Final answer:
The absolute maximum value of the function f(t) = t - 3t on the interval [-1, 3] is 2 at t = -1, and the absolute minimum value is -6 at t = 3.
Step-by-step explanation:
To find the absolute maximum and absolute minimum values of the function f(t) = t - 3t on the interval [-1, 3], we need to evaluate the function at the critical points and endpoints of the interval.
1. First, we find the critical points by taking the derivative of the function: f'(t) = 1 - 3 = -2. Setting this equal to zero, we find that the only critical point is t = 1.
2. Next, we evaluate the function at the critical point and the endpoints of the interval: f(-1) = -1 - 3(-1) = -1 + 3 = 2, f(1) = 1 - 3(1) = 1 - 3 = -2, and f(3) = 3 - 3(3) = 3 - 9 = -6.
Therefore, the absolute maximum value of f(t) on the interval [-1, 3] is 2 at t = -1, and the absolute minimum value is -6 at t = 3.