Sr Examen
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- Gráfico de la función implícita
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- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x^3/3-4*x
-
x^3-6*x^2+9*x+1
-
y=x+2
-
(x^3-x^2)/(4-x^2)
-
Expresiones idénticas
- cosx+|lnx+ uno . cinco |- tres
- coseno de x más módulo de lnx más 1.5| menos 3
- coseno de x más módulo de lnx más uno . cinco | menos tres
-
Expresiones semejantes
- cosx+|lnx-1.5|-3
- cosx+|lnx+1.5|+3
- cosx-|lnx+1.5|-3
-
Expresiones con funciones
- cosx
- cosx+1.5
- cosx/((x-2)(x+10))
- cosx/(1-2cosx+sinx)
- cosx+1/3cosx
- cosx+|cosx|
- lnx
- lnx/x^(1/2)
- lnx/x^1/2
- lnx*sin(3,14/x)^2
- lnx^14
- lnx²+2x+2
- Módulo |
- |x|-x
- |x/(x-1)|
- |x^5|
- |x/3-3/x|+x/3+3/x
- |x|-1
Gráfico de la función y = cosx+|lnx+1.5|-3
Solución
Ha introducido
[src]
f(x) = cos(x) + |log(x) + 3/2| - 3
$$f{\left(x \right)} = \left(\cos{\left(x \right)} + \left|{\log{\left(x \right)} + \frac{3}{2}}\right|\right) - 3$$
f = cos(x) + Abs(log(x) + 3/2) - 3
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:
$$\left(\cos{\left(x \right)} + \left|{\log{\left(x \right)} + \frac{3}{2}}\right|\right) - 3 = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 10.0547482881704$$
$$x_{2} = 8.55735041279717$$
$$x_{3} = 4.67100328296549$$
o sea hay que resolver la ecuación:
$$\left(\cos{\left(x \right)} + \left|{\log{\left(x \right)} + \frac{3}{2}}\right|\right) - 3 = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 10.0547482881704$$
$$x_{2} = 8.55735041279717$$
$$x_{3} = 4.67100328296549$$
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
$$- \sin{\left(x \right)} + \frac{\left(\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} + \frac{3}{2}\right) \operatorname{sign}^{2}{\left(x \right)} \operatorname{sign}{\left(\log{\left(x \right)} + \frac{3}{2} \right)}}{x \left(\log{\left(x \right)} + \frac{3}{2}\right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -72.2444691366352$$
$$x_{2} = -116.231239665117$$
$$x_{3} = -47.1055805937909$$
$$x_{4} = -31.4427791930171$$
$$x_{5} = -25.1658381559897$$
$$x_{6} = 18.9024837303424$$
$$x_{7} = 28.238914350639$$
$$x_{8} = -78.5285915083114$$
$$x_{9} = -65.9601737666156$$
$$x_{10} = 2.77260470826599$$
$$x_{11} = -2.92311362264668$$
$$x_{12} = -6.39754595996566$$
$$x_{13} = 91.0952092079181$$
$$x_{14} = -141.365309497836$$
$$x_{15} = 9.31724294141481$$
$$x_{16} = -1994.91086120507$$
$$x_{17} = -21.9535532543359$$
$$x_{18} = -91.0964577061715$$
$$x_{19} = -4354.24763673601$$
$$x_{20} = -69.1277263309044$$
$$x_{21} = -62.8457548352571$$
$$x_{22} = -50.2826875631558$$
$$x_{23} = 47.1026579842757$$
$$x_{24} = 53.3883433351401$$
$$x_{25} = -18.8927734578501$$
$$x_{26} = 44.0050237846841$$
$$x_{27} = -56.5640437396207$$
$$x_{28} = 62.8477652080076$$
$$x_{29} = -521.50615778176$$
$$x_{30} = -40.8197247770234$$
$$x_{31} = 12.6455325787891$$
$$x_{32} = 25.1724776117344$$
$$x_{33} = -84.8125785483258$$
$$x_{34} = 100.540911278755$$
$$x_{35} = 56.5663470412005$$
$$x_{36} = -53.390827098789$$
$$x_{37} = 69.12950448699$$
$$x_{38} = -81.6922151985267$$
$$x_{39} = 87.9759612866821$$
$$x_{40} = -37.7217215533643$$
$$x_{41} = -1460.83993934185$$
$$x_{42} = 81.6936501525055$$
$$x_{43} = -87.9746561841416$$
$$x_{44} = -59.6756522560516$$
$$x_{45} = -12.6288963090383$$
$$x_{46} = 84.811210479617$$
$$x_{47} = 31.4477306941599$$
$$x_{48} = 40.8162019701941$$
$$x_{49} = -100.539810736949$$
$$x_{50} = 37.7256221319914$$
$$x_{51} = -97.3802494629457$$
$$x_{52} = 15.6439973747731$$
$$x_{53} = 34.5285536017763$$
$$x_{54} = -15.6565750377676$$
$$x_{55} = 6.43911723841725$$
$$x_{56} = 94.2583889420049$$
$$x_{57} = -75.4098945932938$$
$$x_{58} = -28.2446304586779$$
$$x_{59} = -34.5329394828148$$
$$x_{60} = 21.9455654988197$$
$$x_{61} = -130322.688041398$$
$$x_{62} = 50.2853702681169$$
$$x_{63} = 59.6735017769259$$
$$x_{64} = 72.2427883783755$$
$$x_{65} = -44.001833107098$$
$$x_{66} = 78.5270815352651$$
$$x_{67} = 97.3791029371005$$
$$x_{68} = 65.9582840466332$$
$$x_{69} = 75.4114846544819$$
$$x_{70} = -94.2571939610534$$
$$x_{71} = -9.3427781299964$$
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:
$$x_{1} = -72.2444691366352$$
$$x_{2} = -116.231239665117$$
$$x_{3} = -47.1055805937909$$
$$x_{4} = 28.238914350639$$
$$x_{5} = -78.5285915083114$$
$$x_{6} = -65.9601737666156$$
$$x_{7} = 2.77260470826599$$
$$x_{8} = -2.92311362264668$$
$$x_{9} = 91.0952092079181$$
$$x_{10} = -141.365309497836$$
$$x_{11} = 9.31724294141481$$
$$x_{12} = -1994.91086120507$$
$$x_{13} = -21.9535532543359$$
$$x_{14} = -91.0964577061715$$
$$x_{15} = 47.1026579842757$$
$$x_{16} = 53.3883433351401$$
$$x_{17} = -40.8197247770234$$
$$x_{18} = -84.8125785483258$$
$$x_{19} = -53.390827098789$$
$$x_{20} = -1460.83993934185$$
$$x_{21} = -59.6756522560516$$
$$x_{22} = 84.811210479617$$
$$x_{23} = 40.8162019701941$$
$$x_{24} = -97.3802494629457$$
$$x_{25} = 15.6439973747731$$
$$x_{26} = 34.5285536017763$$
$$x_{27} = -15.6565750377676$$
$$x_{28} = -28.2446304586779$$
$$x_{29} = -34.5329394828148$$
$$x_{30} = 21.9455654988197$$
$$x_{31} = -130322.688041398$$
$$x_{32} = 59.6735017769259$$
$$x_{33} = 72.2427883783755$$
$$x_{34} = 78.5270815352651$$
$$x_{35} = 97.3791029371005$$
$$x_{36} = 65.9582840466332$$
$$x_{37} = -9.3427781299964$$
Puntos máximos de la función:
$$x_{37} = -31.4427791930171$$
$$x_{37} = -25.1658381559897$$
$$x_{37} = 18.9024837303424$$
$$x_{37} = -6.39754595996566$$
$$x_{37} = -4354.24763673601$$
$$x_{37} = -69.1277263309044$$
$$x_{37} = -62.8457548352571$$
$$x_{37} = -50.2826875631558$$
$$x_{37} = -18.8927734578501$$
$$x_{37} = 44.0050237846841$$
$$x_{37} = -56.5640437396207$$
$$x_{37} = 62.8477652080076$$
$$x_{37} = -521.50615778176$$
$$x_{37} = 12.6455325787891$$
$$x_{37} = 25.1724776117344$$
$$x_{37} = 100.540911278755$$
$$x_{37} = 56.5663470412005$$
$$x_{37} = 69.12950448699$$
$$x_{37} = -81.6922151985267$$
$$x_{37} = 87.9759612866821$$
$$x_{37} = -37.7217215533643$$
$$x_{37} = 81.6936501525055$$
$$x_{37} = -87.9746561841416$$
$$x_{37} = -12.6288963090383$$
$$x_{37} = 31.4477306941599$$
$$x_{37} = -100.539810736949$$
$$x_{37} = 37.7256221319914$$
$$x_{37} = 6.43911723841725$$
$$x_{37} = 94.2583889420049$$
$$x_{37} = -75.4098945932938$$
$$x_{37} = 50.2853702681169$$
$$x_{37} = -44.001833107098$$
$$x_{37} = 75.4114846544819$$
$$x_{37} = -94.2571939610534$$
Decrece en los intervalos
$$\left[97.3791029371005, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -130322.688041398\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- \sin{\left(x \right)} + \frac{\left(\log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} + \frac{3}{2}\right) \operatorname{sign}^{2}{\left(x \right)} \operatorname{sign}{\left(\log{\left(x \right)} + \frac{3}{2} \right)}}{x \left(\log{\left(x \right)} + \frac{3}{2}\right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -72.2444691366352$$
$$x_{2} = -116.231239665117$$
$$x_{3} = -47.1055805937909$$
$$x_{4} = -31.4427791930171$$
$$x_{5} = -25.1658381559897$$
$$x_{6} = 18.9024837303424$$
$$x_{7} = 28.238914350639$$
$$x_{8} = -78.5285915083114$$
$$x_{9} = -65.9601737666156$$
$$x_{10} = 2.77260470826599$$
$$x_{11} = -2.92311362264668$$
$$x_{12} = -6.39754595996566$$
$$x_{13} = 91.0952092079181$$
$$x_{14} = -141.365309497836$$
$$x_{15} = 9.31724294141481$$
$$x_{16} = -1994.91086120507$$
$$x_{17} = -21.9535532543359$$
$$x_{18} = -91.0964577061715$$
$$x_{19} = -4354.24763673601$$
$$x_{20} = -69.1277263309044$$
$$x_{21} = -62.8457548352571$$
$$x_{22} = -50.2826875631558$$
$$x_{23} = 47.1026579842757$$
$$x_{24} = 53.3883433351401$$
$$x_{25} = -18.8927734578501$$
$$x_{26} = 44.0050237846841$$
$$x_{27} = -56.5640437396207$$
$$x_{28} = 62.8477652080076$$
$$x_{29} = -521.50615778176$$
$$x_{30} = -40.8197247770234$$
$$x_{31} = 12.6455325787891$$
$$x_{32} = 25.1724776117344$$
$$x_{33} = -84.8125785483258$$
$$x_{34} = 100.540911278755$$
$$x_{35} = 56.5663470412005$$
$$x_{36} = -53.390827098789$$
$$x_{37} = 69.12950448699$$
$$x_{38} = -81.6922151985267$$
$$x_{39} = 87.9759612866821$$
$$x_{40} = -37.7217215533643$$
$$x_{41} = -1460.83993934185$$
$$x_{42} = 81.6936501525055$$
$$x_{43} = -87.9746561841416$$
$$x_{44} = -59.6756522560516$$
$$x_{45} = -12.6288963090383$$
$$x_{46} = 84.811210479617$$
$$x_{47} = 31.4477306941599$$
$$x_{48} = 40.8162019701941$$
$$x_{49} = -100.539810736949$$
$$x_{50} = 37.7256221319914$$
$$x_{51} = -97.3802494629457$$
$$x_{52} = 15.6439973747731$$
$$x_{53} = 34.5285536017763$$
$$x_{54} = -15.6565750377676$$
$$x_{55} = 6.43911723841725$$
$$x_{56} = 94.2583889420049$$
$$x_{57} = -75.4098945932938$$
$$x_{58} = -28.2446304586779$$
$$x_{59} = -34.5329394828148$$
$$x_{60} = 21.9455654988197$$
$$x_{61} = -130322.688041398$$
$$x_{62} = 50.2853702681169$$
$$x_{63} = 59.6735017769259$$
$$x_{64} = 72.2427883783755$$
$$x_{65} = -44.001833107098$$
$$x_{66} = 78.5270815352651$$
$$x_{67} = 97.3791029371005$$
$$x_{68} = 65.9582840466332$$
$$x_{69} = 75.4114846544819$$
$$x_{70} = -94.2571939610534$$
$$x_{71} = -9.3427781299964$$
Signos de extremos en los puntos:
________________________
/ 2
(-72.24446913663523, -3.99992604505527 + \/ 33.4090447311325 + pi ) _______________________
/ 2
(-116.23123966511726, -3.99997044349335 + \/ 39.132301807587 + pi ) ________________________
/ 2
(-47.10558059379095, -3.9998323910959 + \/ 28.6480945332137 + pi ) ________________________
/ 2
(-31.442779193017056, -2.00036051093374 + \/ 24.4843800189906 + pi ) ________________________
/ 2
(-25.165838155989658, -2.00054765330274 + \/ 22.3302316031628 + pi )(18.902483730342446, 2.43789297821484)
(28.238914350639, 0.841328174248101)
________________________
/ 2
(-78.52859150831141, -3.99993700224111 + \/ 34.3801957901655 + pi ) ________________________
/ 2
(-65.96017376661558, -3.99991192884799 + \/ 32.3653027837855 + pi )(2.772604708265991, -1.4129056364004)
________________________
/ 2
(-2.9231136226466807, -3.97622824091219 + \/ 6.61852471562641 + pi ) _______________________
/ 2
(-6.397545959965662, -2.00653205574768 + \/ 11.262161950607 + pi )(91.09520920791812, 2.01196546945613)
________________________
/ 2
(-141.36530949783608, -3.99997977581701 + \/ 41.6198831131829 + pi )(9.31724294141481, -0.262356916171007)
________________________
/ 2
(-1994.9108612050723, -3.9999998877452 + \/ 82.7800572866902 + pi ) ________________________
/ 2
(-21.953553254335887, -3.99929337916177 + \/ 21.0582694210868 + pi ) ________________________
/ 2
(-91.09645770617149, -3.99995267124067 + \/ 36.1431690998523 + pi ) ________________________
/ 2
(-4354.247636736007, -2.00000002394997 + \/ 97.5928058038474 + pi ) _______________________
/ 2
(-69.1277263309044, -2.00008049098225 + \/ 32.901190082804 + pi ) ________________________
/ 2
(-62.84575483525714, -2.00009662795747 + \/ 31.8173090878075 + pi ) ________________________
/ 2
(-50.28268756315581, -2.00014800418039 + \/ 29.3510489147046 + pi )(47.102657984275716, 1.35255481883139)
(53.388343335140114, 1.47776786711753)
________________________
/ 2
(-18.892773457850122, -2.00093373237686 + \/ 19.7027633805628 + pi )(44.00502378468411, 3.28404556548434)
________________________
/ 2
(-56.56404373962072, -2.00011820797474 + \/ 30.6403599364138 + pi )(62.8477652080076, 3.64058878148622)
________________________
/ 2
(-521.5061577817598, -2.00000157937209 + \/ 60.1667219470161 + pi ) ________________________
/ 2
(-40.819724777023374, -3.99977993375383 + \/ 27.1354043208619 + pi )(12.645532578789108, 2.03417232459457)
(25.17247761173438, 2.72496185322279)
________________________
/ 2
(-84.81257854832579, -3.99994567999959 + \/ 35.2888732981936 + pi )(100.54091127875482, 4.11051525739672)
(56.566347041200544, 3.53525795879555)
________________________
/ 2
(-53.39082709878904, -3.99986800395309 + \/ 30.0045285107564 + pi )(69.12950448698996, 3.73587698977231)
________________________
/ 2
(-81.69221519852674, -2.00005838646715 + \/ 34.8449215569731 + pi )(87.97596128668211, 3.9769990064728)
________________________
/ 2
(-37.72172155336433, -2.00025558861129 + \/ 26.3193224115452 + pi ) ________________________
/ 2
(-1460.839939341847, -3.99999979225999 + \/ 77.2072716797618 + pi )(81.6936501525055, 3.90290135528057)
________________________
/ 2
(-87.97465618414155, -2.00005062032399 + \/ 35.7251120598212 + pi ) ________________________
/ 2
(-59.67565225605163, -3.99989330269667 + \/ 31.2360726213176 + pi ) ________________________
/ 2
(-12.628896309038288, -2.00195409450149 + \/ 16.2891954721629 + pi )(84.81121047961696, 1.94049724812025)
(31.447730694159905, 2.9478211149426)
(40.81620197019412, 1.20937928157994)
________________________
/ 2
(-100.53981073694867, -2.00003912402898 + \/ 37.3388674469125 + pi )(37.725622131991386, 3.12998811862736)
_______________________
/ 2
(-97.38024946294566, -3.99995838756384 + \/ 36.949662590975 + pi )(15.643997374773143, 0.252132409361107)
(34.52855360177631, 1.04220609566901)
________________________
/ 2
(-15.65657503776763, -3.99867991543891 + \/ 18.0700739450101 + pi )(6.439117238417246, 1.35025868605687)
(94.25838894200487, 4.04598355125005)
________________________
/ 2
(-75.40989459329381, -2.00006810426368 + \/ 33.9066127099384 + pi ) ________________________
/ 2
(-28.244630458677907, -3.99955888574643 + \/ 23.4343454112193 + pi ) ________________________
/ 2
(-34.53293948281483, -3.99969793421845 + \/ 25.4208931144401 + pi )(21.945565498819683, 0.58960382026917)
_______________________
/ 2
(-130322.6880413983, -3.99999999997212 + \/ 176.29914615271 + pi )(50.28537026811692, 3.41751642927177)
(59.67350177692592, 1.58902848827451)
(72.24278837837548, 1.78012831518614)
______________________
/ 2
(-44.001833107098044, -2.00019082073578 + \/ 27.92310037457 + pi )(78.52708153526505, 1.86352463953088)
(97.3791029371005, 2.07866436777537)
(65.95828404663321, 1.68913741834237)
(75.41148465448185, 3.82287165443541)
________________________
/ 2
(-94.25719396105342, -2.00004431469729 + \/ 36.5544443216872 + pi ) ________________________
/ 2
(-9.342778129996402, -3.99663989727954 + \/ 13.9472644400567 + pi )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:
$$x_{1} = -72.2444691366352$$
$$x_{2} = -116.231239665117$$
$$x_{3} = -47.1055805937909$$
$$x_{4} = 28.238914350639$$
$$x_{5} = -78.5285915083114$$
$$x_{6} = -65.9601737666156$$
$$x_{7} = 2.77260470826599$$
$$x_{8} = -2.92311362264668$$
$$x_{9} = 91.0952092079181$$
$$x_{10} = -141.365309497836$$
$$x_{11} = 9.31724294141481$$
$$x_{12} = -1994.91086120507$$
$$x_{13} = -21.9535532543359$$
$$x_{14} = -91.0964577061715$$
$$x_{15} = 47.1026579842757$$
$$x_{16} = 53.3883433351401$$
$$x_{17} = -40.8197247770234$$
$$x_{18} = -84.8125785483258$$
$$x_{19} = -53.390827098789$$
$$x_{20} = -1460.83993934185$$
$$x_{21} = -59.6756522560516$$
$$x_{22} = 84.811210479617$$
$$x_{23} = 40.8162019701941$$
$$x_{24} = -97.3802494629457$$
$$x_{25} = 15.6439973747731$$
$$x_{26} = 34.5285536017763$$
$$x_{27} = -15.6565750377676$$
$$x_{28} = -28.2446304586779$$
$$x_{29} = -34.5329394828148$$
$$x_{30} = 21.9455654988197$$
$$x_{31} = -130322.688041398$$
$$x_{32} = 59.6735017769259$$
$$x_{33} = 72.2427883783755$$
$$x_{34} = 78.5270815352651$$
$$x_{35} = 97.3791029371005$$
$$x_{36} = 65.9582840466332$$
$$x_{37} = -9.3427781299964$$
Puntos máximos de la función:
$$x_{37} = -31.4427791930171$$
$$x_{37} = -25.1658381559897$$
$$x_{37} = 18.9024837303424$$
$$x_{37} = -6.39754595996566$$
$$x_{37} = -4354.24763673601$$
$$x_{37} = -69.1277263309044$$
$$x_{37} = -62.8457548352571$$
$$x_{37} = -50.2826875631558$$
$$x_{37} = -18.8927734578501$$
$$x_{37} = 44.0050237846841$$
$$x_{37} = -56.5640437396207$$
$$x_{37} = 62.8477652080076$$
$$x_{37} = -521.50615778176$$
$$x_{37} = 12.6455325787891$$
$$x_{37} = 25.1724776117344$$
$$x_{37} = 100.540911278755$$
$$x_{37} = 56.5663470412005$$
$$x_{37} = 69.12950448699$$
$$x_{37} = -81.6922151985267$$
$$x_{37} = 87.9759612866821$$
$$x_{37} = -37.7217215533643$$
$$x_{37} = 81.6936501525055$$
$$x_{37} = -87.9746561841416$$
$$x_{37} = -12.6288963090383$$
$$x_{37} = 31.4477306941599$$
$$x_{37} = -100.539810736949$$
$$x_{37} = 37.7256221319914$$
$$x_{37} = 6.43911723841725$$
$$x_{37} = 94.2583889420049$$
$$x_{37} = -75.4098945932938$$
$$x_{37} = 50.2853702681169$$
$$x_{37} = -44.001833107098$$
$$x_{37} = 75.4114846544819$$
$$x_{37} = -94.2571939610534$$
Decrece en los intervalos
$$\left[97.3791029371005, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -130322.688041398\right]$$
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
$$- \cos{\left(x \right)} + \frac{4 \left(2 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} + 3\right) \delta\left(x\right) \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\log{\left(x \right)} + \frac{3}{2} \right)}}{x \left(2 \log{\left(x \right)} + 3\right)} + \frac{\left(2 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} + 3\right) \operatorname{sign}^{2}{\left(x \right)} \frac{d}{d x} \operatorname{sign}{\left(\log{\left(x \right)} + \frac{3}{2} \right)}}{x \left(2 \log{\left(x \right)} + 3\right)} - \frac{2 \left(\frac{2 x \delta\left(x\right)}{\operatorname{sign}{\left(x \right)}} - 1\right) \operatorname{sign}^{2}{\left(x \right)} \operatorname{sign}{\left(\log{\left(x \right)} + \frac{3}{2} \right)}}{x^{2} \left(2 \log{\left(x \right)} + 3\right)} - \frac{\left(2 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} + 3\right) \operatorname{sign}^{2}{\left(x \right)} \operatorname{sign}{\left(\log{\left(x \right)} + \frac{3}{2} \right)}}{x^{2} \left(2 \log{\left(x \right)} + 3\right)} - \frac{2 \left(2 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} + 3\right) \operatorname{sign}^{2}{\left(x \right)} \operatorname{sign}{\left(\log{\left(x \right)} + \frac{3}{2} \right)}}{x^{2} \left(2 \log{\left(x \right)} + 3\right)^{2}} = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \cos{\left(x \right)} + \frac{4 \left(2 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} + 3\right) \delta\left(x\right) \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\log{\left(x \right)} + \frac{3}{2} \right)}}{x \left(2 \log{\left(x \right)} + 3\right)} + \frac{\left(2 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} + 3\right) \operatorname{sign}^{2}{\left(x \right)} \frac{d}{d x} \operatorname{sign}{\left(\log{\left(x \right)} + \frac{3}{2} \right)}}{x \left(2 \log{\left(x \right)} + 3\right)} - \frac{2 \left(\frac{2 x \delta\left(x\right)}{\operatorname{sign}{\left(x \right)}} - 1\right) \operatorname{sign}^{2}{\left(x \right)} \operatorname{sign}{\left(\log{\left(x \right)} + \frac{3}{2} \right)}}{x^{2} \left(2 \log{\left(x \right)} + 3\right)} - \frac{\left(2 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} + 3\right) \operatorname{sign}^{2}{\left(x \right)} \operatorname{sign}{\left(\log{\left(x \right)} + \frac{3}{2} \right)}}{x^{2} \left(2 \log{\left(x \right)} + 3\right)} - \frac{2 \left(2 \log{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} + 3\right) \operatorname{sign}^{2}{\left(x \right)} \operatorname{sign}{\left(\log{\left(x \right)} + \frac{3}{2} \right)}}{x^{2} \left(2 \log{\left(x \right)} + 3\right)^{2}} = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\left(\cos{\left(x \right)} + \left|{\log{\left(x \right)} + \frac{3}{2}}\right|\right) - 3\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
$$\lim_{x \to \infty}\left(\left(\cos{\left(x \right)} + \left|{\log{\left(x \right)} + \frac{3}{2}}\right|\right) - 3\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
$$\lim_{x \to -\infty}\left(\left(\cos{\left(x \right)} + \left|{\log{\left(x \right)} + \frac{3}{2}}\right|\right) - 3\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
$$\lim_{x \to \infty}\left(\left(\cos{\left(x \right)} + \left|{\log{\left(x \right)} + \frac{3}{2}}\right|\right) - 3\right) = \infty$$
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 cos(x) + Abs(log(x) + 3/2) - 3, dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(\cos{\left(x \right)} + \left|{\log{\left(x \right)} + \frac{3}{2}}\right|\right) - 3}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\left(\cos{\left(x \right)} + \left|{\log{\left(x \right)} + \frac{3}{2}}\right|\right) - 3}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\left(\cos{\left(x \right)} + \left|{\log{\left(x \right)} + \frac{3}{2}}\right|\right) - 3}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\left(\cos{\left(x \right)} + \left|{\log{\left(x \right)} + \frac{3}{2}}\right|\right) - 3}{x}\right) = 0$$
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:
$$\left(\cos{\left(x \right)} + \left|{\log{\left(x \right)} + \frac{3}{2}}\right|\right) - 3 = \cos{\left(x \right)} + \left|{\log{\left(- x \right)} + \frac{3}{2}}\right| - 3$$
- No
$$\left(\cos{\left(x \right)} + \left|{\log{\left(x \right)} + \frac{3}{2}}\right|\right) - 3 = - \cos{\left(x \right)} - \left|{\log{\left(- x \right)} + \frac{3}{2}}\right| + 3$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\left(\cos{\left(x \right)} + \left|{\log{\left(x \right)} + \frac{3}{2}}\right|\right) - 3 = \cos{\left(x \right)} + \left|{\log{\left(- x \right)} + \frac{3}{2}}\right| - 3$$
- No
$$\left(\cos{\left(x \right)} + \left|{\log{\left(x \right)} + \frac{3}{2}}\right|\right) - 3 = - \cos{\left(x \right)} - \left|{\log{\left(- x \right)} + \frac{3}{2}}\right| + 3$$
- No
es decir, función
no es
par ni impar