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Find the maximum and minimum values of the expression cos²(theta) - 6 sin(theta) cos(theta) + 3 sin²(theta) + 2.

1. max = 4 + sqrt(10), min = 4 - sqrt(10)
2. max = 6 + sqrt(10), min = 6 - sqrt(10)
3. max = 5 + 3, min = 5 - 3
4. max = 4 + sqrt(13), min = 4 - sqrt(13)

Correct answer: max = 4 + sqrt(10), min = 4 - sqrt(10)

Solution

Rewriting gives 4 - cos2t - 3 sin2t, whose amplitude is sqrt(1 + 9) = sqrt(10), so the values range over 4 - sqrt(10) to 4 + sqrt(10).

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