Final answer:
To find the absolute extrema of the function h(t)=(t)/(t-3) on the interval [4,5], we need to evaluate the function at the endpoints and any critical points. By comparing the values, we can determine the absolute maximum and minimum.
Step-by-step explanation:
The equation for the function h(t) is h(t) = t / (t-3) within the interval [4,5]. To find the absolute extrema of this function, we need to find the maximum and minimum values within this interval. To do this, we can take the derivative of h(t) and set it equal to zero to find any critical points. We can also evaluate h(t) at the endpoints of the interval. By comparing these values, we can determine the absolute extrema.
Taking the derivative of h(t), we get h'(t) = 3 / (t-3)^2. Setting this equal to zero, we find that t = 3 is a critical point.
Evaluating h(t) at the endpoints and the critical point, we find that h(4) = 4/1 = 4, h(5) = 5/2 = 2.5, and h(3) = undefined. Comparing these values, we can see that the absolute maximum value occurs at t=4 with a value of 4, and the absolute minimum value occurs at t=5 with a value of 2.5.