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Trazado el gráfico de la función/ 2^(3*x+1)-3^x-6^(3*x-1)

Gráfico de la función y = 2^(3*x+1)-3^x-6^(3*x-1)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        3*x + 1    x    3*x - 1
f(x) = 2        - 3  - 6
f(x)=−63x−1+(23x+1−3x)f{\left(x \right)} = - 6^{3 x - 1} + \left(2^{3 x + 1} - 3^{x}\right) 
f = -6^(3*x - 1) + 2^(3*x + 1) - 3^x
Gráfico de la función
02468-8-6-4-2-1010-5e225e22
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:
−63x−1+(23x+1−3x)=0- 6^{3 x - 1} + \left(2^{3 x + 1} - 3^{x}\right) = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−76.9855570613729x_{1} = -76.9855570613729
x2=−110.985557061373x_{2} = -110.985557061373
x3=0.662111936702832x_{3} = 0.662111936702832
x4=−60.9855570613729x_{4} = -60.9855570613729
x5=−96.9855570613729x_{5} = -96.9855570613729
x6=−48.9855570613734x_{6} = -48.9855570613734
x7=−100.985557061373x_{7} = -100.985557061373
x8=−38.985557070945x_{8} = -38.985557070945
x9=−98.9855570613729x_{9} = -98.9855570613729
x10=−78.9855570613729x_{10} = -78.9855570613729
x11=−74.9855570613729x_{11} = -74.9855570613729
x12=−56.9855570613729x_{12} = -56.9855570613729
x13=−118.985557061373x_{13} = -118.985557061373
x14=−112.985557061373x_{14} = -112.985557061373
x15=−50.985557061373x_{15} = -50.985557061373
x16=−84.9855570613729x_{16} = -84.9855570613729
x17=−0.698149538775665x_{17} = -0.698149538775665
x18=−46.9855570613766x_{18} = -46.9855570613766
x19=−26.986795840981x_{19} = -26.986795840981
x20=−114.985557061373x_{20} = -114.985557061373
x21=−92.9855570613729x_{21} = -92.9855570613729
x22=−80.9855570613729x_{22} = -80.9855570613729
x23=−36.9855571294411x_{23} = -36.9855571294411
x24=−104.985557061373x_{24} = -104.985557061373
x25=−28.9857311397934x_{25} = -28.9857311397934
x26=−102.985557061373x_{26} = -102.985557061373
x27=−58.9855570613729x_{27} = -58.9855570613729
x28=−66.9855570613729x_{28} = -66.9855570613729
x29=−106.985557061373x_{29} = -106.985557061373
x30=−70.9855570613729x_{30} = -70.9855570613729
x31=−64.9855570613729x_{31} = -64.9855570613729
x32=−82.9855570613729x_{32} = -82.9855570613729
x33=−44.9855570613995x_{33} = -44.9855570613995
x34=−54.9855570613729x_{34} = -54.9855570613729
x35=−88.9855570613729x_{35} = -88.9855570613729
x36=−108.985557061373x_{36} = -108.985557061373
x37=−62.9855570613729x_{37} = -62.9855570613729
x38=−72.9855570613729x_{38} = -72.9855570613729
x39=−90.9855570613729x_{39} = -90.9855570613729
x40=−86.9855570613729x_{40} = -86.9855570613729
x41=−40.9855570627189x_{41} = -40.9855570627189
x42=−30.9855815386827x_{42} = -30.9855815386827
x43=−116.985557061373x_{43} = -116.985557061373
x44=−34.9855575454134x_{44} = -34.9855575454134
x45=−52.9855570613729x_{45} = -52.9855570613729
x46=−32.9855605034458x_{46} = -32.9855605034458
x47=−24.9944115035602x_{47} = -24.9944115035602
x48=−42.9855570615622x_{48} = -42.9855570615622
x49=−68.9855570613729x_{49} = -68.9855570613729
x50=−94.9855570613729x_{50} = -94.9855570613729
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en 2^(3*x + 1) - 3^x - 6^(3*x - 1).
−16+(−30+20⋅3+1)- \frac{1}{6} + \left(- 3^{0} + 2^{0 \cdot 3 + 1}\right)
Resultado:
f(0)=56f{\left(0 \right)} = \frac{5}{6}
Punto:
(0, 5/6)
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
3⋅23x+1log⁡(2)−3xlog⁡(3)−3⋅63x−1log⁡(6)=03 \cdot 2^{3 x + 1} \log{\left(2 \right)} - 3^{x} \log{\left(3 \right)} - 3 \cdot 6^{3 x - 1} \log{\left(6 \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−64.9855570613729x_{1} = -64.9855570613729
x2=−100.985557061373x_{2} = -100.985557061373
x3=−38.9855570794908x_{3} = -38.9855570794908
x4=−98.9855570613729x_{4} = -98.9855570613729
x5=−108.985557061373x_{5} = -108.985557061373
x6=−58.9855570613729x_{6} = -58.9855570613729
x7=−92.9855570613729x_{7} = -92.9855570613729
x8=−104.985557061373x_{8} = -104.985557061373
x9=−50.985557061373x_{9} = -50.985557061373
x10=−70.9855570613729x_{10} = -70.9855570613729
x11=−68.9855570613729x_{11} = -68.9855570613729
x12=−86.9855570613729x_{12} = -86.9855570613729
x13=0.406588491722199x_{13} = 0.406588491722199
x14=−54.9855570613729x_{14} = -54.9855570613729
x15=−42.9855570617312x_{15} = -42.9855570617312
x16=−44.9855570614233x_{16} = -44.9855570614233
x17=−60.9855570613729x_{17} = -60.9855570613729
x18=−36.9855571902116x_{18} = -36.9855571902116
x19=−114.985557061373x_{19} = -114.985557061373
x20=−74.9855570613729x_{20} = -74.9855570613729
x21=−1.35468731928638x_{21} = -1.35468731928638
x22=−52.9855570613729x_{22} = -52.9855570613729
x23=−76.9855570613729x_{23} = -76.9855570613729
x24=−66.9855570613729x_{24} = -66.9855570613729
x25=−48.9855570613739x_{25} = -48.9855570613739
x26=−28.98588658966x_{26} = -28.98588658966
x27=−34.9855579775599x_{27} = -34.9855579775599
x28=−82.9855570613729x_{28} = -82.9855570613729
x29=−96.9855570613729x_{29} = -96.9855570613729
x30=−40.9855570639207x_{30} = -40.9855570639207
x31=−110.985557061373x_{31} = -110.985557061373
x32=−26.987903561424x_{32} = -26.987903561424
x33=−118.985557061373x_{33} = -118.985557061373
x34=−30.9856033924444x_{34} = -30.9856033924444
x35=−94.9855570613729x_{35} = -94.9855570613729
x36=−32.985563576505x_{36} = -32.985563576505
x37=−84.9855570613729x_{37} = -84.9855570613729
x38=−80.9855570613729x_{38} = -80.9855570613729
x39=−78.9855570613729x_{39} = -78.9855570613729
x40=−62.9855570613729x_{40} = -62.9855570613729
x41=−116.985557061373x_{41} = -116.985557061373
x42=−90.9855570613729x_{42} = -90.9855570613729
x43=−88.9855570613729x_{43} = -88.9855570613729
x44=−106.985557061373x_{44} = -106.985557061373
x45=−102.985557061373x_{45} = -102.985557061373
x46=−72.9855570613729x_{46} = -72.9855570613729
x47=−112.985557061373x_{47} = -112.985557061373
x48=−56.9855570613729x_{48} = -56.9855570613729
x49=−46.98555706138x_{49} = -46.98555706138
Signos de extremos en los puntos:
(-64.98555706137287, -9.86301006293521e-32)

(-100.98555706137287, -6.57119426248568e-49)

(-38.98555707949082, -2.50704477444229e-19)

(-98.98555706137287, -5.91407483623711e-48)

(-108.98555706137287, -1.00155376657303e-52)

(-58.98555706137287, -7.19013433587977e-29)

(-92.98555706137287, -4.31136055561685e-45)

(-104.98555706137287, -8.11258550924158e-51)

(-50.985557061373015, -4.71744713776998e-25)

(-70.98555706137287, -1.35295062591704e-34)

(-68.98555706137287, -1.21765556332533e-33)

(-86.98555706137287, -3.14298184504469e-42)

(0.4065884917221987, 1.61250327852661)

(-54.98555706137288, -5.82400881206257e-27)

(-42.985557061731164, -3.09511706587306e-21)

(-44.98555706142326, -3.43901896324449e-22)

(-60.98555706137287, -7.98903815097752e-30)

(-36.9855571902116, -2.25634002253862e-18)

(-114.98555706137287, -1.37387347952405e-55)

(-74.98555706137287, -1.67030941471239e-36)

(-1.3546873192863802, -0.106305273005093)

(-52.985557061372894, -5.24160793085623e-26)

(-76.98555706137287, -1.85589934968044e-37)

(-66.98555706137287, -1.0958900069928e-32)

(-48.985557061373875, -4.24570242398897e-24)

(-28.98588658966001, -1.47984906071999e-14)

(-34.98555797755987, -2.03070426374413e-17)

(-82.98555706137287, -2.5458152944862e-40)

(-96.98555706137287, -5.3226673526134e-47)

(-40.98555706392071, -2.78560535258509e-20)

(-110.98555706137287, -1.11283751841448e-53)

(-26.987903561423952, -1.32891618422772e-13)

(-118.98555706137287, -1.69614009817784e-57)

(-30.98560339244445, -1.64478838758211e-15)

(-94.98555706137287, -4.79040061735206e-46)

(-32.98556357650499, -1.82762259550076e-16)

(-84.98555706137287, -2.82868366054022e-41)

(-80.98555706137287, -2.29123376503758e-39)

(-78.98555706137287, -2.06211038853382e-38)

(-62.98555706137287, -8.87670905664169e-31)

(-116.98555706137287, -1.52652608836006e-56)

(-90.98555706137287, -3.88022450005517e-44)

(-88.98555706137287, -3.49220205004965e-43)

(-106.98555706137287, -9.01398389915731e-52)

(-102.98555706137287, -7.30132695831742e-50)

(-72.98555706137287, -1.50327847324115e-35)

(-112.98555706137287, -1.23648613157165e-54)

(-56.98555706137287, -6.47112090229179e-28)

(-46.98555706137996, -3.82113218156454e-23)


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:
x1=−1.35468731928638x_{1} = -1.35468731928638
Puntos máximos de la función:
x1=0.406588491722199x_{1} = 0.406588491722199
Decrece en los intervalos
[−1.35468731928638,0.406588491722199]\left[-1.35468731928638, 0.406588491722199\right]
Crece en los intervalos
(−∞,−1.35468731928638]∪[0.406588491722199,∞)\left(-\infty, -1.35468731928638\right] \cup \left[0.406588491722199, \infty\right)
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
18⋅23xlog⁡(2)2−3xlog⁡(3)2−3⋅63xlog⁡(6)22=018 \cdot 2^{3 x} \log{\left(2 \right)}^{2} - 3^{x} \log{\left(3 \right)}^{2} - \frac{3 \cdot 6^{3 x} \log{\left(6 \right)}^{2}}{2} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−76.9855570613729x_{1} = -76.9855570613729
x2=−2.00697879867335x_{2} = -2.00697879867335
x3=−34.9855587955227x_{3} = -34.9855587955227
x4=−110.985557061373x_{4} = -110.985557061373
x5=−60.9855570613729x_{5} = -60.9855570613729
x6=−96.9855570613729x_{6} = -96.9855570613729
x7=−100.985557061373x_{7} = -100.985557061373
x8=−98.9855570613729x_{8} = -98.9855570613729
x9=−78.9855570613729x_{9} = -78.9855570613729
x10=−32.9855693931933x_{10} = -32.9855693931933
x11=−74.9855570613729x_{11} = -74.9855570613729
x12=−54.9855570613729x_{12} = -54.9855570613729
x13=−56.9855570613729x_{13} = -56.9855570613729
x14=−118.985557061373x_{14} = -118.985557061373
x15=−112.985557061373x_{15} = -112.985557061373
x16=−28.9861809127146x_{16} = -28.9861809127146
x17=−30.9856447587725x_{17} = -30.9856447587725
x18=0.138212478149236x_{18} = 0.138212478149236
x19=−44.9855570614682x_{19} = -44.9855570614682
x20=−84.9855570613729x_{20} = -84.9855570613729
x21=−50.9855570613731x_{21} = -50.9855570613731
x22=−26.9900047867738x_{22} = -26.9900047867738
x23=−36.9855573052375x_{23} = -36.9855573052375
x24=−42.985557062051x_{24} = -42.985557062051
x25=−114.985557061373x_{25} = -114.985557061373
x26=−92.9855570613729x_{26} = -92.9855570613729
x27=−80.9855570613729x_{27} = -80.9855570613729
x28=−48.9855570613748x_{28} = -48.9855570613748
x29=−52.9855570613729x_{29} = -52.9855570613729
x30=−104.985557061373x_{30} = -104.985557061373
x31=−46.9855570613863x_{31} = -46.9855570613863
x32=−94.9855570613729x_{32} = -94.9855570613729
x33=−102.985557061373x_{33} = -102.985557061373
x34=−58.9855570613729x_{34} = -58.9855570613729
x35=−66.9855570613729x_{35} = -66.9855570613729
x36=−40.9855570661954x_{36} = -40.9855570661954
x37=−70.9855570613729x_{37} = -70.9855570613729
x38=−106.985557061373x_{38} = -106.985557061373
x39=−38.9855570956663x_{39} = -38.9855570956663
x40=−82.9855570613729x_{40} = -82.9855570613729
x41=−88.9855570613729x_{41} = -88.9855570613729
x42=−108.985557061373x_{42} = -108.985557061373
x43=−62.9855570613729x_{43} = -62.9855570613729
x44=−72.9855570613729x_{44} = -72.9855570613729
x45=−90.9855570613729x_{45} = -90.9855570613729
x46=−86.9855570613729x_{46} = -86.9855570613729
x47=−116.985557061373x_{47} = -116.985557061373
x48=−68.9855570613729x_{48} = -68.9855570613729
x49=−64.9855570613729x_{49} = -64.9855570613729

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
[−2.00697879867335,0.138212478149236]\left[-2.00697879867335, 0.138212478149236\right]
Convexa en los intervalos
(−∞,−2.00697879867335]∪[0.138212478149236,∞)\left(-\infty, -2.00697879867335\right] \cup \left[0.138212478149236, \infty\right)
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(−63x−1+(23x+1−3x))=0\lim_{x \to -\infty}\left(- 6^{3 x - 1} + \left(2^{3 x + 1} - 3^{x}\right)\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(−63x−1+(23x+1−3x))=−∞\lim_{x \to \infty}\left(- 6^{3 x - 1} + \left(2^{3 x + 1} - 3^{x}\right)\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función 2^(3*x + 1) - 3^x - 6^(3*x - 1), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(−63x−1+(23x+1−3x)x)=0\lim_{x \to -\infty}\left(\frac{- 6^{3 x - 1} + \left(2^{3 x + 1} - 3^{x}\right)}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(−63x−1+(23x+1−3x)x)=−∞\lim_{x \to \infty}\left(\frac{- 6^{3 x - 1} + \left(2^{3 x + 1} - 3^{x}\right)}{x}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la derecha
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:
−63x−1+(23x+1−3x)=21−3x−6−3x−1−3−x- 6^{3 x - 1} + \left(2^{3 x + 1} - 3^{x}\right) = 2^{1 - 3 x} - 6^{- 3 x - 1} - 3^{- x}
- No
−63x−1+(23x+1−3x)=−21−3x+6−3x−1+3−x- 6^{3 x - 1} + \left(2^{3 x + 1} - 3^{x}\right) = - 2^{1 - 3 x} + 6^{- 3 x - 1} + 3^{- x}
- No
es decir, función
no es
par ni impar
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