Sr Examen
Otras calculadoras
- Gráfico de la función implícita
- Derivadas paso a paso
- Integrales paso a paso
- Gráfico de la función paramétrica
- Gráfico en coordenadas polares
- Superficie especificada paramétricamente
- Superficie dada por la ecuación
- Curva en el espacio especificada paramétricamente
- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x^3+3*x^2+2
-
x*(-1-log(x))
-
e^3*x+10*e^2*x
-
-cos(2*x)-sin(2*x)
-
Expresiones idénticas
- sqrt(x- uno)/(x(x- dos)(x- cuatro)(x- seis))
- raíz cuadrada de (x menos 1) dividir por (x(x menos 2)(x menos 4)(x menos 6))
- raíz cuadrada de (x menos uno) dividir por (x(x menos dos)(x menos cuatro)(x menos seis))
- √(x-1)/(x(x-2)(x-4)(x-6))
- sqrtx-1/xx-2x-4x-6
- sqrt(x-1) dividir por (x(x-2)(x-4)(x-6))
-
Expresiones semejantes
- sqrt(x-1)/(x(x-2)(x+4)(x-6))
- sqrt(x-1)/(x(x+2)(x-4)(x-6))
- sqrt(x+1)/(x(x-2)(x-4)(x-6))
- sqrt(x-1)/(x(x-2)(x-4)(x+6))
-
Expresiones con funciones
- Raíz cuadrada sqrt
- sqrt((x-4)/(x+1))
- sqrt(x^2)-x+1
- sqrt(x^2-4x+3)
- sqrt(x^2+4*x)
- sqrt(x^2-8*x+160)
Gráfico de la función y = sqrt(x-1)/(x(x-2)(x-4)(x-6))
Solución
Ha introducido
[src]
_______
\/ x - 1
f(x) = -------------------------
x*(x - 2)*(x - 4)*(x - 6)
$$f{\left(x \right)} = \frac{\sqrt{x - 1}}{x \left(x - 2\right) \left(x - 4\right) \left(x - 6\right)}$$
f = sqrt(x - 1)/((((x*(x - 2))*(x - 4))*(x - 6)))
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 0$$
$$x_{2} = 2$$
$$x_{3} = 4$$
$$x_{4} = 6$$
$$x_{1} = 0$$
$$x_{2} = 2$$
$$x_{3} = 4$$
$$x_{4} = 6$$
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:
$$\frac{\sqrt{x - 1}}{x \left(x - 2\right) \left(x - 4\right) \left(x - 6\right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 1$$
Solución numérica
$$x_{1} = -10463.3520231334$$
$$x_{2} = -18355.8192081777$$
$$x_{3} = -10858.1352691136$$
$$x_{4} = 10418.4460670948$$
$$x_{5} = 4092.27301132531$$
$$x_{6} = -15988.557495318$$
$$x_{7} = 6863.54106929506$$
$$x_{8} = 15943.5730944369$$
$$x_{9} = 11997.3909563043$$
$$x_{10} = -17566.7589971481$$
$$x_{11} = -2541.05574791508$$
$$x_{12} = -19539.3681093536$$
$$x_{13} = 13181.3856980523$$
$$x_{14} = -10068.5379312492$$
$$x_{15} = -19144.8568733011$$
$$x_{16} = -13226.3430210557$$
$$x_{17} = -5721.68041642362$$
$$x_{18} = 12786.7374039534$$
$$x_{19} = 13970.6394351944$$
$$x_{20} = -20328.3768703791$$
$$x_{21} = 9628.80657458026$$
$$x_{22} = 12392.0730165096$$
$$x_{23} = -11252.890949987$$
$$x_{24} = 3695.10777751089$$
$$x_{25} = -9673.68916977333$$
$$x_{26} = -3340.95974714954$$
$$x_{27} = -13620.9815192031$$
$$x_{28} = 2898.52935941839$$
$$x_{29} = 17916.2955322612$$
$$x_{30} = 14365.2472213818$$
$$x_{31} = -14804.8182828537$$
$$x_{32} = -2941.8021254212$$
$$x_{33} = -5325.80598005326$$
$$x_{34} = -15593.9872157644$$
$$x_{35} = -9278.8012548172$$
$$x_{36} = 6468.19172768099$$
$$x_{37} = -14410.2180637099$$
$$x_{38} = -6908.26867253397$$
$$x_{39} = -19933.8746841156$$
$$x_{40} = -16383.1192341825$$
$$x_{41} = -4533.25966811153$$
$$x_{42} = 16338.1320419569$$
$$x_{43} = -6512.87964715932$$
$$x_{44} = 4488.95354451896$$
$$x_{45} = -8093.84440311863$$
$$x_{46} = -8883.86889313524$$
$$x_{47} = 9233.93265012983$$
$$x_{48} = -7303.54827820532$$
$$x_{49} = -12831.6893655994$$
$$x_{50} = 8839.01619570758$$
$$x_{51} = -8488.88579054538$$
$$x_{52} = 15549.0058212758$$
$$x_{53} = 14759.8436400939$$
$$x_{54} = 10023.6429659865$$
$$x_{55} = -6117.35951234089$$
$$x_{56} = 8444.05128452171$$
$$x_{57} = 11207.9663931036$$
$$x_{58} = -17961.2921077183$$
$$x_{59} = 3297.28347966006$$
$$x_{60} = 18310.8206573651$$
$$x_{61} = 13576.0192953367$$
$$x_{62} = -12437.0190967089$$
$$x_{63} = 5281.29539464981$$
$$x_{64} = 15154.4295752189$$
$$x_{65} = -14015.6061507218$$
$$x_{66} = 16732.6832502719$$
$$x_{67} = -11647.621896795$$
$$x_{68} = -12042.3305654038$$
$$x_{69} = 7258.78730287627$$
$$x_{70} = -15199.4077255852$$
$$x_{71} = 19099.854732456$$
$$x_{72} = 5677.09823025008$$
$$x_{73} = 7653.94630137329$$
$$x_{74} = -4136.42930203729$$
$$x_{75} = -17172.2194596259$$
$$x_{76} = -16777.6730388327$$
$$x_{77} = 11602.6894300542$$
$$x_{78} = 6072.71929150701$$
$$x_{79} = 17521.7645318927$$
$$x_{80} = -3739.06298804957$$
$$x_{81} = 8049.03083262451$$
$$x_{82} = -7698.73561010356$$
$$x_{83} = -18750.3406800699$$
$$x_{84} = -4929.68774390662$$
$$x_{85} = 17127.227251982$$
$$x_{86} = 18705.3402774955$$
$$x_{87} = 4885.26692970721$$
$$x_{88} = 10813.2195037302$$
o sea hay que resolver la ecuación:
$$\frac{\sqrt{x - 1}}{x \left(x - 2\right) \left(x - 4\right) \left(x - 6\right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 1$$
Solución numérica
$$x_{1} = -10463.3520231334$$
$$x_{2} = -18355.8192081777$$
$$x_{3} = -10858.1352691136$$
$$x_{4} = 10418.4460670948$$
$$x_{5} = 4092.27301132531$$
$$x_{6} = -15988.557495318$$
$$x_{7} = 6863.54106929506$$
$$x_{8} = 15943.5730944369$$
$$x_{9} = 11997.3909563043$$
$$x_{10} = -17566.7589971481$$
$$x_{11} = -2541.05574791508$$
$$x_{12} = -19539.3681093536$$
$$x_{13} = 13181.3856980523$$
$$x_{14} = -10068.5379312492$$
$$x_{15} = -19144.8568733011$$
$$x_{16} = -13226.3430210557$$
$$x_{17} = -5721.68041642362$$
$$x_{18} = 12786.7374039534$$
$$x_{19} = 13970.6394351944$$
$$x_{20} = -20328.3768703791$$
$$x_{21} = 9628.80657458026$$
$$x_{22} = 12392.0730165096$$
$$x_{23} = -11252.890949987$$
$$x_{24} = 3695.10777751089$$
$$x_{25} = -9673.68916977333$$
$$x_{26} = -3340.95974714954$$
$$x_{27} = -13620.9815192031$$
$$x_{28} = 2898.52935941839$$
$$x_{29} = 17916.2955322612$$
$$x_{30} = 14365.2472213818$$
$$x_{31} = -14804.8182828537$$
$$x_{32} = -2941.8021254212$$
$$x_{33} = -5325.80598005326$$
$$x_{34} = -15593.9872157644$$
$$x_{35} = -9278.8012548172$$
$$x_{36} = 6468.19172768099$$
$$x_{37} = -14410.2180637099$$
$$x_{38} = -6908.26867253397$$
$$x_{39} = -19933.8746841156$$
$$x_{40} = -16383.1192341825$$
$$x_{41} = -4533.25966811153$$
$$x_{42} = 16338.1320419569$$
$$x_{43} = -6512.87964715932$$
$$x_{44} = 4488.95354451896$$
$$x_{45} = -8093.84440311863$$
$$x_{46} = -8883.86889313524$$
$$x_{47} = 9233.93265012983$$
$$x_{48} = -7303.54827820532$$
$$x_{49} = -12831.6893655994$$
$$x_{50} = 8839.01619570758$$
$$x_{51} = -8488.88579054538$$
$$x_{52} = 15549.0058212758$$
$$x_{53} = 14759.8436400939$$
$$x_{54} = 10023.6429659865$$
$$x_{55} = -6117.35951234089$$
$$x_{56} = 8444.05128452171$$
$$x_{57} = 11207.9663931036$$
$$x_{58} = -17961.2921077183$$
$$x_{59} = 3297.28347966006$$
$$x_{60} = 18310.8206573651$$
$$x_{61} = 13576.0192953367$$
$$x_{62} = -12437.0190967089$$
$$x_{63} = 5281.29539464981$$
$$x_{64} = 15154.4295752189$$
$$x_{65} = -14015.6061507218$$
$$x_{66} = 16732.6832502719$$
$$x_{67} = -11647.621896795$$
$$x_{68} = -12042.3305654038$$
$$x_{69} = 7258.78730287627$$
$$x_{70} = -15199.4077255852$$
$$x_{71} = 19099.854732456$$
$$x_{72} = 5677.09823025008$$
$$x_{73} = 7653.94630137329$$
$$x_{74} = -4136.42930203729$$
$$x_{75} = -17172.2194596259$$
$$x_{76} = -16777.6730388327$$
$$x_{77} = 11602.6894300542$$
$$x_{78} = 6072.71929150701$$
$$x_{79} = 17521.7645318927$$
$$x_{80} = -3739.06298804957$$
$$x_{81} = 8049.03083262451$$
$$x_{82} = -7698.73561010356$$
$$x_{83} = -18750.3406800699$$
$$x_{84} = -4929.68774390662$$
$$x_{85} = 17127.227251982$$
$$x_{86} = 18705.3402774955$$
$$x_{87} = 4885.26692970721$$
$$x_{88} = 10813.2195037302$$
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
$$\frac{\frac{1}{x} \frac{1}{x - 6} \frac{1}{x - 4} \frac{1}{x - 2}}{2 \sqrt{x - 1}} + \frac{\sqrt{x - 1} \left(- x \left(x - 2\right) \left(x - 4\right) - \left(x - 6\right) \left(x \left(x - 2\right) + \left(x - 4\right) \left(2 x - 2\right)\right)\right)}{x^{2} \left(x - 6\right)^{2} \left(x - 4\right)^{2} \left(x - 2\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = - \sqrt{- 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} - \frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + \frac{432}{343 \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}}} + \frac{102}{49}} + \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}} + \frac{17}{7}$$
$$x_{2} = \sqrt{- 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} - \frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + \frac{432}{343 \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}}} + \frac{102}{49}} + \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}} + \frac{17}{7}$$
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:
$$x_{1} = - \sqrt{- 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} - \frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + \frac{432}{343 \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}}} + \frac{102}{49}} + \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}} + \frac{17}{7}$$
Puntos máximos de la función:
$$x_{1} = \sqrt{- 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} - \frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + \frac{432}{343 \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}}} + \frac{102}{49}} + \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}} + \frac{17}{7}$$
Decrece en los intervalos
$$\left[- \sqrt{- 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} - \frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + \frac{432}{343 \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}}} + \frac{102}{49}} + \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}} + \frac{17}{7}, \sqrt{- 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} - \frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + \frac{432}{343 \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}}} + \frac{102}{49}} + \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}} + \frac{17}{7}\right]$$
Crece en los intervalos
$$\left(-\infty, - \sqrt{- 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} - \frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + \frac{432}{343 \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}}} + \frac{102}{49}} + \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}} + \frac{17}{7}\right] \cup \left[\sqrt{- 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} - \frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + \frac{432}{343 \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}}} + \frac{102}{49}} + \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}} + \frac{17}{7}, \infty\right)$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{\frac{1}{x} \frac{1}{x - 6} \frac{1}{x - 4} \frac{1}{x - 2}}{2 \sqrt{x - 1}} + \frac{\sqrt{x - 1} \left(- x \left(x - 2\right) \left(x - 4\right) - \left(x - 6\right) \left(x \left(x - 2\right) + \left(x - 4\right) \left(2 x - 2\right)\right)\right)}{x^{2} \left(x - 6\right)^{2} \left(x - 4\right)^{2} \left(x - 2\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = - \sqrt{- 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} - \frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + \frac{432}{343 \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}}} + \frac{102}{49}} + \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}} + \frac{17}{7}$$
$$x_{2} = \sqrt{- 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} - \frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + \frac{432}{343 \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}}} + \frac{102}{49}} + \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}} + \frac{17}{7}$$
Signos de extremos en los puntos:
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
/ ______________________________________________________________ ______________________________________________________________________________________________________________________________________________
/ / __________________ / __________________
/ / / _______ / / _______
/ 10 / 51 / 177 \/ 51393 83 / 102 / 177 \/ 51393 83 432
/ -- + / -- + 2*3 / ---- + --------- + --------------------------- - / --- - 2*3 / ---- + --------- - --------------------------- + ----------------------------------------------------------------------------
/ 7 / 49 \/ 2744 6174 __________________ / 49 \/ 2744 6174 __________________ ______________________________________________________________
/ / / _______ / / _______ / __________________
/ / / 177 \/ 51393 / / 177 \/ 51393 / / _______
/ / 294*3 / ---- + --------- / 294*3 / ---- + --------- / 51 / 177 \/ 51393 83
/ \/ \/ 2744 6174 / \/ 2744 6174 343* / -- + 2*3 / ---- + --------- + ---------------------------
/ / / 49 \/ 2744 6174 __________________
______________________________________________________________ ______________________________________________________________________________________________________________________________________________ / / / / _______
/ __________________ / __________________ / / / / 177 \/ 51393
/ / _______ / / _______ / / / 294*3 / ---- + ---------
17 / 51 / 177 \/ 51393 83 / 102 / 177 \/ 51393 83 432 \/ \/ \/ \/ 2744 6174
(-- + / -- + 2*3 / ---- + --------- + --------------------------- - / --- - 2*3 / ---- + --------- - --------------------------- + ----------------------------------------------------------------------------, --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------)
7 / 49 \/ 2744 6174 __________________ / 49 \/ 2744 6174 __________________ ______________________________________________________________ / ______________________________________________________________ ______________________________________________________________________________________________________________________________________________\ / ______________________________________________________________ ______________________________________________________________________________________________________________________________________________\ / ______________________________________________________________ ______________________________________________________________________________________________________________________________________________\ / ______________________________________________________________ ______________________________________________________________________________________________________________________________________________\
/ / _______ / / _______ / __________________ | / __________________ / __________________ | | / __________________ / __________________ | | / __________________ / __________________ | | / __________________ / __________________ |
/ / 177 \/ 51393 / / 177 \/ 51393 / / _______ | / / _______ / / _______ | | / / _______ / / _______ | | / / _______ / / _______ | | / / _______ / / _______ |
/ 294*3 / ---- + --------- / 294*3 / ---- + --------- / 51 / 177 \/ 51393 83 | 25 / 51 / 177 \/ 51393 83 / 102 / 177 \/ 51393 83 432 | | 11 / 51 / 177 \/ 51393 83 / 102 / 177 \/ 51393 83 432 | |3 / 51 / 177 \/ 51393 83 / 102 / 177 \/ 51393 83 432 | |17 / 51 / 177 \/ 51393 83 / 102 / 177 \/ 51393 83 432 |
\/ \/ 2744 6174 / \/ 2744 6174 343* / -- + 2*3 / ---- + --------- + --------------------------- |- -- + / -- + 2*3 / ---- + --------- + --------------------------- - / --- - 2*3 / ---- + --------- - --------------------------- + ---------------------------------------------------------------------------- |*|- -- + / -- + 2*3 / ---- + --------- + --------------------------- - / --- - 2*3 / ---- + --------- - --------------------------- + ---------------------------------------------------------------------------- |*|- + / -- + 2*3 / ---- + --------- + --------------------------- - / --- - 2*3 / ---- + --------- - --------------------------- + ---------------------------------------------------------------------------- |*|-- + / -- + 2*3 / ---- + --------- + --------------------------- - / --- - 2*3 / ---- + --------- - --------------------------- + ---------------------------------------------------------------------------- |
/ / 49 \/ 2744 6174 __________________ | 7 / 49 \/ 2744 6174 __________________ / 49 \/ 2744 6174 __________________ ______________________________________________________________ | | 7 / 49 \/ 2744 6174 __________________ / 49 \/ 2744 6174 __________________ ______________________________________________________________ | |7 / 49 \/ 2744 6174 __________________ / 49 \/ 2744 6174 __________________ ______________________________________________________________ | |7 / 49 \/ 2744 6174 __________________ / 49 \/ 2744 6174 __________________ ______________________________________________________________ |
/ / / _______ | / / _______ / / _______ / __________________ | | / / _______ / / _______ / __________________ | | / / _______ / / _______ / __________________ | | / / _______ / / _______ / __________________ |
/ / / 177 \/ 51393 | / / 177 \/ 51393 / / 177 \/ 51393 / / _______ | | / / 177 \/ 51393 / / 177 \/ 51393 / / _______ | | / / 177 \/ 51393 / / 177 \/ 51393 / / _______ | | / / 177 \/ 51393 / / 177 \/ 51393 / / _______ |
/ / 294*3 / ---- + --------- | / 294*3 / ---- + --------- / 294*3 / ---- + --------- / 51 / 177 \/ 51393 83 | | / 294*3 / ---- + --------- / 294*3 / ---- + --------- / 51 / 177 \/ 51393 83 | | / 294*3 / ---- + --------- / 294*3 / ---- + --------- / 51 / 177 \/ 51393 83 | | / 294*3 / ---- + --------- / 294*3 / ---- + --------- / 51 / 177 \/ 51393 83 |
\/ \/ \/ 2744 6174 | \/ \/ 2744 6174 / \/ 2744 6174 343* / -- + 2*3 / ---- + --------- + --------------------------- | | \/ \/ 2744 6174 / \/ 2744 6174 343* / -- + 2*3 / ---- + --------- + --------------------------- | | \/ \/ 2744 6174 / \/ 2744 6174 343* / -- + 2*3 / ---- + --------- + --------------------------- | | \/ \/ 2744 6174 / \/ 2744 6174 343* / -- + 2*3 / ---- + --------- + --------------------------- |
| / / 49 \/ 2744 6174 __________________ | | / / 49 \/ 2744 6174 __________________ | | / / 49 \/ 2744 6174 __________________ | | / / 49 \/ 2744 6174 __________________ |
| / / / _______ | | / / / _______ | | / / / _______ | | / / / _______ |
| / / / 177 \/ 51393 | | / / / 177 \/ 51393 | | / / / 177 \/ 51393 | | / / / 177 \/ 51393 |
| / / 294*3 / ---- + --------- | | / / 294*3 / ---- + --------- | | / / 294*3 / ---- + --------- | | / / 294*3 / ---- + --------- |
\ \/ \/ \/ 2744 6174 / \ \/ \/ \/ 2744 6174 / \ \/ \/ \/ 2744 6174 / \ \/ \/ \/ 2744 6174 / _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
/ ______________________________________________________________ ______________________________________________________________________________________________________________________________________________
/ / __________________ / __________________
/ / / _______ / / _______
/ 10 / 51 / 177 \/ 51393 83 / 102 / 177 \/ 51393 83 432
/ -- + / -- + 2*3 / ---- + --------- + --------------------------- + / --- - 2*3 / ---- + --------- - --------------------------- + ----------------------------------------------------------------------------
/ 7 / 49 \/ 2744 6174 __________________ / 49 \/ 2744 6174 __________________ ______________________________________________________________
/ / / _______ / / _______ / __________________
/ / / 177 \/ 51393 / / 177 \/ 51393 / / _______
/ / 294*3 / ---- + --------- / 294*3 / ---- + --------- / 51 / 177 \/ 51393 83
/ \/ \/ 2744 6174 / \/ 2744 6174 343* / -- + 2*3 / ---- + --------- + ---------------------------
/ / / 49 \/ 2744 6174 __________________
______________________________________________________________ ______________________________________________________________________________________________________________________________________________ / / / / _______
/ __________________ / __________________ / / / / 177 \/ 51393
/ / _______ / / _______ / / / 294*3 / ---- + ---------
17 / 51 / 177 \/ 51393 83 / 102 / 177 \/ 51393 83 432 \/ \/ \/ \/ 2744 6174
(-- + / -- + 2*3 / ---- + --------- + --------------------------- + / --- - 2*3 / ---- + --------- - --------------------------- + ----------------------------------------------------------------------------, --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------)
7 / 49 \/ 2744 6174 __________________ / 49 \/ 2744 6174 __________________ ______________________________________________________________ / ______________________________________________________________ ______________________________________________________________________________________________________________________________________________\ / ______________________________________________________________ ______________________________________________________________________________________________________________________________________________\ / ______________________________________________________________ ______________________________________________________________________________________________________________________________________________\ / ______________________________________________________________ ______________________________________________________________________________________________________________________________________________\
/ / _______ / / _______ / __________________ | / __________________ / __________________ | | / __________________ / __________________ | | / __________________ / __________________ | | / __________________ / __________________ |
/ / 177 \/ 51393 / / 177 \/ 51393 / / _______ | / / _______ / / _______ | | / / _______ / / _______ | | / / _______ / / _______ | | / / _______ / / _______ |
/ 294*3 / ---- + --------- / 294*3 / ---- + --------- / 51 / 177 \/ 51393 83 | 25 / 51 / 177 \/ 51393 83 / 102 / 177 \/ 51393 83 432 | | 11 / 51 / 177 \/ 51393 83 / 102 / 177 \/ 51393 83 432 | |3 / 51 / 177 \/ 51393 83 / 102 / 177 \/ 51393 83 432 | |17 / 51 / 177 \/ 51393 83 / 102 / 177 \/ 51393 83 432 |
\/ \/ 2744 6174 / \/ 2744 6174 343* / -- + 2*3 / ---- + --------- + --------------------------- |- -- + / -- + 2*3 / ---- + --------- + --------------------------- + / --- - 2*3 / ---- + --------- - --------------------------- + ---------------------------------------------------------------------------- |*|- -- + / -- + 2*3 / ---- + --------- + --------------------------- + / --- - 2*3 / ---- + --------- - --------------------------- + ---------------------------------------------------------------------------- |*|- + / -- + 2*3 / ---- + --------- + --------------------------- + / --- - 2*3 / ---- + --------- - --------------------------- + ---------------------------------------------------------------------------- |*|-- + / -- + 2*3 / ---- + --------- + --------------------------- + / --- - 2*3 / ---- + --------- - --------------------------- + ---------------------------------------------------------------------------- |
/ / 49 \/ 2744 6174 __________________ | 7 / 49 \/ 2744 6174 __________________ / 49 \/ 2744 6174 __________________ ______________________________________________________________ | | 7 / 49 \/ 2744 6174 __________________ / 49 \/ 2744 6174 __________________ ______________________________________________________________ | |7 / 49 \/ 2744 6174 __________________ / 49 \/ 2744 6174 __________________ ______________________________________________________________ | |7 / 49 \/ 2744 6174 __________________ / 49 \/ 2744 6174 __________________ ______________________________________________________________ |
/ / / _______ | / / _______ / / _______ / __________________ | | / / _______ / / _______ / __________________ | | / / _______ / / _______ / __________________ | | / / _______ / / _______ / __________________ |
/ / / 177 \/ 51393 | / / 177 \/ 51393 / / 177 \/ 51393 / / _______ | | / / 177 \/ 51393 / / 177 \/ 51393 / / _______ | | / / 177 \/ 51393 / / 177 \/ 51393 / / _______ | | / / 177 \/ 51393 / / 177 \/ 51393 / / _______ |
/ / 294*3 / ---- + --------- | / 294*3 / ---- + --------- / 294*3 / ---- + --------- / 51 / 177 \/ 51393 83 | | / 294*3 / ---- + --------- / 294*3 / ---- + --------- / 51 / 177 \/ 51393 83 | | / 294*3 / ---- + --------- / 294*3 / ---- + --------- / 51 / 177 \/ 51393 83 | | / 294*3 / ---- + --------- / 294*3 / ---- + --------- / 51 / 177 \/ 51393 83 |
\/ \/ \/ 2744 6174 | \/ \/ 2744 6174 / \/ 2744 6174 343* / -- + 2*3 / ---- + --------- + --------------------------- | | \/ \/ 2744 6174 / \/ 2744 6174 343* / -- + 2*3 / ---- + --------- + --------------------------- | | \/ \/ 2744 6174 / \/ 2744 6174 343* / -- + 2*3 / ---- + --------- + --------------------------- | | \/ \/ 2744 6174 / \/ 2744 6174 343* / -- + 2*3 / ---- + --------- + --------------------------- |
| / / 49 \/ 2744 6174 __________________ | | / / 49 \/ 2744 6174 __________________ | | / / 49 \/ 2744 6174 __________________ | | / / 49 \/ 2744 6174 __________________ |
| / / / _______ | | / / / _______ | | / / / _______ | | / / / _______ |
| / / / 177 \/ 51393 | | / / / 177 \/ 51393 | | / / / 177 \/ 51393 | | / / / 177 \/ 51393 |
| / / 294*3 / ---- + --------- | | / / 294*3 / ---- + --------- | | / / 294*3 / ---- + --------- | | / / 294*3 / ---- + --------- |
\ \/ \/ \/ 2744 6174 / \ \/ \/ \/ 2744 6174 / \ \/ \/ \/ 2744 6174 / \ \/ \/ \/ 2744 6174 / 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:
$$x_{1} = - \sqrt{- 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} - \frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + \frac{432}{343 \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}}} + \frac{102}{49}} + \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}} + \frac{17}{7}$$
Puntos máximos de la función:
$$x_{1} = \sqrt{- 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} - \frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + \frac{432}{343 \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}}} + \frac{102}{49}} + \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}} + \frac{17}{7}$$
Decrece en los intervalos
$$\left[- \sqrt{- 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} - \frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + \frac{432}{343 \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}}} + \frac{102}{49}} + \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}} + \frac{17}{7}, \sqrt{- 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} - \frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + \frac{432}{343 \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}}} + \frac{102}{49}} + \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}} + \frac{17}{7}\right]$$
Crece en los intervalos
$$\left(-\infty, - \sqrt{- 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} - \frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + \frac{432}{343 \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}}} + \frac{102}{49}} + \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}} + \frac{17}{7}\right] \cup \left[\sqrt{- 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} - \frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + \frac{432}{343 \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}}} + \frac{102}{49}} + \sqrt{\frac{83}{294 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}}} + 2 \sqrt[3]{\frac{\sqrt{51393}}{6174} + \frac{177}{2744}} + \frac{51}{49}} + \frac{17}{7}, \infty\right)$$
Asíntotas verticales
Hay:
$$x_{1} = 0$$
$$x_{2} = 2$$
$$x_{3} = 4$$
$$x_{4} = 6$$
$$x_{1} = 0$$
$$x_{2} = 2$$
$$x_{3} = 4$$
$$x_{4} = 6$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x - 1}}{x \left(x - 2\right) \left(x - 4\right) \left(x - 6\right)}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sqrt{x - 1}}{x \left(x - 2\right) \left(x - 4\right) \left(x - 6\right)}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x - 1}}{x \left(x - 2\right) \left(x - 4\right) \left(x - 6\right)}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sqrt{x - 1}}{x \left(x - 2\right) \left(x - 4\right) \left(x - 6\right)}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sqrt(x - 1)/((((x*(x - 2))*(x - 4))*(x - 6))), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{1}{x \left(x - 6\right) \left(x - 4\right) \left(x - 2\right)} \sqrt{x - 1}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\frac{1}{x \left(x - 6\right) \left(x - 4\right) \left(x - 2\right)} \sqrt{x - 1}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\frac{1}{x \left(x - 6\right) \left(x - 4\right) \left(x - 2\right)} \sqrt{x - 1}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\frac{1}{x \left(x - 6\right) \left(x - 4\right) \left(x - 2\right)} \sqrt{x - 1}}{x}\right) = 0$$
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:
$$\frac{\sqrt{x - 1}}{x \left(x - 2\right) \left(x - 4\right) \left(x - 6\right)} = - \frac{\sqrt{- x - 1}}{x \left(- x - 6\right) \left(- x - 4\right) \left(- x - 2\right)}$$
- No
$$\frac{\sqrt{x - 1}}{x \left(x - 2\right) \left(x - 4\right) \left(x - 6\right)} = \frac{\sqrt{- x - 1}}{x \left(- x - 6\right) \left(- x - 4\right) \left(- x - 2\right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\sqrt{x - 1}}{x \left(x - 2\right) \left(x - 4\right) \left(x - 6\right)} = - \frac{\sqrt{- x - 1}}{x \left(- x - 6\right) \left(- x - 4\right) \left(- x - 2\right)}$$
- No
$$\frac{\sqrt{x - 1}}{x \left(x - 2\right) \left(x - 4\right) \left(x - 6\right)} = \frac{\sqrt{- x - 1}}{x \left(- x - 6\right) \left(- x - 4\right) \left(- x - 2\right)}$$
- No
es decir, función
no es
par ni impar