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How to find the minimum value of cos2x - cosx ? Please elaborate the method

How to find the minimum value of cos2x - cosx ? Please elaborate the method

Grade:12

2 Answers

Arnav Prasad
44 Points
5 years ago
I guess this was a question from recent FIITJEE Reshuffling test. JEE Mains level.
 
Let Y = Cos2x – Cosx.
Differentiate and put dY/dx = 0 to get x at which minimum value exists.
dY/dx = – 2Sin2x + Sinx = 0.
-2(2SinxCosx) + Sinx = 0
Sinx(1 – 4Cosx) = 0
  1. Either Sinx = 0 ---> x = 0 or Pi (3.14) Gives minumum.
  1. OR Cosx = ¼  ----> Gives minimum.
Simplfying Y, we get...
Y = Cos2x – Sin2x – Cosx. (Cos2x = Cos2x – Sin2x)
And then..
Y = 2Cos2x – Cosx – 1. (Sin2x = 1 – Cos2x)
 
After putting Sinx = 0, in the equation,
Y = -1.
After putting Cosx = ¼ in the equation,
Y = -9/8. 
Himanshh Yadav
13 Points
2 years ago
cos2x-cosx
Using cos2x=2cos^x-1,we get
2cos^2x-1-cosx
2(cos^2x-cosx/2-1/2)
By making perfect square,we get
2(cosx-1/4)^2-9/8>-9/8

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