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For which values of a does the function f(x)=(a²-3a+2) ((cos² x)/(4)-(sin² x)/(4))+(a-1)x+sin 1 have no critical points?

1. [1,∞)
2. (0,1)∪(1,4)
3. (-2,4)
4. (1,3)∪(3,5)

Correct answer: (0,1)∪(1,4)

Solution

f(x) = (a^2-3a+2)(cos2x)/4 + (a-1)x + sin1, so f'(x) = (a-1)[1 - (a-2)(sin2x)/2]. For no critical point f' must never be 0. Excluding a=1 (where f'==0), need |a-1| > |(a-1)(a-2)/2|, i.e. |a-2| < 2 -> 0<a<4. Removing a=1 gives (0,1) U (1,4).

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