Determine the absolute extrema of each function on the given interval. Illustrate your results by
sketching the graph of each function.
a) f(x)���� = ��x^2 − 4x� + 3 , 0 ≤ x� ≤ 3
b) ����f(x) = ��(x − 1)^2�� , 0 ≤ x� ≤ 2
To determine the absolute extrema of a function on a given interval, you need to find the maximum and minimum values of the function within that interval. Here's how you can find the absolute extrema for each function:
a) We have the function f(x) = x^2 - 4x + 3, and the interval 0 ≤ x ≤ 3.
Step 1: Find the critical points by taking the derivative of the function and setting it equal to zero.
Find f'(x) = 2x - 4.
Setting the derivative equal to zero, we get 2x - 4 = 0.
Solving for x, we find x = 2.
Step 2: Evaluate the function at the critical points and the endpoints of the interval.
Evaluate f(x) at x = 0, 2, and 3.
f(0) = (0)^2 - 4(0) + 3 = 3.
f(2) = (2)^2 - 4(2) + 3 = -1.
f(3) = (3)^2 - 4(3) + 3 = 0.
Step 3: Compare the values obtained in Step 2 to determine the absolute extrema.
The absolute maximum is f(0) = 3, and it occurs at x = 0.
The absolute minimum is f(2) = -1, and it occurs at x = 2.
To illustrate these results, you can sketch the graph of the function f(x) = x^2 - 4x + 3 between x = 0 to x = 3.
b) We have the function f(x) = (x - 1)^2, and the interval 0 ≤ x ≤ 2.
Step 1: Find the critical points by taking the derivative of the function and setting it equal to zero.
Find f'(x) = 2(x - 1).
Setting the derivative equal to zero, we get 2(x - 1) = 0.
Solving for x, we find x = 1.
Step 2: Evaluate the function at the critical point and the endpoints of the interval.
Evaluate f(x) at x = 0, 1, and 2.
f(0) = (0 - 1)^2 = 1.
f(1) = (1 - 1)^2 = 0.
f(2) = (2 - 1)^2 = 1.
Step 3: Compare the values obtained in Step 2 to determine the absolute extrema.
The absolute maximum is f(0) = 1, and it occurs at x = 0 and x = 2.
The absolute minimum is f(1) = 0, and it occurs at x = 1.
To illustrate these results, you can sketch the graph of the function f(x) = (x - 1)^2 between x = 0 to x = 2.