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Trazado el gráfico de la función/ (x-1)(x+2)/(x-3)^(x+4)

Gráfico de la función y = (x-1)(x+2)/(x-3)^(x+4)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       (x - 1)*(x + 2)
f(x) = ---------------
                x + 4 
         (x - 3)
f(x)=(x−1)(x+2)(x−3)x+4f{\left(x \right)} = \frac{\left(x - 1\right) \left(x + 2\right)}{\left(x - 3\right)^{x + 4}} 
f = ((x - 1)*(x + 2))/(x - 3)^(x + 4)
Gráfico de la función
02468-8-6-4-2-101005000000000
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=3x_{1} = 3
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:
(x−1)(x+2)(x−3)x+4=0\frac{\left(x - 1\right) \left(x + 2\right)}{\left(x - 3\right)^{x + 4}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−2x_{1} = -2
x2=1x_{2} = 1
Solución numérica
x1=25.129395707092x_{1} = 25.129395707092
x2=101.916056883239x_{2} = 101.916056883239
x3=38.685611913773x_{3} = 38.685611913773
x4=13.7238011127816x_{4} = 13.7238011127816
x5=44.5547890310757x_{5} = 44.5547890310757
x6=40.6391696990262x_{6} = 40.6391696990262
x7=84.0476741265208x_{7} = 84.0476741265208
x8=46.5162961986722x_{8} = 46.5162961986722
x9=99.9290850253909x_{9} = 99.9290850253909
x10=88.0153372103801x_{10} = 88.0153372103801
x11=60.2976298652206x_{11} = 60.2976298652206
x12=58.3244529622891x_{12} = 58.3244529622891
x13=54.3820140974427x_{13} = 54.3820140974427
x14=76.1190224147724x_{14} = 76.1190224147724
x15=28.975870133988x_{15} = 28.975870133988
x16=103.903366540475x_{16} = 103.903366540475
x17=70.1795179820527x_{17} = 70.1795179820527
x18=86.0312531029396x_{18} = 86.0312531029396
x19=15.636503227009x_{19} = 15.636503227009
x20=1x_{20} = 1
x21=21.3126407828175x_{21} = 21.3126407828175
x22=32.846239787784x_{22} = 32.846239787784
x23=64.2474179630578x_{23} = 64.2474179630578
x24=36.7353298981528x_{24} = 36.7353298981528
x25=56.3525453345966x_{25} = 56.3525453345966
x26=80.0821490697791x_{26} = 80.0821490697791
x27=48.4799575834948x_{27} = 48.4799575834948
x28=50.4455763156202x_{28} = 50.4455763156202
x29=23.2169822773919x_{29} = 23.2169822773919
x30=91.9849162392321x_{30} = 91.9849162392321
x31=82.0646290623517x_{31} = 82.0646290623517
x32=42.5956609359943x_{32} = 42.5956609359943
x33=78.1002679449919x_{33} = 78.1002679449919
x34=11.7040783570685x_{34} = 11.7040783570685
x35=30.9084203277257x_{35} = 30.9084203277257
x36=89.9998998172416x_{36} = 89.9998998172416
x37=−2x_{37} = -2
x38=74.138452470952x_{38} = 74.138452470952
x39=97.9424671271873x_{39} = 97.9424671271873
x40=19.4163965950949x_{40} = 19.4163965950949
x41=17.5265666277313x_{41} = 17.5265666277313
x42=95.9562204701949x_{42} = 95.9562204701949
x43=62.2719802056795x_{43} = 62.2719802056795
x44=68.2012534702347x_{44} = 68.2012534702347
x45=52.4129793690297x_{45} = 52.4129793690297
x46=34.7887196706439x_{46} = 34.7887196706439
x47=72.1586017523229x_{47} = 72.1586017523229
x48=93.9703635597743x_{48} = 93.9703635597743
x49=66.2238656931341x_{49} = 66.2238656931341
x50=27.0492777889211x_{50} = 27.0492777889211
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 ((x - 1)*(x + 2))/(x - 3)^(x + 4).
(−1)2(−3)4\frac{\left(-1\right) 2}{\left(-3\right)^{4}}
Resultado:
f(0)=−281f{\left(0 \right)} = - \frac{2}{81}
Punto:
(0, -2/81)
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
−(x−3)−2x−8(x−3)x+4(x−1)(x+2)(log⁡(x−3)+x+4x−3)+(x−3)−x−4(2x+1)=0- \left(x - 3\right)^{- 2 x - 8} \left(x - 3\right)^{x + 4} \left(x - 1\right) \left(x + 2\right) \left(\log{\left(x - 3 \right)} + \frac{x + 4}{x - 3}\right) + \left(x - 3\right)^{- x - 4} \left(2 x + 1\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=74.1403165534919x_{1} = 74.1403165534919
x2=82.066222308221x_{2} = 82.066222308221
x3=86.0327346319684x_{3} = 86.0327346319684
x4=72.1605457219322x_{4} = 72.1605457219322
x5=68.2033754234951x_{5} = 68.2033754234951
x6=56.3554035756928x_{6} = 56.3554035756928
x7=40.6439441303037x_{7} = 40.6439441303037
x8=48.4835768318452x_{8} = 48.4835768318452
x9=99.9302627392761x_{9} = 99.9302627392761
x10=70.1815477042959x_{10} = 70.1815477042959
x11=13.7314823252768x_{11} = 13.7314823252768
x12=15.6501286975936x_{12} = 15.6501286975936
x13=78.1019877473629x_{13} = 78.1019877473629
x14=84.0492098151872x_{14} = 84.0492098151872
x15=25.1392882214227x_{15} = 25.1392882214227
x16=76.1208119346711x_{16} = 76.1208119346711
x17=90.0012821569151x_{17} = 90.0012821569151
x18=50.4489765807413x_{18} = 50.4489765807413
x19=38.6907692597463x_{19} = 38.6907692597463
x20=19.4298401219163x_{20} = 19.4298401219163
x21=28.9839150400384x_{21} = 28.9839150400384
x22=27.0581829418993x_{22} = 27.0581829418993
x23=36.7409211208176x_{23} = 36.7409211208176
x24=32.8528896924642x_{24} = 32.8528896924642
x25=95.9574735157058x_{25} = 95.9574735157058
x26=52.416181474456x_{26} = 52.416181474456
x27=103.904476242198x_{27} = 103.904476242198
x28=93.9716573655828x_{28} = 93.9716573655828
x29=17.5408488720846x_{29} = 17.5408488720846
x30=46.520158331053x_{30} = 46.520158331053
x31=64.2497466958954x_{31} = 64.2497466958954
x32=66.2260870520326x_{32} = 66.2260870520326
x33=23.2279928592706x_{33} = 23.2279928592706
x34=34.7948048009259x_{34} = 34.7948048009259
x35=54.3850362648255x_{35} = 54.3850362648255
x36=97.9436815252634x_{36} = 97.9436815252634
x37=58.3271614034562x_{37} = 58.3271614034562
x38=91.9862530826948x_{38} = 91.9862530826948
x39=21.3248730357537x_{39} = 21.3248730357537
x40=80.0838035739241x_{40} = 80.0838035739241
x41=44.558921604036x_{41} = 44.558921604036
x42=30.9157186366283x_{42} = 30.9157186366283
x43=42.6000958675282x_{43} = 42.6000958675282
x44=60.3002010171713x_{44} = 60.3002010171713
x45=101.917199739831x_{45} = 101.917199739831
x46=11.6849844809559x_{46} = 11.6849844809559
x47=62.2744251887052x_{47} = 62.2744251887052
x48=88.0167677048301x_{48} = 88.0167677048301
Signos de extremos en los puntos:
(74.14031655349194, 1.04919721384824e-141)

(82.06622230822104, 3.02375814289022e-160)

(86.03273463196841, 1.1993990319105e-169)

(72.16054572193221, 3.96087846997681e-137)

(68.20337542349509, 4.75691584488604e-128)

(56.355403575692776, 1.83946344894248e-101)

(40.6439441303037, 7.63465311273189e-68)

(48.483576831845234, 2.34182116880967e-84)

(99.93026273927615, 3.55128877791139e-203)

(70.1815477042959, 1.41313310051309e-132)

(13.731482325276827, 1.06297758070912e-16)

(15.650128697593637, 5.70534396098732e-20)

(78.10198774736287, 6.25169812519158e-151)

(84.04920981518721, 6.16964434832579e-165)

(25.139288221422664, 4.16113196450975e-37)

(76.12081193467111, 2.63055997779423e-146)

(90.0012821569151, 3.93814759460495e-179)

(50.44897658074128, 1.39655436359407e-88)

(38.690769259746276, 8.05239080864271e-64)

(19.429840121916264, 1.30150475057789e-26)

(28.983915040038415, 1.88603107639648e-44)

(27.058182941899325, 9.54331227090084e-41)

(36.74092112081756, 7.63509687555102e-60)

(32.85288969246422, 4.87390891160736e-52)

(95.9574735157058, 1.67432508300353e-193)

(52.416181474455996, 7.67709028435055e-93)

(103.9044762421975, 6.40387623634888e-213)

(93.97165736558277, 1.07948259524076e-188)

(17.540848872084574, 2.92638110623919e-23)

(46.520158331053, 3.60764786139785e-80)

(64.24974669589538, 4.49601162020611e-119)

(66.22608705203255, 1.50824226579431e-123)

(23.227992859270568, 1.54992135625485e-33)

(34.7948048009259, 6.47175933370676e-56)

(54.38503626482549, 3.90231693663052e-97)

(97.94368152526344, 2.48945226075914e-198)

(58.32716140345617, 8.06252436165412e-106)

(91.98625308269479, 6.66556449031336e-184)

(21.32487303575374, 4.89209426203078e-30)

(80.08380357392414, 1.41028646339702e-155)

(44.558921604036, 5.08731531752322e-76)

(30.91571863662828, 3.23930958515106e-48)

(42.60009586752822, 6.54082151025351e-72)

(60.30020101717132, 3.29421572478862e-110)

(101.91719973983122, 4.86462575989279e-208)

(11.684984480955926, 2.75702227842627e-13)

(62.27442518870523, 1.25762978991666e-114)

(88.01676770483007, 2.22408194136399e-174)


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:
La función no tiene puntos mínimos
La función no tiene puntos máximos
Decrece en todo el eje numérico
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
(x−3)−x−4(−(x−1)(x+2)(2−x+4x−3x−3−(log⁡(x−3)+x+4x−3)2)−2(2x+1)(log⁡(x−3)+x+4x−3)+2)=0\left(x - 3\right)^{- x - 4} \left(- \left(x - 1\right) \left(x + 2\right) \left(\frac{2 - \frac{x + 4}{x - 3}}{x - 3} - \left(\log{\left(x - 3 \right)} + \frac{x + 4}{x - 3}\right)^{2}\right) - 2 \left(2 x + 1\right) \left(\log{\left(x - 3 \right)} + \frac{x + 4}{x - 3}\right) + 2\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=74.1421887946616x_{1} = 74.1421887946616
x2=88.0182032625657x_{2} = 88.0182032625657
x3=103.905589143571x_{3} = 103.905589143571
x4=46.524050419812x_{4} = 46.524050419812
x5=68.2055076741785x_{5} = 68.2055076741785
x6=93.972955394772x_{6} = 93.972955394772
x7=44.5630879335167x_{7} = 44.5630879335167
x8=90.0026692563464x_{8} = 90.0026692563464
x9=86.0342215549278x_{9} = 86.0342215549278
x10=91.987594406754x_{10} = 91.987594406754
x11=72.1624984895949x_{11} = 72.1624984895949
x12=99.9314440163957x_{12} = 99.9314440163957
x13=32.8596165823695x_{13} = 32.8596165823695
x14=84.0507512586791x_{14} = 84.0507512586791
x15=82.0678217036397x_{15} = 82.0678217036397
x16=95.9587305471902x_{16} = 95.9587305471902
x17=19.4434587668191x_{17} = 19.4434587668191
x18=76.1226090354505x_{18} = 76.1226090354505
x19=38.6959763615845x_{19} = 38.6959763615845
x20=52.4194050663191x_{20} = 52.4194050663191
x21=36.746569575393x_{21} = 36.746569575393
x22=97.9448996898235x_{22} = 97.9448996898235
x23=42.604569019705x_{23} = 42.604569019705
x24=66.2283195899602x_{24} = 66.2283195899602
x25=56.3582793666291x_{25} = 56.3582793666291
x26=25.1493256320743x_{26} = 25.1493256320743
x27=28.9920654891976x_{27} = 28.9920654891976
x28=62.2768834472712x_{28} = 62.2768834472712
x29=50.4524007517238x_{29} = 50.4524007517238
x30=101.918345971157x_{30} = 101.918345971157
x31=21.3372881687658x_{31} = 21.3372881687658
x32=80.0854646602727x_{32} = 80.0854646602727
x33=27.0672120358277x_{33} = 27.0672120358277
x34=60.3027866953792x_{34} = 60.3027866953792
x35=30.9231069187223x_{35} = 30.9231069187223
x36=70.1835869342035x_{36} = 70.1835869342035
x37=15.6634905177281x_{37} = 15.6634905177281
x38=64.2520875950174x_{38} = 64.2520875950174
x39=58.3298857863275x_{39} = 58.3298857863275
x40=40.6487620551517x_{40} = 40.6487620551517
x41=23.2391700386789x_{41} = 23.2391700386789
x42=78.1037146072358x_{42} = 78.1037146072358
x43=48.4872227841919x_{43} = 48.4872227841919
x44=17.555218049996x_{44} = 17.555218049996
x45=54.3880778166341x_{45} = 54.3880778166341
x46=13.7376308872189x_{46} = 13.7376308872189
x47=34.8009561289646x_{47} = 34.8009561289646
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=3x_{1} = 3

lim⁡x→3−((x−3)−x−4(−(x−1)(x+2)(2−x+4x−3x−3−(log⁡(x−3)+x+4x−3)2)−2(2x+1)(log⁡(x−3)+x+4x−3)+2))=−∞\lim_{x \to 3^-}\left(\left(x - 3\right)^{- x - 4} \left(- \left(x - 1\right) \left(x + 2\right) \left(\frac{2 - \frac{x + 4}{x - 3}}{x - 3} - \left(\log{\left(x - 3 \right)} + \frac{x + 4}{x - 3}\right)^{2}\right) - 2 \left(2 x + 1\right) \left(\log{\left(x - 3 \right)} + \frac{x + 4}{x - 3}\right) + 2\right)\right) = -\infty
lim⁡x→3+((x−3)−x−4(−(x−1)(x+2)(2−x+4x−3x−3−(log⁡(x−3)+x+4x−3)2)−2(2x+1)(log⁡(x−3)+x+4x−3)+2))=∞\lim_{x \to 3^+}\left(\left(x - 3\right)^{- x - 4} \left(- \left(x - 1\right) \left(x + 2\right) \left(\frac{2 - \frac{x + 4}{x - 3}}{x - 3} - \left(\log{\left(x - 3 \right)} + \frac{x + 4}{x - 3}\right)^{2}\right) - 2 \left(2 x + 1\right) \left(\log{\left(x - 3 \right)} + \frac{x + 4}{x - 3}\right) + 2\right)\right) = \infty
- los límites no son iguales, signo
x1=3x_{1} = 3
- es el punto de flexión

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:
No tiene corvaduras en todo el eje numérico
Asíntotas verticales
Hay:
x1=3x_{1} = 3
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((x−1)(x+2)(x−3)x+4)=0\lim_{x \to -\infty}\left(\frac{\left(x - 1\right) \left(x + 2\right)}{\left(x - 3\right)^{x + 4}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞((x−1)(x+2)(x−3)x+4)=0\lim_{x \to \infty}\left(\frac{\left(x - 1\right) \left(x + 2\right)}{\left(x - 3\right)^{x + 4}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función ((x - 1)*(x + 2))/(x - 3)^(x + 4), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((x−3)−x−4(x−1)(x+2)x)=0\lim_{x \to -\infty}\left(\frac{\left(x - 3\right)^{- x - 4} \left(x - 1\right) \left(x + 2\right)}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((x−3)−x−4(x−1)(x+2)x)=0\lim_{x \to \infty}\left(\frac{\left(x - 3\right)^{- x - 4} \left(x - 1\right) \left(x + 2\right)}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
(x−1)(x+2)(x−3)x+4=(2−x)(−x−3)x−4(−x−1)\frac{\left(x - 1\right) \left(x + 2\right)}{\left(x - 3\right)^{x + 4}} = \left(2 - x\right) \left(- x - 3\right)^{x - 4} \left(- x - 1\right)
- No
(x−1)(x+2)(x−3)x+4=−(2−x)(−x−3)x−4(−x−1)\frac{\left(x - 1\right) \left(x + 2\right)}{\left(x - 3\right)^{x + 4}} = - \left(2 - x\right) \left(- x - 3\right)^{x - 4} \left(- x - 1\right)
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор