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

Gráfico de la función y = 2*cos(x)^(3)+log(3*x)-2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
            3                  
f(x) = 2*cos (x) + log(3*x) - 2
f(x)=(log⁡(3x)+2cos⁡3(x))−2f{\left(x \right)} = \left(\log{\left(3 x \right)} + 2 \cos^{3}{\left(x \right)}\right) - 2 
f = log(3*x) + 2*cos(x)^3 - 2
Gráfico de la función
02468-8-6-4-2-10105-5
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⁡(3x)+2cos⁡3(x))−2=0\left(\log{\left(3 x \right)} + 2 \cos^{3}{\left(x \right)}\right) - 2 = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=9.90656313688425x_{1} = 9.90656313688425
x2=15.4714788640343x_{2} = 15.4714788640343
x3=15.9219022858052x_{3} = 15.9219022858052
x4=4.03442491300886x_{4} = 4.03442491300886
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*cos(x)^3 + log(3*x) - 2.
(log⁡(0⋅3)+2cos⁡3(0))−2\left(\log{\left(0 \cdot 3 \right)} + 2 \cos^{3}{\left(0 \right)}\right) - 2
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
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
−6sin⁡(x)cos⁡2(x)+1x=0- 6 \sin{\left(x \right)} \cos^{2}{\left(x \right)} + \frac{1}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−7.99960692517173x_{1} = -7.99960692517173
x2=−83.2969738178074x_{2} = -83.2969738178074
x3=72.2543243512315x_{3} = 72.2543243512315
x4=−39.3351166347041x_{4} = -39.3351166347041
x5=−21.9835666459728x_{5} = -21.9835666459728
x6=−51.7794681257968x_{6} = -51.7794681257968
x7=−75.4004341192736x_{7} = -75.4004341192736
x8=−43.9860862906991x_{8} = -43.9860862906991
x9=−89.5785583974263x_{9} = -89.5785583974263
x10=−28.268437786183x_{10} = -28.268437786183
x11=45.6136332143689x_{11} = 45.6136332143689
x12=58.1730540345467x_{12} = 58.1730540345467
x13=−70.7344235192801x_{13} = -70.7344235192801
x14=−70.6372124037413x_{14} = -70.6372124037413
x15=20.3294956968854x_{15} = 20.3294956968854
x16=78.5376942053477x_{16} = 78.5376942053477
x17=−50.268798009246x_{17} = -50.268798009246
x18=65.9709193408908x_{18} = 65.9709193408908
x19=−95.7768304451449x_{19} = -95.7768304451449
x20=−45.6136332143689x_{20} = -45.6136332143689
x21=−65.9709193408908x_{21} = -65.9709193408908
x22=−59.6874680701942x_{22} = -59.6874680701942
x23=12.5796222695488x_{23} = 12.5796222695488
x24=81.6834494002643x_{24} = 81.6834494002643
x25=34.5526955081915x_{25} = 34.5526955081915
x26=−81.6834494002643x_{26} = -81.6834494002643
x27=−53.4039542080325x_{27} = -53.4039542080325
x28=−9.40705426568954x_{28} = -9.40705426568954
x29=56.5516149546433x_{29} = 56.5516149546433
x30=43.9860862906991x_{30} = 43.9860862906991
x31=−6.30962154068306x_{31} = -6.30962154068306
x32=89.5785583974263x_{32} = 89.5785583974263
x33=−72.2543243512315x_{33} = -72.2543243512315
x34=6.30962154068306x_{34} = 6.30962154068306
x35=−87.9664889693436x_{35} = -87.9664889693436
x36=14.2458647707014x_{36} = 14.2458647707014
x37=1.88077893345191x_{37} = 1.88077893345191
x38=37.7035323961616x_{38} = 37.7035323961616
x39=50.268798009246x_{39} = 50.268798009246
x40=87.9664889693436x_{40} = 87.9664889693436
x41=−95.8603032179737x_{41} = -95.8603032179737
x42=−15.6973443638602x_{42} = -15.6973443638602
x43=94.2495479692224x_{43} = 94.2495479692224
x44=26.6242097273469x_{44} = 26.6242097273469
x45=−14.0276182978701x_{45} = -14.0276182978701
x46=−1.88077893345191x_{46} = -1.88077893345191
x47=15.6973443638602x_{47} = 15.6973443638602
x48=−94.2495479692224x_{48} = -94.2495479692224
x49=95.8603032179737x_{49} = 95.8603032179737
x50=21.9835666459728x_{50} = 21.9835666459728
x51=51.8930270808787x_{51} = 51.8930270808787
x52=−64.3517029200816x_{52} = -64.3517029200816
x53=−37.7035323961616x_{53} = -37.7035323961616
x54=28.268437786183x_{54} = 28.268437786183
x55=64.3517029200816x_{55} = 64.3517029200816
x56=59.6874680701942x_{56} = 59.6874680701942
x57=70.6372124037413x_{57} = 70.6372124037413
x58=3.08742504955347x_{58} = 3.08742504955347
x59=−58.0658245702513x_{59} = -58.0658245702513
x60=−97.3876608818886x_{60} = -97.3876608818886
x61=937.75207456793x_{61} = 937.75207456793
x62=39.2045908573273x_{62} = 39.2045908573273
x63=−31.4212309791897x_{63} = -31.4212309791897
x64=100.532622756841x_{64} = 100.532622756841
x65=7.99960692517173x_{65} = 7.99960692517173
x66=−20.3294956968854x_{66} = -20.3294956968854
x67=32.9154136883458x_{67} = 32.9154136883458
x68=97.3876608818886x_{68} = 97.3876608818886
Signos de extremos en los puntos:
(-7.999606925171734, 1.17189341489522 + pi*I)

(-83.2969738178074, 3.52084523652315 + pi*I)

(72.25432435123146, 1.37882042876854)

(-39.33511663470405, 2.77017658798915 + pi*I)

(-21.98356664597275, 0.189079943433468 + pi*I)

(-51.779468125796804, 3.04597210700263 + pi*I)

(-75.40043411927357, 5.42141066321356 + pi*I)

(-43.98608629069909, 4.88244257951702 + pi*I)

(-89.57855839742632, 3.59356754373643 + pi*I)

(-28.268437786182986, 0.440462488098656 + pi*I)

(45.613633214368875, 2.91837598355398)

(58.17305403454667, 3.16172718082829)

(-70.7344235192801, 3.35731548534626 + pi*I)

(-70.63721240374134, 3.35639900927717 + pi*I)

(20.32949569688541, 2.11217895575701)

(78.53769420534775, 1.46220448949404)

(-50.268798009245955, 5.01596387694949 + pi*I)

(65.97091934089076, 1.28784546533221)

(-95.77683044514492, 3.66077846282114 + pi*I)

(-45.613633214368875, 2.91837598355398 + pi*I)

(-65.97091934089076, 1.28784546533221 + pi*I)

(-59.68746807019421, 1.18775776347515 + pi*I)

(12.57962226954883, 3.63016374808887)

(81.68344940026425, 5.5014512025413)

(34.552695508191476, 0.641167656198109)

(-81.68344940026425, 5.5014512025413 + pi*I)

(-53.403954208032545, 1.07652630067281 + pi*I)

(-9.407054265689542, -0.65898563427642 + pi*I)

(56.551614954643334, 5.13373999116279)

(43.98608629069909, 4.88244257951702)

(-6.309621540683057, 2.9385922169633 + pi*I)

(89.57855839742632, 3.59356754373643)

(-72.25432435123146, 1.37882042876854 + pi*I)

(6.309621540683057, 2.9385922169633)

(-87.96648896934359, 5.57555745416694 + pi*I)

(14.245864770701424, 1.75252551996435)

(1.8807789334519098, -0.326470429795601)

(37.70353239616164, 4.7283074530432)

(50.268798009245955, 5.01596387694949)

(87.96648896934359, 5.57555745416694)

(-95.86030321797365, 3.6613590637131 + pi*I)

(-15.697344363860205, -0.147557900962119 + pi*I)

(94.24954796922239, 5.64454893761183)

(26.624209727346887, 2.38142849836729)

(-14.02761829787005, 1.74225403991162 + pi*I)

(-1.8807789334519098, -0.326470429795601 + pi*I)

(15.697344363860205, -0.147557900962119)

(-94.24954796922239, 5.64454893761183 + pi*I)

(95.86030321797365, 3.6613590637131)

(21.98356664597275, 0.189079943433468)

(51.893027080878745, 3.04743180402748)

(-64.35170292008156, 3.26323981035881 + pi*I)

(-37.70353239616164, 4.7283074530432 + pi*I)

(28.268437786182986, 0.440462488098656)

(64.35170292008156, 3.26323981035881)

(59.68746807019421, 1.18775776347515)

(70.63721240374134, 3.35639900927717)

(3.0874250495534685, -1.7652629502561)

(-58.065824570251344, 3.16049778138377 + pi*I)

(-97.38766088188859, 1.67732059274379 + pi*I)

(937.75207456793, 5.94210262937136)

(39.20459085732729, 2.76796228772829)

(-31.421230979189666, 4.54601168896233 + pi*I)

(100.53262275684067, 5.70908632273352)

(7.999606925171734, 1.17189341489522)

(-20.32949569688541, 2.11217895575701 + pi*I)

(32.91541368834582, 2.59327671013379)

(97.38766088188859, 1.67732059274379)


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=72.2543243512315x_{1} = 72.2543243512315
x2=20.3294956968854x_{2} = 20.3294956968854
x3=78.5376942053477x_{3} = 78.5376942053477
x4=65.9709193408908x_{4} = 65.9709193408908
x5=34.5526955081915x_{5} = 34.5526955081915
x6=26.6242097273469x_{6} = 26.6242097273469
x7=15.6973443638602x_{7} = 15.6973443638602
x8=21.9835666459728x_{8} = 21.9835666459728
x9=28.268437786183x_{9} = 28.268437786183
x10=64.3517029200816x_{10} = 64.3517029200816
x11=59.6874680701942x_{11} = 59.6874680701942
x12=70.6372124037413x_{12} = 70.6372124037413
x13=3.08742504955347x_{13} = 3.08742504955347
x14=937.75207456793x_{14} = 937.75207456793
x15=39.2045908573273x_{15} = 39.2045908573273
x16=32.9154136883458x_{16} = 32.9154136883458
x17=97.3876608818886x_{17} = 97.3876608818886
Puntos máximos de la función:
x17=45.6136332143689x_{17} = 45.6136332143689
x17=58.1730540345467x_{17} = 58.1730540345467
x17=12.5796222695488x_{17} = 12.5796222695488
x17=81.6834494002643x_{17} = 81.6834494002643
x17=56.5516149546433x_{17} = 56.5516149546433
x17=43.9860862906991x_{17} = 43.9860862906991
x17=89.5785583974263x_{17} = 89.5785583974263
x17=6.30962154068306x_{17} = 6.30962154068306
x17=14.2458647707014x_{17} = 14.2458647707014
x17=1.88077893345191x_{17} = 1.88077893345191
x17=37.7035323961616x_{17} = 37.7035323961616
x17=50.268798009246x_{17} = 50.268798009246
x17=87.9664889693436x_{17} = 87.9664889693436
x17=94.2495479692224x_{17} = 94.2495479692224
x17=95.8603032179737x_{17} = 95.8603032179737
x17=51.8930270808787x_{17} = 51.8930270808787
x17=100.532622756841x_{17} = 100.532622756841
x17=7.99960692517173x_{17} = 7.99960692517173
Decrece en los intervalos
[937.75207456793,∞)\left[937.75207456793, \infty\right)
Crece en los intervalos
(−∞,3.08742504955347]\left(-\infty, 3.08742504955347\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
12sin⁡2(x)cos⁡(x)−6cos⁡3(x)−1x2=012 \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 6 \cos^{3}{\left(x \right)} - \frac{1}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−5.66545536021399x_{1} = -5.66545536021399
x2=32.0314765852011x_{2} = 32.0314765852011
x3=−93.6322916671375x_{3} = -93.6322916671375
x4=−86.3938091385852x_{4} = -86.3938091385852
x5=−32.0314765852011x_{5} = -32.0314765852011
x6=95.8185668579678x_{6} = 95.8185668579678
x7=−64.402629307128x_{7} = -64.402629307128
x8=−33.9421021223389x_{8} = -33.9421021223389
x9=33.9421021223389x_{9} = 33.9421021223389
x10=5.66545536021399x_{10} = 5.66545536021399
x11=54.0225300913121x_{11} = 54.0225300913121
x12=25.7483297973015x_{12} = 25.7483297973015
x13=−957.570279714923x_{13} = -957.570279714923
x14=71.6411653850805x_{14} = 71.6411653850805
x15=−26.7034206903774x_{15} = -26.7034206903774
x16=−18.2338591299968x_{16} = -18.2338591299968
x17=−60.3057202828718x_{17} = -60.3057202828718
x18=−47.7393378466241x_{18} = -47.7393378466241
x19=−58.1194394210265x_{19} = -58.1194394210265
x20=36.1283793605336x_{20} = 36.1283793605336
x21=55.933164987723x_{21} = 55.933164987723
x22=−77.9243485161296x_{22} = -77.9243485161296
x23=−67.5442603181133x_{23} = -67.5442603181133
x24=−95.8185668579678x_{24} = -95.8185668579678
x25=−23.5620950054738x_{25} = -23.5620950054738
x26=−42.411547152206x_{26} = -42.411547152206
x27=22.6064870750663x_{27} = 22.6064870750663
x28=−91.7216580844143x_{28} = -91.7216580844143
x29=103.057084654842x_{29} = 103.057084654842
x30=−99.915477977091x_{30} = -99.915477977091
x31=60.3057202828718x_{31} = 60.3057202828718
x32=10.0395418359485x_{32} = 10.0395418359485
x33=11.9503854729973x_{33} = 11.9503854729973
x34=68.4995432895946x_{34} = 68.4995432895946
x35=29.8452237645508x_{35} = 29.8452237645508
x36=84.2075321158673x_{36} = 84.2075321158673
x37=−82.2968993577563x_{37} = -82.2968993577563
x38=−62.2163547189525x_{38} = -62.2163547189525
x39=90.4907160587626x_{39} = 90.4907160587626
x40=66.5889091582283x_{40} = 66.5889091582283
x41=27.6589485066224x_{41} = 27.6589485066224
x42=−38.3146407135078x_{42} = -38.3146407135078
x43=−89.5353802321919x_{43} = -89.5353802321919
x44=−84.2075321158673x_{44} = -84.2075321158673
x45=−14.1367499568317x_{45} = -14.1367499568317
x46=24.5171414518022x_{46} = 24.5171414518022
x47=51.8362477706702x_{47} = 51.8362477706702
x48=47.7393378466241x_{48} = 47.7393378466241
x49=77.9243485161296x_{49} = 77.9243485161296
x50=−71.6411653850805x_{50} = -71.6411653850805
x51=44.5978131439863x_{51} = 44.5978131439863
x52=46.5084434593717x_{52} = 46.5084434593717
x53=−49.6499734724898x_{53} = -49.6499734724898
x54=−27.6589485066224x_{54} = -27.6589485066224
x55=−55.933164987723x_{55} = -55.933164987723
x56=20.420152399865x_{56} = 20.420152399865
x57=−80.1106256514367x_{57} = -80.1106256514367
x58=−25.7483297973015x_{58} = -25.7483297973015
x59=80.1106256514367x_{59} = 80.1106256514367
x60=−45.5530533179705x_{60} = -45.5530533179705
x61=93.6322916671375x_{61} = 93.6322916671375
x62=76.0137158849675x_{62} = 76.0137158849675
x63=−54.0225300913121x_{63} = -54.0225300913121
x64=−98.0048444562662x_{64} = -98.0048444562662
x65=−10.0395418359485x_{65} = -10.0395418359485
x66=−7.85263021571567x_{66} = -7.85263021571567
x67=98.0048444562662x_{67} = 98.0048444562662
x68=3.7519547754225x_{68} = 3.7519547754225
x69=88.5800832068717x_{69} = 88.5800832068717
x70=18.2338591299968x_{70} = 18.2338591299968
x71=−1.53537211492886x_{71} = -1.53537211492886
x72=62.2163547189525x_{72} = 62.2163547189525
x73=42.411547152206x_{73} = 42.411547152206
x74=−69.7305329300993x_{74} = -69.7305329300993
x75=82.2968993577563x_{75} = 82.2968993577563
x76=−36.1283793605336x_{76} = -36.1283793605336
x77=99.915477977091x_{77} = 99.915477977091
x78=40.225269388999x_{78} = 40.225269388999
x79=58.1194394210265x_{79} = 58.1194394210265
x80=−51.8362477706702x_{80} = -51.8362477706702
x81=38.3146407135078x_{81} = 38.3146407135078
x82=−11.9503854729973x_{82} = -11.9503854729973
x83=14.1367499568317x_{83} = 14.1367499568317
x84=16.3231721453993x_{84} = 16.3231721453993
x85=−3.7519547754225x_{85} = -3.7519547754225
x86=−29.8452237645508x_{86} = -29.8452237645508
x87=49.6499734724898x_{87} = 49.6499734724898
x88=−40.225269388999x_{88} = -40.225269388999
x89=64.402629307128x_{89} = 64.402629307128
x90=69.7305329300993x_{90} = 69.7305329300993
x91=86.3938091385852x_{91} = 86.3938091385852
x92=7.85263021571567x_{92} = 7.85263021571567
x93=−76.0137158849675x_{93} = -76.0137158849675
x94=73.8274426485018x_{94} = 73.8274426485018
x95=−16.3231721453993x_{95} = -16.3231721453993
x96=−73.8274426485018x_{96} = -73.8274426485018

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
[103.057084654842,∞)\left[103.057084654842, \infty\right)
Convexa en los intervalos
(−∞,−99.915477977091]\left(-\infty, -99.915477977091\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((log⁡(3x)+2cos⁡3(x))−2)=∞\lim_{x \to -\infty}\left(\left(\log{\left(3 x \right)} + 2 \cos^{3}{\left(x \right)}\right) - 2\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞((log⁡(3x)+2cos⁡3(x))−2)=∞\lim_{x \to \infty}\left(\left(\log{\left(3 x \right)} + 2 \cos^{3}{\left(x \right)}\right) - 2\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*cos(x)^3 + log(3*x) - 2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((log⁡(3x)+2cos⁡3(x))−2x)=0\lim_{x \to -\infty}\left(\frac{\left(\log{\left(3 x \right)} + 2 \cos^{3}{\left(x \right)}\right) - 2}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((log⁡(3x)+2cos⁡3(x))−2x)=0\lim_{x \to \infty}\left(\frac{\left(\log{\left(3 x \right)} + 2 \cos^{3}{\left(x \right)}\right) - 2}{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⁡(3x)+2cos⁡3(x))−2=log⁡(−3x)+2cos⁡3(x)−2\left(\log{\left(3 x \right)} + 2 \cos^{3}{\left(x \right)}\right) - 2 = \log{\left(- 3 x \right)} + 2 \cos^{3}{\left(x \right)} - 2
- No
(log⁡(3x)+2cos⁡3(x))−2=−log⁡(−3x)−2cos⁡3(x)+2\left(\log{\left(3 x \right)} + 2 \cos^{3}{\left(x \right)}\right) - 2 = - \log{\left(- 3 x \right)} - 2 \cos^{3}{\left(x \right)} + 2
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор