Answer:
- increasing: (0, π/2) ∪ (3π/2, 2π)
- decreasing: (π/2, 3π/2)
- relative maximum: (π/2, 1/2)
- relative minimum: (3π/3, -1/2)
Explanation:
You want to know the intervals on which f(x) = sin(x)/(2+cos(x)²) is increasing and decreasing, and the relative extremes.
Derivative
The quotient rule can be used to find the derivative of f(x). Where the derivative is positive, the function is increasing.
f'(x) = ((2+cos(x)²)cos(x) +sin(x)(2cos(x)sin(x)))/(2+cos(x)²)²
f'(x) = (cos(x)(2 +cos(x)² +2sin(x)²)/(2+cos(x)²)²
f'(x) = cos(x)(3+sin(x)²)/(2+cos(x)²)²
We observe that the factors (3+sin(x)²) and (2+cos(x)²) are both positive for all x. This means the sign of the derivative will match the sign of cos(x).
Increasing
The function is increasing where cos(x) > 0, on the intervals ...
(0, π/2) ∪ (3π/2, 2π)
Decreasing
The function is decreasing where cos(x) < 0, on the interval ...
(π/2, 3π/2)
Relative maximum
The first derivative test tells us the function will have a relative maximum where the function goes from increasing to decreasing, at x = π/2. The function value at that point is ...
f(π/2) = sin(π/2)/(2 +cos(π/2)²) = 1/2
The relative maximum is at (π/2, 1/2).
Relative minimum
The first derivative test tells us the function will have a relative minimum where the function goes from decreasing to increasing, at x = 3π/2. The function value at that point is ...
f(3π/2) = sin(3π/2)/(2 +cos(3π/2)²) = -1/2
The relative minimum is at (3π/2, -1/2).