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

1 Answer
MattyMatty
May 25, 2018

#"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."#

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