Question

What is the range of cos^2x-cosx+1?

Answer 1

`"The range is "[3/4, 3].`

Explanation:

`"The biggest value is 3, this is if"`
`" "cos(x) = -1 => x = (2k+1) * pi`
`" "=> cos^2(x) = 1`
`" so we have 1+1+1=3."`
`"(this is the biggest value possible as "-1 <= cos(x) <= 1).`
`"The smallest value is more difficult to find."`
`"We take the derivative to find the minimum."`

`- 2 cos(x) sin(x) + sin(x) = 0`
`=> sin(x) (1 - 2 cos(x)) = 0`
`=> sin(x) = 0 " or "cos(x) = 1/2`

`"if "cos(x) = 1/2 => x = pm pi/3 + 2 k pi`
`=> cos^2(x) - cos(x) + 1 = 1/4 - 1/2 + 1 = 3/4`
`"This is the minimum."`