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Trazado el gráfico de la función/ acot(5*x)^(2)*log(x-4)

Gráfico de la función y = acot(5*x)^(2)*log(x-4)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
           2                
f(x) = acot (5*x)*log(x - 4)
f(x)=log⁡(x−4)acot⁡2(5x)f{\left(x \right)} = \log{\left(x - 4 \right)} \operatorname{acot}^{2}{\left(5 x \right)} 
f = log(x - 4)*acot(5*x)^2
Gráfico de la función
02468-8-6-4-2-10100.010-0.010
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:
log⁡(x−4)acot⁡2(5x)=0\log{\left(x - 4 \right)} \operatorname{acot}^{2}{\left(5 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=5x_{1} = 5
Solución numérica
x1=760141.997003234x_{1} = 760141.997003234
x2=5x_{2} = 5
x3=715189.92165217x_{3} = 715189.92165217
x4=775099.588440047x_{4} = 775099.588440047
x5=745171.400543652x_{5} = 745171.400543652
x6=670111.866690648x_{6} = 670111.866690648
x7=730187.484585667x_{7} = 730187.484585667
x8=655056.156593733x_{8} = 655056.156593733
x9=685152.475825446x_{9} = 685152.475825446
x10=790044.476977954x_{10} = 790044.476977954
x11=804976.953006942x_{11} = 804976.953006942
x12=700178.370583401x_{12} = 700178.370583401
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 acot(5*x)^2*log(x - 4).
log⁡(−4)acot⁡2(0⋅5)\log{\left(-4 \right)} \operatorname{acot}^{2}{\left(0 \cdot 5 \right)}
Resultado:
f(0)=π2(log⁡(4)+iπ)4f{\left(0 \right)} = \frac{\pi^{2} \left(\log{\left(4 \right)} + i \pi\right)}{4}
Punto:
(0, pi^2*(pi*i + log(4))/4)
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
−10log⁡(x−4)acot⁡(5x)25x2+1+acot⁡2(5x)x−4=0- \frac{10 \log{\left(x - 4 \right)} \operatorname{acot}{\left(5 x \right)}}{25 x^{2} + 1} + \frac{\operatorname{acot}^{2}{\left(5 x \right)}}{x - 4} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=9611.84053835698x_{1} = 9611.84053835698
x2=42413.5683484109x_{2} = 42413.5683484109
x3=29493.9398872642x_{3} = 29493.9398872642
x4=48833.3664888115x_{4} = 48833.3664888115
x5=15408.0280671425x_{5} = 15408.0280671425
x6=54165.7543471659x_{6} = 54165.7543471659
x7=26245.9453016179x_{7} = 26245.9453016179
x8=27329.3794864507x_{8} = 27329.3794864507
x9=14337.7301145638x_{9} = 14337.7301145638
x10=49901.0501089827x_{10} = 49901.0501089827
x11=43485.2360945014x_{11} = 43485.2360945014
x12=32734.6864836606x_{12} = 32734.6864836606
x13=40268.0740407141x_{13} = 40268.0740407141
x14=34891.0686096907x_{14} = 34891.0686096907
x15=38119.6029796687x_{15} = 38119.6029796687
x16=31655.262374402x_{16} = 31655.262374402
x17=39194.2178946115x_{17} = 39194.2178946115
x18=56294.6607802403x_{18} = 56294.6607802403
x19=19734.6989562826x_{19} = 19734.6989562826
x20=13278.7443201098x_{20} = 13278.7443201098
x21=45626.4866169943x_{21} = 45626.4866169943
x22=35968.0411974386x_{22} = 35968.0411974386
x23=37044.21520896x_{23} = 37044.21520896
x24=28412.0554959634x_{24} = 28412.0554959634
x25=25161.8043242596x_{25} = 25161.8043242596
x26=12239.4561489534x_{26} = 12239.4561489534
x27=20820.296559896x_{27} = 20820.296559896
x28=18649.7686333039x_{28} = 18649.7686333039
x29=53100.457664144x_{29} = 53100.457664144
x30=7.12721878090273x_{30} = 7.12721878090273
x31=21906.0860007187x_{31} = 21906.0860007187
x32=24077.0325904846x_{32} = 24077.0325904846
x33=33813.286614885x_{33} = 33813.286614885
x34=44556.2039659933x_{34} = 44556.2039659933
x35=46696.0985008394x_{35} = 46696.0985008394
x36=10313.4454621241x_{36} = 10313.4454621241
x37=16485.077701001x_{37} = 16485.077701001
x38=47765.0538198603x_{38} = 47765.0538198603
x39=52034.5829422324x_{39} = 52034.5829422324
x40=41341.1859405396x_{40} = 41341.1859405396
x41=55230.4848734819x_{41} = 55230.4848734819
x42=30575.012374393x_{42} = 30575.012374393
x43=10608.6135636748x_{43} = 10608.6135636748
x44=17566.2079670701x_{44} = 17566.2079670701
x45=50968.1179506774x_{45} = 50968.1179506774
x46=22991.7403596272x_{46} = 22991.7403596272
x47=11237.0503173276x_{47} = 11237.0503173276
Signos de extremos en los puntos:
(9611.840538356983, 3.97037993011726e-9)

(42413.56834841089, 2.36924018742695e-10)

(29493.939887264165, 4.73244893157346e-10)

(48833.366488811494, 1.81089406203148e-10)

(15408.028067142517, 1.6246144312175e-9)

(54165.754347165945, 1.48602512289146e-10)

(26245.945301617925, 5.90846751656658e-10)

(27329.379486450718, 5.47095451511591e-10)

(14337.730114563796, 1.86220699206495e-9)

(49901.0501089827, 1.73770577247695e-10)

(43485.23609450142, 2.25918099752669e-10)

(32734.686483660567, 3.88072390146042e-10)

(40268.074040714106, 2.61562715438744e-10)

(34891.06860969066, 3.43682801235254e-10)

(38119.602979668685, 2.90368151234317e-10)

(31655.262374402013, 4.13651016560067e-10)

(39194.21789461153, 2.75387941797556e-10)

(56294.66078024027, 1.38062193845164e-10)

(19734.698956282617, 1.01576265643813e-9)

(13278.74432010984, 2.153663032117e-9)

(45626.48661699426, 2.06134657201003e-10)

(35968.041197438615, 3.24349538938298e-10)

(37044.21520896002, 3.06637316314249e-10)

(28412.05549596342, 5.08119806492439e-10)

(25161.80432425957, 6.40193398145826e-10)

(12239.456148953417, 2.51317060558657e-9)

(20820.296559896007, 9.1754014168478e-10)

(18649.76863330391, 1.13087778972143e-9)

(53100.45766414402, 1.54343018381941e-10)

(7.127218780902727, 0.000897328830949463)

(21906.08600071868, 8.33075593289527e-10)

(24077.032590484556, 6.9613845983863e-10)

(33813.28661488504, 3.6484355763632e-10)

(44556.20396599328, 2.15678393390042e-10)

(46696.09850083937, 1.9722456195479e-10)

(10313.44546212413, 3.47506355209164e-9)

(16485.07770100099, 1.42920919527646e-9)

(47765.053819860346, 1.88892642249023e-10)

(52034.58294223241, 1.60431314724117e-10)

(41341.18594053956, 2.48775547875599e-10)

(55230.48487348186, 1.43183514210459e-10)

(30575.012374392958, 4.41911032636565e-10)

(10608.61356367477, 3.29441036288958e-9)

(17566.207967070062, 1.26693482125914e-9)

(50968.11795067739, 1.66896446061025e-10)

(22991.740359627212, 7.59919333674734e-10)

(11237.05031732756, 2.9544688756137e-9)


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
Puntos máximos de la función:
x47=7.12721878090273x_{47} = 7.12721878090273
Decrece en los intervalos
(−∞,7.12721878090273]\left(-\infty, 7.12721878090273\right]
Crece en los intervalos
[7.12721878090273,∞)\left[7.12721878090273, \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
50(10xacot⁡(5x)+1)log⁡(x−4)(25x2+1)2−20acot⁡(5x)(x−4)(25x2+1)−acot⁡2(5x)(x−4)2=0\frac{50 \left(10 x \operatorname{acot}{\left(5 x \right)} + 1\right) \log{\left(x - 4 \right)}}{\left(25 x^{2} + 1\right)^{2}} - \frac{20 \operatorname{acot}{\left(5 x \right)}}{\left(x - 4\right) \left(25 x^{2} + 1\right)} - \frac{\operatorname{acot}^{2}{\left(5 x \right)}}{\left(x - 4\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=6914.64070942284x_{1} = 6914.64070942284
x2=2680.28026976098x_{2} = 2680.28026976098
x3=5123.30908612897x_{3} = 5123.30908612897
x4=2888.44222329043x_{4} = 2888.44222329043
x5=5378.90973604347x_{5} = 5378.90973604347
x6=11761.641128454x_{6} = 11761.641128454
x7=3854.12863224073x_{7} = 3854.12863224073
x8=10489.574824385x_{8} = 10489.574824385
x9=9725.14094149203x_{9} = 9725.14094149203
x10=6658.66278563147x_{10} = 6658.66278563147
x11=5634.71254972053x_{11} = 5634.71254972053
x12=4359.0225673344x_{12} = 4359.0225673344
x13=12777.5883147809x_{13} = 12777.5883147809
x14=3604.62369365567x_{14} = 3604.62369365567
x15=9470.12583085793x_{15} = 9470.12583085793
x16=11253.1067314629x_{16} = 11253.1067314629
x17=7170.56764909649x_{17} = 7170.56764909649
x18=8449.04802338807x_{18} = 8449.04802338807
x19=12269.7979960085x_{19} = 12269.7979960085
x20=7682.2155805248x_{20} = 7682.2155805248
x21=12523.7382756682x_{21} = 12523.7382756682
x22=4868.01743615581x_{22} = 4868.01743615581
x23=3063.02641891737x_{23} = 3063.02641891737
x24=10234.8649137348x_{24} = 10234.8649137348
x25=4105.84573321937x_{25} = 4105.84573321937
x26=9215.00841648872x_{26} = 9215.00841648872
x27=10998.6946753141x_{27} = 10998.6946753141
x28=8193.53008707241x_{28} = 8193.53008707241
x29=7937.91778891367x_{29} = 7937.91778891367
x30=5890.6397679809x_{30} = 5890.6397679809
x31=6402.65241550942x_{31} = 6402.65241550942
x32=2533.97469817704x_{32} = 2533.97469817704
x33=8959.78900519307x_{33} = 8959.78900519307
x34=13031.3495180793x_{34} = 13031.3495180793
x35=6146.63399938728x_{35} = 6146.63399938728
x36=13285.0232884845x_{36} = 13285.0232884845
x37=7426.42950673344x_{37} = 7426.42950673344
x38=12015.7660759017x_{38} = 12015.7660759017
x39=4613.18414560867x_{39} = 4613.18414560867
x40=11507.4217898116x_{40} = 11507.4217898116
x41=9980.05383564866x_{41} = 9980.05383564866
x42=9.34420933964016x_{42} = 9.34420933964016
x43=10744.1844121669x_{43} = 10744.1844121669
x44=3118.31939912779x_{44} = 3118.31939912779
x45=3358.59225392568x_{45} = 3358.59225392568
x46=8704.46839915151x_{46} = 8704.46839915151

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
[9.34420933964016,∞)\left[9.34420933964016, \infty\right)
Convexa en los intervalos
(−∞,9.34420933964016]\left(-\infty, 9.34420933964016\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(log⁡(x−4)acot⁡2(5x))=0\lim_{x \to -\infty}\left(\log{\left(x - 4 \right)} \operatorname{acot}^{2}{\left(5 x \right)}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(log⁡(x−4)acot⁡2(5x))=0\lim_{x \to \infty}\left(\log{\left(x - 4 \right)} \operatorname{acot}^{2}{\left(5 x \right)}\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 acot(5*x)^2*log(x - 4), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(log⁡(x−4)acot⁡2(5x)x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(x - 4 \right)} \operatorname{acot}^{2}{\left(5 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(log⁡(x−4)acot⁡2(5x)x)=0\lim_{x \to \infty}\left(\frac{\log{\left(x - 4 \right)} \operatorname{acot}^{2}{\left(5 x \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:
log⁡(x−4)acot⁡2(5x)=log⁡(−x−4)acot⁡2(5x)\log{\left(x - 4 \right)} \operatorname{acot}^{2}{\left(5 x \right)} = \log{\left(- x - 4 \right)} \operatorname{acot}^{2}{\left(5 x \right)}
- No
log⁡(x−4)acot⁡2(5x)=−log⁡(−x−4)acot⁡2(5x)\log{\left(x - 4 \right)} \operatorname{acot}^{2}{\left(5 x \right)} = - \log{\left(- x - 4 \right)} \operatorname{acot}^{2}{\left(5 x \right)}
- No
es decir, función
no es
par ni impar
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