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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
- absolute(cosx- uno)
- absolute( coseno de x menos 1)
- absolute( coseno de x menos uno)
- absolutecosx-1
-
Expresiones semejantes
- absolute(cosx+1)
-
Expresiones con funciones
- absolute
- absolute((x+2)^2-4)
- absolute(x+6)-3x
- absolute(sin(x))/sin(x)
- absolute(3x+1)cbrt(1/(3x)+1)
- absolute((9-(absolute(x)))/(absolute(x)-3))
- cosx
- cosx-(cosx)^2
- cosx+1.5
- cosx/((x-2)(x+10))
- cosx/(1-2cosx+sinx)
- cosx+1/3cosx
Gráfico de la función y = absolute(cosx-1)
Solución
Ha introducido
[src]
f(x) = |cos(x) - 1|
$$f{\left(x \right)} = \left|{\cos{\left(x \right)} - 1}\right|$$
f = Abs(cos(x) - 1)
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:
$$\left|{\cos{\left(x \right)} - 1}\right| = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = 2 \pi$$
Solución numérica
$$x_{1} = 62.8318535568358$$
$$x_{2} = 37.6991115173992$$
$$x_{3} = -31.4159260507536$$
$$x_{4} = -37.6991113479743$$
$$x_{5} = -50.2654829667315$$
$$x_{6} = -18.8495563230046$$
$$x_{7} = 94.2477800892631$$
$$x_{8} = 100.530965157364$$
$$x_{9} = 31.4159268459961$$
$$x_{10} = -94.2477801171671$$
$$x_{11} = 12.5663710110881$$
$$x_{12} = -43.9822976246252$$
$$x_{13} = -94.2477797298079$$
$$x_{14} = -87.964593928489$$
$$x_{15} = -62.8318534787248$$
$$x_{16} = 81.6814084860076$$
$$x_{17} = 6.28318579821791$$
$$x_{18} = 100.530964759815$$
$$x_{19} = -75.3982238741744$$
$$x_{20} = 56.5486668532011$$
$$x_{21} = 87.964594335905$$
$$x_{22} = -56.5486682426592$$
$$x_{23} = -12.5663710889626$$
$$x_{24} = -43.9822971745392$$
$$x_{25} = 81.6814085860518$$
$$x_{26} = -56.5486674685864$$
$$x_{27} = -69.1150379045123$$
$$x_{28} = -62.831852673202$$
$$x_{29} = -25.1327407505866$$
$$x_{30} = 56.5486682809363$$
$$x_{31} = 50.2654824463392$$
$$x_{32} = -31.4159267157965$$
$$x_{33} = 56.5486680806249$$
$$x_{34} = 6.28318626747926$$
$$x_{35} = 37.6991120311338$$
$$x_{36} = -18.8495552124105$$
$$x_{37} = 69.115038794053$$
$$x_{38} = -87.9645947692094$$
$$x_{39} = 0$$
$$x_{40} = 87.9645946044253$$
$$x_{41} = -50.265482641087$$
$$x_{42} = 25.1327416384075$$
$$x_{43} = -12.5663703112531$$
$$x_{44} = -6.2831858160515$$
$$x_{45} = -75.3982231720141$$
$$x_{46} = -69.1150386869085$$
$$x_{47} = -25.1327415297174$$
$$x_{48} = 37.6991113348642$$
$$x_{49} = 69.1150379887504$$
$$x_{50} = 12.5663704426592$$
$$x_{51} = -6.2831851275477$$
$$x_{52} = 87.9645938121814$$
$$x_{53} = -37.6991121287155$$
$$x_{54} = -81.6814075578313$$
$$x_{55} = -100.530964626003$$
$$x_{56} = -81.6814092565354$$
$$x_{57} = -18.8495555173448$$
$$x_{58} = -37.6991118772631$$
$$x_{59} = 75.3982232188727$$
$$x_{60} = -43.9822967932182$$
$$x_{61} = -31.4159260208155$$
$$x_{62} = -94.2477794452815$$
$$x_{63} = 6.28318528420851$$
$$x_{64} = 94.2477792651059$$
$$x_{65} = -81.6814090382277$$
$$x_{66} = 94.2477796093523$$
$$x_{67} = -6.28318555849548$$
$$x_{68} = 50.2654821322586$$
$$x_{69} = 25.1327408328211$$
$$x_{70} = -50.2654822863493$$
$$x_{71} = 62.8318527849002$$
$$x_{72} = 43.9822974733639$$
$$x_{73} = 81.6814091897036$$
$$x_{74} = 6.28318500093652$$
$$x_{75} = -81.6814084945807$$
$$x_{76} = 12.5663711301703$$
$$x_{77} = 43.9822971694647$$
$$x_{78} = 75.3982240031607$$
$$x_{79} = -75.3982231045728$$
$$x_{80} = 43.9822966661001$$
$$x_{81} = 31.4159260648825$$
$$x_{82} = 18.8495556275525$$
$$x_{83} = 18.8495564031971$$
$$x_{84} = 50.2654829439723$$
$$x_{85} = 56.5486676011951$$
$$x_{86} = -87.9645943586158$$
o sea hay que resolver la ecuación:
$$\left|{\cos{\left(x \right)} - 1}\right| = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = 2 \pi$$
Solución numérica
$$x_{1} = 62.8318535568358$$
$$x_{2} = 37.6991115173992$$
$$x_{3} = -31.4159260507536$$
$$x_{4} = -37.6991113479743$$
$$x_{5} = -50.2654829667315$$
$$x_{6} = -18.8495563230046$$
$$x_{7} = 94.2477800892631$$
$$x_{8} = 100.530965157364$$
$$x_{9} = 31.4159268459961$$
$$x_{10} = -94.2477801171671$$
$$x_{11} = 12.5663710110881$$
$$x_{12} = -43.9822976246252$$
$$x_{13} = -94.2477797298079$$
$$x_{14} = -87.964593928489$$
$$x_{15} = -62.8318534787248$$
$$x_{16} = 81.6814084860076$$
$$x_{17} = 6.28318579821791$$
$$x_{18} = 100.530964759815$$
$$x_{19} = -75.3982238741744$$
$$x_{20} = 56.5486668532011$$
$$x_{21} = 87.964594335905$$
$$x_{22} = -56.5486682426592$$
$$x_{23} = -12.5663710889626$$
$$x_{24} = -43.9822971745392$$
$$x_{25} = 81.6814085860518$$
$$x_{26} = -56.5486674685864$$
$$x_{27} = -69.1150379045123$$
$$x_{28} = -62.831852673202$$
$$x_{29} = -25.1327407505866$$
$$x_{30} = 56.5486682809363$$
$$x_{31} = 50.2654824463392$$
$$x_{32} = -31.4159267157965$$
$$x_{33} = 56.5486680806249$$
$$x_{34} = 6.28318626747926$$
$$x_{35} = 37.6991120311338$$
$$x_{36} = -18.8495552124105$$
$$x_{37} = 69.115038794053$$
$$x_{38} = -87.9645947692094$$
$$x_{39} = 0$$
$$x_{40} = 87.9645946044253$$
$$x_{41} = -50.265482641087$$
$$x_{42} = 25.1327416384075$$
$$x_{43} = -12.5663703112531$$
$$x_{44} = -6.2831858160515$$
$$x_{45} = -75.3982231720141$$
$$x_{46} = -69.1150386869085$$
$$x_{47} = -25.1327415297174$$
$$x_{48} = 37.6991113348642$$
$$x_{49} = 69.1150379887504$$
$$x_{50} = 12.5663704426592$$
$$x_{51} = -6.2831851275477$$
$$x_{52} = 87.9645938121814$$
$$x_{53} = -37.6991121287155$$
$$x_{54} = -81.6814075578313$$
$$x_{55} = -100.530964626003$$
$$x_{56} = -81.6814092565354$$
$$x_{57} = -18.8495555173448$$
$$x_{58} = -37.6991118772631$$
$$x_{59} = 75.3982232188727$$
$$x_{60} = -43.9822967932182$$
$$x_{61} = -31.4159260208155$$
$$x_{62} = -94.2477794452815$$
$$x_{63} = 6.28318528420851$$
$$x_{64} = 94.2477792651059$$
$$x_{65} = -81.6814090382277$$
$$x_{66} = 94.2477796093523$$
$$x_{67} = -6.28318555849548$$
$$x_{68} = 50.2654821322586$$
$$x_{69} = 25.1327408328211$$
$$x_{70} = -50.2654822863493$$
$$x_{71} = 62.8318527849002$$
$$x_{72} = 43.9822974733639$$
$$x_{73} = 81.6814091897036$$
$$x_{74} = 6.28318500093652$$
$$x_{75} = -81.6814084945807$$
$$x_{76} = 12.5663711301703$$
$$x_{77} = 43.9822971694647$$
$$x_{78} = 75.3982240031607$$
$$x_{79} = -75.3982231045728$$
$$x_{80} = 43.9822966661001$$
$$x_{81} = 31.4159260648825$$
$$x_{82} = 18.8495556275525$$
$$x_{83} = 18.8495564031971$$
$$x_{84} = 50.2654829439723$$
$$x_{85} = 56.5486676011951$$
$$x_{86} = -87.9645943586158$$
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
$$- \sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} - 1 \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -267.035375555132$$
$$x_{2} = -31.4159265358979$$
$$x_{3} = 47.1238898038469$$
$$x_{4} = -75.398223686155$$
$$x_{5} = 9.42477796076938$$
$$x_{6} = 97.3893722612836$$
$$x_{7} = -34.5575191894877$$
$$x_{8} = -97.3893722612836$$
$$x_{9} = -62.8318530717959$$
$$x_{10} = 18.8495559215388$$
$$x_{11} = 87.9645943005142$$
$$x_{12} = -3.14159265358979$$
$$x_{13} = -87.9645943005142$$
$$x_{14} = 6.28318530717959$$
$$x_{15} = 59.6902604182061$$
$$x_{16} = -47.1238898038469$$
$$x_{17} = 100.530964914873$$
$$x_{18} = -40.8407044966673$$
$$x_{19} = 62.8318530717959$$
$$x_{20} = -75.398223686155$$
$$x_{21} = 3.14159265358979$$
$$x_{22} = 28.2743338823081$$
$$x_{23} = -69.1150383789755$$
$$x_{24} = 94.2477796076938$$
$$x_{25} = -50.2654824574367$$
$$x_{26} = 6.28318530717959$$
$$x_{27} = 31.4159265358979$$
$$x_{28} = 43.9822971502571$$
$$x_{29} = -37.6991118430775$$
$$x_{30} = -94.2477796076938$$
$$x_{31} = -59.6902604182061$$
$$x_{32} = -31.4159265358979$$
$$x_{33} = -56.5486677646163$$
$$x_{34} = 25.1327412287183$$
$$x_{35} = 81.6814089933346$$
$$x_{36} = 21.9911485751286$$
$$x_{37} = 69.1150383789755$$
$$x_{38} = 15.707963267949$$
$$x_{39} = 34.5575191894877$$
$$x_{40} = -91.106186954104$$
$$x_{41} = 40.8407044966673$$
$$x_{42} = 37.6991118430775$$
$$x_{43} = 65.9734457253857$$
$$x_{44} = -72.2566310325652$$
$$x_{45} = 56.5486677646163$$
$$x_{46} = -21.9911485751286$$
$$x_{47} = 53.4070751110265$$
$$x_{48} = 91.106186954104$$
$$x_{49} = -28.2743338823081$$
$$x_{50} = 56.5486677646163$$
$$x_{51} = -65.9734457253857$$
$$x_{52} = -53.4070751110265$$
$$x_{53} = -100.530964914873$$
$$x_{54} = -18.8495559215388$$
$$x_{55} = -113.097335529233$$
$$x_{56} = -15.707963267949$$
$$x_{57} = -6.28318530717957$$
$$x_{58} = 84.8230016469244$$
$$x_{59} = 18.8495559215388$$
$$x_{60} = 72.2566310325652$$
$$x_{61} = -87.9645943005142$$
$$x_{62} = -232.477856365645$$
$$x_{63} = 0$$
$$x_{64} = -43.9822971502571$$
$$x_{65} = -78.5398163397448$$
$$x_{66} = 12.5663706143592$$
$$x_{67} = -12.5663706143592$$
$$x_{68} = 75.398223686155$$
$$x_{69} = -84.8230016469244$$
$$x_{70} = -25.1327412287183$$
$$x_{71} = 78.5398163397448$$
$$x_{72} = -81.6814089933346$$
$$x_{73} = -9.42477796076938$$
$$x_{74} = 50.2654824574367$$
$$x_{75} = -2642.07942166902$$
$$x_{76} = -25.1327412287183$$
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:
$$x_{1} = -31.4159265358979$$
$$x_{2} = -75.398223686155$$
$$x_{3} = -62.8318530717959$$
$$x_{4} = 18.8495559215388$$
$$x_{5} = 87.9645943005142$$
$$x_{6} = -87.9645943005142$$
$$x_{7} = 6.28318530717959$$
$$x_{8} = 100.530964914873$$
$$x_{9} = 62.8318530717959$$
$$x_{10} = -75.398223686155$$
$$x_{11} = -69.1150383789755$$
$$x_{12} = 94.2477796076938$$
$$x_{13} = -50.2654824574367$$
$$x_{14} = 6.28318530717959$$
$$x_{15} = 31.4159265358979$$
$$x_{16} = 43.9822971502571$$
$$x_{17} = -37.6991118430775$$
$$x_{18} = -94.2477796076938$$
$$x_{19} = -31.4159265358979$$
$$x_{20} = -56.5486677646163$$
$$x_{21} = 25.1327412287183$$
$$x_{22} = 81.6814089933346$$
$$x_{23} = 69.1150383789755$$
$$x_{24} = 37.6991118430775$$
$$x_{25} = 56.5486677646163$$
$$x_{26} = 56.5486677646163$$
$$x_{27} = -100.530964914873$$
$$x_{28} = -18.8495559215388$$
$$x_{29} = -113.097335529233$$
$$x_{30} = -6.28318530717957$$
$$x_{31} = 18.8495559215388$$
$$x_{32} = -87.9645943005142$$
$$x_{33} = -232.477856365645$$
$$x_{34} = 0$$
$$x_{35} = -43.9822971502571$$
$$x_{36} = 12.5663706143592$$
$$x_{37} = -12.5663706143592$$
$$x_{38} = 75.398223686155$$
$$x_{39} = -25.1327412287183$$
$$x_{40} = -81.6814089933346$$
$$x_{41} = 50.2654824574367$$
$$x_{42} = -25.1327412287183$$
Puntos máximos de la función:
$$x_{42} = -267.035375555132$$
$$x_{42} = 47.1238898038469$$
$$x_{42} = 9.42477796076938$$
$$x_{42} = 97.3893722612836$$
$$x_{42} = -34.5575191894877$$
$$x_{42} = -97.3893722612836$$
$$x_{42} = -3.14159265358979$$
$$x_{42} = 59.6902604182061$$
$$x_{42} = -47.1238898038469$$
$$x_{42} = -40.8407044966673$$
$$x_{42} = 3.14159265358979$$
$$x_{42} = 28.2743338823081$$
$$x_{42} = -59.6902604182061$$
$$x_{42} = 21.9911485751286$$
$$x_{42} = 15.707963267949$$
$$x_{42} = 34.5575191894877$$
$$x_{42} = -91.106186954104$$
$$x_{42} = 40.8407044966673$$
$$x_{42} = 65.9734457253857$$
$$x_{42} = -72.2566310325652$$
$$x_{42} = -21.9911485751286$$
$$x_{42} = 53.4070751110265$$
$$x_{42} = 91.106186954104$$
$$x_{42} = -28.2743338823081$$
$$x_{42} = -65.9734457253857$$
$$x_{42} = -53.4070751110265$$
$$x_{42} = -15.707963267949$$
$$x_{42} = 84.8230016469244$$
$$x_{42} = 72.2566310325652$$
$$x_{42} = -78.5398163397448$$
$$x_{42} = -84.8230016469244$$
$$x_{42} = 78.5398163397448$$
$$x_{42} = -9.42477796076938$$
$$x_{42} = -2642.07942166902$$
Decrece en los intervalos
$$\left[100.530964914873, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -232.477856365645\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- \sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} - 1 \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -267.035375555132$$
$$x_{2} = -31.4159265358979$$
$$x_{3} = 47.1238898038469$$
$$x_{4} = -75.398223686155$$
$$x_{5} = 9.42477796076938$$
$$x_{6} = 97.3893722612836$$
$$x_{7} = -34.5575191894877$$
$$x_{8} = -97.3893722612836$$
$$x_{9} = -62.8318530717959$$
$$x_{10} = 18.8495559215388$$
$$x_{11} = 87.9645943005142$$
$$x_{12} = -3.14159265358979$$
$$x_{13} = -87.9645943005142$$
$$x_{14} = 6.28318530717959$$
$$x_{15} = 59.6902604182061$$
$$x_{16} = -47.1238898038469$$
$$x_{17} = 100.530964914873$$
$$x_{18} = -40.8407044966673$$
$$x_{19} = 62.8318530717959$$
$$x_{20} = -75.398223686155$$
$$x_{21} = 3.14159265358979$$
$$x_{22} = 28.2743338823081$$
$$x_{23} = -69.1150383789755$$
$$x_{24} = 94.2477796076938$$
$$x_{25} = -50.2654824574367$$
$$x_{26} = 6.28318530717959$$
$$x_{27} = 31.4159265358979$$
$$x_{28} = 43.9822971502571$$
$$x_{29} = -37.6991118430775$$
$$x_{30} = -94.2477796076938$$
$$x_{31} = -59.6902604182061$$
$$x_{32} = -31.4159265358979$$
$$x_{33} = -56.5486677646163$$
$$x_{34} = 25.1327412287183$$
$$x_{35} = 81.6814089933346$$
$$x_{36} = 21.9911485751286$$
$$x_{37} = 69.1150383789755$$
$$x_{38} = 15.707963267949$$
$$x_{39} = 34.5575191894877$$
$$x_{40} = -91.106186954104$$
$$x_{41} = 40.8407044966673$$
$$x_{42} = 37.6991118430775$$
$$x_{43} = 65.9734457253857$$
$$x_{44} = -72.2566310325652$$
$$x_{45} = 56.5486677646163$$
$$x_{46} = -21.9911485751286$$
$$x_{47} = 53.4070751110265$$
$$x_{48} = 91.106186954104$$
$$x_{49} = -28.2743338823081$$
$$x_{50} = 56.5486677646163$$
$$x_{51} = -65.9734457253857$$
$$x_{52} = -53.4070751110265$$
$$x_{53} = -100.530964914873$$
$$x_{54} = -18.8495559215388$$
$$x_{55} = -113.097335529233$$
$$x_{56} = -15.707963267949$$
$$x_{57} = -6.28318530717957$$
$$x_{58} = 84.8230016469244$$
$$x_{59} = 18.8495559215388$$
$$x_{60} = 72.2566310325652$$
$$x_{61} = -87.9645943005142$$
$$x_{62} = -232.477856365645$$
$$x_{63} = 0$$
$$x_{64} = -43.9822971502571$$
$$x_{65} = -78.5398163397448$$
$$x_{66} = 12.5663706143592$$
$$x_{67} = -12.5663706143592$$
$$x_{68} = 75.398223686155$$
$$x_{69} = -84.8230016469244$$
$$x_{70} = -25.1327412287183$$
$$x_{71} = 78.5398163397448$$
$$x_{72} = -81.6814089933346$$
$$x_{73} = -9.42477796076938$$
$$x_{74} = 50.2654824574367$$
$$x_{75} = -2642.07942166902$$
$$x_{76} = -25.1327412287183$$
Signos de extremos en los puntos:
(-267.0353755551324, 2)
(-31.41592653589793, 0)
(47.1238898038469, 2)
(-75.39822368615503, 0)
(9.42477796076938, 2)
(97.3893722612836, 2)
(-34.55751918948773, 2)
(-97.3893722612836, 2)
(-62.83185307179586, 0)
(18.849555921538755, 0)
(87.96459430051421, 0)
(-3.141592653589793, 2)
(-87.96459430051421, 0)
(6.283185307179586, 0)
(59.69026041820607, 2)
(-47.1238898038469, 2)
(100.53096491487338, 0)
(-40.840704496667314, 2)
(62.83185307179586, 0)
(-75.39822368615505, 0)
(3.141592653589793, 2)
(28.274333882308138, 2)
(-69.11503837897546, 0)
(94.2477796076938, 0)
(-50.26548245743668, 0)
(6.283185307179591, 0)
(31.41592653589793, 0)
(43.98229715025708, 0)
(-37.69911184307752, 0)
(-94.2477796076938, 0)
(-59.69026041820607, 2)
(-31.41592653589794, 0)
(-56.548667764616276, 0)
(25.132741228718345, 0)
(81.68140899333463, 0)
(21.991148575128552, 2)
(69.11503837897546, 0)
(15.707963267948966, 2)
(34.55751918948773, 2)
(-91.106186954104, 2)
(40.840704496667314, 2)
(37.69911184307752, 0)
(65.97344572538566, 2)
(-72.25663103256524, 2)
(56.5486677646163, 0)
(-21.991148575128552, 2)
(53.40707511102649, 2)
(91.106186954104, 2)
(-28.274333882308138, 2)
(56.548667764616276, 0)
(-65.97344572538566, 2)
(-53.40707511102649, 2)
(-100.53096491487338, 0)
(-18.84955592153876, 0)
(-113.09733552923255, 0)
(-15.707963267948966, 2)
(-6.283185307179572, 0)
(84.82300164692441, 2)
(18.84955592153876, 0)
(72.25663103256524, 2)
(-87.9645943005142, 0)
(-232.4778563656447, 0)
(0, 0)
(-43.982297150257104, 0)
(-78.53981633974483, 2)
(12.56637061435917, 0)
(-12.566370614359172, 0)
(75.39822368615503, 0)
(-84.82300164692441, 2)
(-25.13274122871832, 0)
(78.53981633974483, 2)
(-81.68140899333463, 0)
(-9.42477796076938, 2)
(50.26548245743669, 0)
(-2642.079421669016, 2)
(-25.132741228718345, 0)
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:
$$x_{1} = -31.4159265358979$$
$$x_{2} = -75.398223686155$$
$$x_{3} = -62.8318530717959$$
$$x_{4} = 18.8495559215388$$
$$x_{5} = 87.9645943005142$$
$$x_{6} = -87.9645943005142$$
$$x_{7} = 6.28318530717959$$
$$x_{8} = 100.530964914873$$
$$x_{9} = 62.8318530717959$$
$$x_{10} = -75.398223686155$$
$$x_{11} = -69.1150383789755$$
$$x_{12} = 94.2477796076938$$
$$x_{13} = -50.2654824574367$$
$$x_{14} = 6.28318530717959$$
$$x_{15} = 31.4159265358979$$
$$x_{16} = 43.9822971502571$$
$$x_{17} = -37.6991118430775$$
$$x_{18} = -94.2477796076938$$
$$x_{19} = -31.4159265358979$$
$$x_{20} = -56.5486677646163$$
$$x_{21} = 25.1327412287183$$
$$x_{22} = 81.6814089933346$$
$$x_{23} = 69.1150383789755$$
$$x_{24} = 37.6991118430775$$
$$x_{25} = 56.5486677646163$$
$$x_{26} = 56.5486677646163$$
$$x_{27} = -100.530964914873$$
$$x_{28} = -18.8495559215388$$
$$x_{29} = -113.097335529233$$
$$x_{30} = -6.28318530717957$$
$$x_{31} = 18.8495559215388$$
$$x_{32} = -87.9645943005142$$
$$x_{33} = -232.477856365645$$
$$x_{34} = 0$$
$$x_{35} = -43.9822971502571$$
$$x_{36} = 12.5663706143592$$
$$x_{37} = -12.5663706143592$$
$$x_{38} = 75.398223686155$$
$$x_{39} = -25.1327412287183$$
$$x_{40} = -81.6814089933346$$
$$x_{41} = 50.2654824574367$$
$$x_{42} = -25.1327412287183$$
Puntos máximos de la función:
$$x_{42} = -267.035375555132$$
$$x_{42} = 47.1238898038469$$
$$x_{42} = 9.42477796076938$$
$$x_{42} = 97.3893722612836$$
$$x_{42} = -34.5575191894877$$
$$x_{42} = -97.3893722612836$$
$$x_{42} = -3.14159265358979$$
$$x_{42} = 59.6902604182061$$
$$x_{42} = -47.1238898038469$$
$$x_{42} = -40.8407044966673$$
$$x_{42} = 3.14159265358979$$
$$x_{42} = 28.2743338823081$$
$$x_{42} = -59.6902604182061$$
$$x_{42} = 21.9911485751286$$
$$x_{42} = 15.707963267949$$
$$x_{42} = 34.5575191894877$$
$$x_{42} = -91.106186954104$$
$$x_{42} = 40.8407044966673$$
$$x_{42} = 65.9734457253857$$
$$x_{42} = -72.2566310325652$$
$$x_{42} = -21.9911485751286$$
$$x_{42} = 53.4070751110265$$
$$x_{42} = 91.106186954104$$
$$x_{42} = -28.2743338823081$$
$$x_{42} = -65.9734457253857$$
$$x_{42} = -53.4070751110265$$
$$x_{42} = -15.707963267949$$
$$x_{42} = 84.8230016469244$$
$$x_{42} = 72.2566310325652$$
$$x_{42} = -78.5398163397448$$
$$x_{42} = -84.8230016469244$$
$$x_{42} = 78.5398163397448$$
$$x_{42} = -9.42477796076938$$
$$x_{42} = -2642.07942166902$$
Decrece en los intervalos
$$\left[100.530964914873, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -232.477856365645\right]$$
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
$$2 \sin^{2}{\left(x \right)} \delta\left(\cos{\left(x \right)} - 1\right) - \cos{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} - 1 \right)} = 0$$
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
$$2 \sin^{2}{\left(x \right)} \delta\left(\cos{\left(x \right)} - 1\right) - \cos{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} - 1 \right)} = 0$$
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
$$\lim_{x \to -\infty} \left|{\cos{\left(x \right)} - 1}\right| = \left|{\left\langle -2, 0\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left|{\left\langle -2, 0\right\rangle}\right|$$
$$\lim_{x \to \infty} \left|{\cos{\left(x \right)} - 1}\right| = \left|{\left\langle -2, 0\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left|{\left\langle -2, 0\right\rangle}\right|$$
$$\lim_{x \to -\infty} \left|{\cos{\left(x \right)} - 1}\right| = \left|{\left\langle -2, 0\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left|{\left\langle -2, 0\right\rangle}\right|$$
$$\lim_{x \to \infty} \left|{\cos{\left(x \right)} - 1}\right| = \left|{\left\langle -2, 0\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left|{\left\langle -2, 0\right\rangle}\right|$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función Abs(cos(x) - 1), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left|{\cos{\left(x \right)} - 1}\right|}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\left|{\cos{\left(x \right)} - 1}\right|}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\left|{\cos{\left(x \right)} - 1}\right|}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\left|{\cos{\left(x \right)} - 1}\right|}{x}\right) = 0$$
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:
$$\left|{\cos{\left(x \right)} - 1}\right| = \left|{\cos{\left(x \right)} - 1}\right|$$
- Sí
$$\left|{\cos{\left(x \right)} - 1}\right| = - \left|{\cos{\left(x \right)} - 1}\right|$$
- No
es decir, función
es
par
Pues, comprobamos:
$$\left|{\cos{\left(x \right)} - 1}\right| = \left|{\cos{\left(x \right)} - 1}\right|$$
- Sí
$$\left|{\cos{\left(x \right)} - 1}\right| = - \left|{\cos{\left(x \right)} - 1}\right|$$
- No
es decir, función
es
par