\rangle x$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(3 x \right)}}{5}\right) = \left\langle - \frac{1}{5}, \frac{1}{5}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle - \frac{1}{5}, \frac{1}{5}\right\rangle x$$
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(3 x \right)}}{5}\right) = \left\langle - \frac{1}{5}, \frac{1}{5}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle - \frac{1}{5}, \frac{1}{5}\right\rangle x$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(3 x \right)}}{5}\right) = \left\langle - \frac{1}{5}, \frac{1}{5}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle - \frac{1}{5}, \frac{1}{5}\right\rangle x$$
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:
$$\frac{x \sin{\left(3 x \right)}}{5} = \frac{x \sin{\left(3 x \right)}}{5}$$
- Sí
$$\frac{x \sin{\left(3 x \right)}}{5} = - \frac{x \sin{\left(3 x \right)}}{5}$$
- No
es decir, función
es
par
Pues, comprobamos:
$$\frac{x \sin{\left(3 x \right)}}{5} = \frac{x \sin{\left(3 x \right)}}{5}$$
- Sí
$$\frac{x \sin{\left(3 x \right)}}{5} = - \frac{x \sin{\left(3 x \right)}}{5}$$
- No
es decir, función
es
par