Final answer:
The maximum value of the function g(θ) = 4θ - 8sin(θ) on the interval [0, π/2] is 2π at θ = π/2, and the minimum value is less than 0 at θ = π/3 due to the negative sinusoidal term.
Step-by-step explanation:
To find the maximum and minimum values of the function g(θ) = 4θ - 8sin(θ) on the interval [0, π/2], we need to first calculate its derivative and then determine where that derivative is zero or does not exist within the given interval.
The derivative g'(θ) of the function g(θ) is 4 - 8cos(θ). At the endpoints of the interval θ = 0 and θ = π/2, we evaluate the function g(θ) to find the values 0 and π, respectively. Inside the interval, we set the derivative equal to zero to find the critical points: 4 - 8cos(θ) = 0. Solving for θ, we get cos(θ) = 1/2, which within the interval [0, π/2] occurs at θ = π/3. Substituting π/3 into g(θ), we find another potential maximum or minimum value.
Testing these values, g(0) = 0, g(π/3) = 4π/3 - 8(√3/2), and g(π/2) = 2π. The maximum value of g(θ) on the interval is 2π which occurs at θ = π/2, and the minimum value is g(π/3) which is less than g(0) =