2) Let g(s)= t(4−t)^1/2 on the interval [0,2].

Find the absolute maximum and absolute minimum of g(t) on this interval.

Enter DNE if the absolute maximum or minimum does not exist.

The absolute max occurs at t= .
The absolute min occurs at t=

To find the absolute maximum and minimum of a given function on a closed interval, we need to follow these steps:

1. Find the critical points within the interval by taking the derivative of the function and finding where it is equal to zero or undefined.
2. Evaluate the function at the critical points as well as at the endpoints of the interval.
3. Compare the values obtained in step 2, and the highest value will be the absolute maximum, while the lowest value will be the absolute minimum.

Given g(t) = t(4−t)^(1/2) on the interval [0,2], let's proceed with the steps mentioned above:

1. Taking the derivative of g(t), we get:

g'(t) = (4−t)^(1/2) - (t/(2√(4−t)))

To find the critical points, we need to solve the equation g'(t) = 0:

(4−t)^(1/2) - (t/(2√(4−t))) = 0

Simplifying and rearranging the equation, we have:

(4−t)^(1/2) = (t/(2√(4−t)))

Squaring both sides of the equation, we get:

4 - t = (t^2)/(4 - t)

Multiplying both sides of the equation by (4 - t), we have:

4(4 - t) - t(4 - t) = t^2

Expanding and rearranging the equation, we get:

16 - 4t - 4t + t^2 = t^2

Simplifying further, we have:

16 - 8t = 0
8t = 16
t = 2

So, the critical point within the interval [0,2] is t = 2.

2. Now, let's evaluate g(t) at the critical point and the endpoints of the interval:

g(0) = 0(4−0)^(1/2) = 0
g(2) = 2(4−2)^(1/2) = 2(2)^(1/2) = 2√2

3. Comparing the values obtained:

g(0) = 0
g(2) = 2√2

Since there are no other critical points within the interval and we have evaluated the function at the endpoints, we can conclude that the absolute maximum occurs at t = 2 with a value of 2√2, and the absolute minimum occurs at t = 0 with a value of 0.

Therefore:
The absolute max occurs at t=2.
The absolute min occurs at t=0.