Determine the absolute extrema of each function on the given interval. Illustrate your results by

sketching the graph of each function.

����f(x) = ��(x − 1)^2�� , 0 ≤ x� ≤ 2

clearly a simple parabola

f ' (x) = 2(1-x) = 0 for a max/min
x=1
f(1) = (1-1)^2 = 0

endpoints:
f(0) = (-1)^2 = 1
f(2) = (162 = 1

parabola opens up, so the minimum is 0 when x = 1

or
(the vertex is (1,0) )

You have posted several of these rather straightforward problems. What is it exactly that you are having difficulties with ?

To determine the absolute extrema (maximum and minimum) of the function f(x) on the given interval [0, 2], we can follow these steps:

1. Find the critical points of the function on the interval by setting the derivative of f(x) equal to zero and solving for x. This will give us the potential locations of maximum and minimum points.

2. Determine the values of f(x) at the critical points and at the endpoints of the interval.

3. Compare these values to identify the absolute maximum and minimum.

Now let's go through these steps for the given function f(x) = (x - 1)^2.

Step 1: Find the critical points
To find the critical points, we need to calculate the derivative of f(x) and set it equal to zero:
f'(x) = 2(x - 1)

Setting f'(x) = 0, we get:
2(x - 1) = 0
x - 1 = 0
x = 1

So, we have a critical point at x = 1.

Step 2: Determine the function values
Evaluate f(x) at the critical point and the endpoints of the interval:
f(0) = (0 - 1)^2 = 1
f(1) = (1 - 1)^2 = 0
f(2) = (2 - 1)^2 = 1

Step 3: Identify the absolute extrema
Comparing the values of f(x), we see that f(x) has a minimum at x = 1, where f(1) = 0. The values of f(x) at the endpoints are f(0) = 1 and f(2) = 1.

Therefore, the absolute minimum of f(x) on the interval [0, 2] is 0 at x = 1, and the absolute maximum is 1, which occurs at the endpoints x = 0 and x = 2.

To illustrate these results, you can sketch the graph of the function f(x) = (x - 1)^2 on the interval [0, 2], showing a minimum at x = 1 and the function values at the endpoints being equal to 1.