Final answer:
The absolute maximum value of f(t) = 2 cos(t) sin(2t) on the interval [0, π/2] is 1.73, which occurs at t=π/6. The absolute minimum value is 0, which occurs at t=0 and t=π/2.
Step-by-step explanation:
To find the absolute maximum and absolute minimum values of the function f(t) = 2 cos(t) sin(2t) on the interval [0, π/2], we can start by finding the critical points of the function within the given interval. We do this by taking the derivative of the function and setting it equal to zero.
Taking the derivative of f(t) with respect to t, we get f'(t) = 2[cos(t)cos(2t) - sin(t)sin(2t)]. Setting this equal to zero and solving for t, we find that t = 0 and t = π/6 are the critical points within the interval.
Next, we evaluate the function f(t) at the critical points and endpoints of the interval to find the maximum and minimum values:
- At t=0: f(0) = 2 cos(0) sin(0) = 0
- At t=π/6: f(π/6) = 2 cos(π/6) sin(π/3) ≈ 1.73
- At t=π/2: f(π/2) = 2 cos(π/2) sin(π) = 0
Therefore, the absolute maximum value of the function on the interval [0, π/2] is 1.73, which occurs at t=π/6. The absolute minimum value is 0, which occurs at t=0 and t=π/2.