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cos^2*2x-2cos2x>=0
Resolver desigualdades/ cos^2*2x-2cos2x>=0

cos^2*2x-2cos2x>=0 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src] 
   2                       
cos (2)*x - 2*cos(2*x) >= 0
xcos⁡2(2)−2cos⁡(2x)≥0x \cos^{2}{\left(2 \right)} - 2 \cos{\left(2 x \right)} \geq 0 
x*cos(2)^2 - 2*cos(2*x) >= 0
Solución detallada
Se da la desigualdad:
xcos⁡2(2)−2cos⁡(2x)≥0x \cos^{2}{\left(2 \right)} - 2 \cos{\left(2 x \right)} \geq 0
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
xcos⁡2(2)−2cos⁡(2x)=0x \cos^{2}{\left(2 \right)} - 2 \cos{\left(2 x \right)} = 0
Resolvemos:
x1=6.75613855781251x_{1} = 6.75613855781251
x2=−2.2578099808939x_{2} = -2.2578099808939
x3=−4.10886672045165x_{3} = -4.10886672045165
x4=−8.2420187646535x_{4} = -8.2420187646535
x5=9.09260518693471x_{5} = 9.09260518693471
x6=0.752783614200585x_{6} = 0.752783614200585
x7=−5.26128117540242x_{7} = -5.26128117540242
x8=−11.1305493593001x_{8} = -11.1305493593001
x9=−2.2578099808939x_{9} = -2.2578099808939
x10=3.76112921313398x_{10} = 3.76112921313398
x11=5.75880741322772x_{11} = 5.75880741322772
x12=−7.41730347807894x_{12} = -7.41730347807894
x13=−10.8165708300838x_{13} = -10.8165708300838
x14=9.71075448678832x_{14} = 9.71075448678832
x15=−0.820971768777298x_{15} = -0.820971768777298
x16=2.46368466215011x_{16} = 2.46368466215011
x1=6.75613855781251x_{1} = 6.75613855781251
x2=−2.2578099808939x_{2} = -2.2578099808939
x3=−4.10886672045165x_{3} = -4.10886672045165
x4=−8.2420187646535x_{4} = -8.2420187646535
x5=9.09260518693471x_{5} = 9.09260518693471
x6=0.752783614200585x_{6} = 0.752783614200585
x7=−5.26128117540242x_{7} = -5.26128117540242
x8=−11.1305493593001x_{8} = -11.1305493593001
x9=−2.2578099808939x_{9} = -2.2578099808939
x10=3.76112921313398x_{10} = 3.76112921313398
x11=5.75880741322772x_{11} = 5.75880741322772
x12=−7.41730347807894x_{12} = -7.41730347807894
x13=−10.8165708300838x_{13} = -10.8165708300838
x14=9.71075448678832x_{14} = 9.71075448678832
x15=−0.820971768777298x_{15} = -0.820971768777298
x16=2.46368466215011x_{16} = 2.46368466215011
Las raíces dadas
x8=−11.1305493593001x_{8} = -11.1305493593001
x13=−10.8165708300838x_{13} = -10.8165708300838
x4=−8.2420187646535x_{4} = -8.2420187646535
x12=−7.41730347807894x_{12} = -7.41730347807894
x7=−5.26128117540242x_{7} = -5.26128117540242
x3=−4.10886672045165x_{3} = -4.10886672045165
x2=−2.2578099808939x_{2} = -2.2578099808939
x9=−2.2578099808939x_{9} = -2.2578099808939
x15=−0.820971768777298x_{15} = -0.820971768777298
x6=0.752783614200585x_{6} = 0.752783614200585
x16=2.46368466215011x_{16} = 2.46368466215011
x10=3.76112921313398x_{10} = 3.76112921313398
x11=5.75880741322772x_{11} = 5.75880741322772
x1=6.75613855781251x_{1} = 6.75613855781251
x5=9.09260518693471x_{5} = 9.09260518693471
x14=9.71075448678832x_{14} = 9.71075448678832
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0≤x8x_{0} \leq x_{8}
Consideremos, por ejemplo, el punto
x0=x8−110x_{0} = x_{8} - \frac{1}{10}
=
−11.1305493593001+−110-11.1305493593001 + - \frac{1}{10}
=
−11.2305493593001-11.2305493593001
lo sustituimos en la expresión
xcos⁡2(2)−2cos⁡(2x)≥0x \cos^{2}{\left(2 \right)} - 2 \cos{\left(2 x \right)} \geq 0
(−11.2305493593001)cos⁡2(2)−2cos⁡((−11.2305493593001)2)≥0\left(-11.2305493593001\right) \cos^{2}{\left(2 \right)} - 2 \cos{\left(\left(-11.2305493593001\right) 2 \right)} \geq 0
                                       2        
1.78318173285034 - 11.2305493593001*cos (2) >= 0

pero
                                       2       
1.78318173285034 - 11.2305493593001*cos (2) < 0

Entonces
x≤−11.1305493593001x \leq -11.1305493593001
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
x≥−11.1305493593001∧x≤−10.8165708300838x \geq -11.1305493593001 \wedge x \leq -10.8165708300838
         _____           _____           _____           _____           _____           _____           _____           _____  
        /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \  
-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------
       x8      x13      x4      x12      x7      x3      x2      x9      x15      x6      x16      x10      x11      x1      x5      x14

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x≥−11.1305493593001∧x≤−10.8165708300838x \geq -11.1305493593001 \wedge x \leq -10.8165708300838
x≥−8.2420187646535∧x≤−7.41730347807894x \geq -8.2420187646535 \wedge x \leq -7.41730347807894
x≥−5.26128117540242∧x≤−4.10886672045165x \geq -5.26128117540242 \wedge x \leq -4.10886672045165
x≥−2.2578099808939∧x≤−2.2578099808939x \geq -2.2578099808939 \wedge x \leq -2.2578099808939
x≥−0.820971768777298∧x≤0.752783614200585x \geq -0.820971768777298 \wedge x \leq 0.752783614200585
x≥2.46368466215011∧x≤3.76112921313398x \geq 2.46368466215011 \wedge x \leq 3.76112921313398
x≥5.75880741322772∧x≤6.75613855781251x \geq 5.75880741322772 \wedge x \leq 6.75613855781251
x≥9.09260518693471∧x≤9.71075448678832x \geq 9.09260518693471 \wedge x \leq 9.71075448678832
Solución de la desigualdad en el gráfico
0-80-60-40-2020406080-5050
Gráfico
cos^2*2x-2cos2x>=0 desigualdades
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