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For the function f(x)=3 cos⁴ x + 10 cos³ x + 6 cos² x - 3, defined on 0≤x≤π, which statement is true?
- It increases on (π/2,2π/3).
- It increases on (0,π/2)∪(2π/3,π).
- It decreases on (π/2,2π/3).
- All of the above.
Correct answer: It increases on (π/2,2π/3).
Solution
f'(x) = -6 sinx cosx (2cosx+1)(cosx+2). On (0,pi) sinx>0 and cosx+2>0, so the sign is set by -cosx(2cosx+1). On (pi/2,2pi/3): cosx<0 but 2cosx+1>0, giving f'>0, so f increases there. Hence the true statement is 'It increases on (pi/2,2pi/3)'.
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