Locate the absolute extrema of the function on the closed interval
f(x) = 2(3-x) [-1,2]
6 - 2x is a straight line with a negative slope. In the interval [-1,2], it is highest when x is smallest (-1) and lowest when x is highest (2).
You don't need to use calculus to answer this question.
drwls is right no need for calculus here but if youd like to use it
f(x)=2(3-x) [-1,2]
f'(x)=-2
no critical values therefore only test endpoints
f(-1)=8 <--- abs. max (-1,8)
f(2)=2 <---- abs. min (2,2)
To locate the absolute extrema of the function f(x) = 2(3-x) on the closed interval [-1,2], follow the steps below:
1. Find the critical points of the function:
To find the critical points, set the derivative of f(x) equal to zero and solve for x.
f'(x) = -2
Setting f'(x) = 0, we get:
-2 = 0
There are no critical points in this case.
2. Evaluate the function at the endpoints of the interval:
- Evaluate f(-1):
f(-1) = 2(3-(-1)) = 2(3+1) = 2*4 = 8
- Evaluate f(2):
f(2) = 2(3-2) = 2(1) = 2
3. Compare the values obtained in step 2 to determine the absolute extrema:
- The maximum value of f(x) = 8 occurs at x = -1.
- The minimum value of f(x) = 2 occurs at x = 2.
Therefore, the absolute maximum value of f(x) = 2(3-x) on the interval [-1,2] is 8, and it occurs at x = -1.
The absolute minimum value of f(x) = 2(3-x) on the interval [-1,2] is 2, and it occurs at x = 2.
To locate the absolute extrema of a function on a closed interval, you need to do the following steps:
Step 1: Find the critical points of the function on the interval.
To do this, you need to find where the derivative of the function is either 0 or does not exist. Let's start by finding the derivative of the given function:
f(x) = 2(3 - x)
To find the derivative, apply the power rule and consider the chain rule:
f'(x) = -2
Since the derivative of the function is constant (-2), it means the function has no critical points on the interval [-1, 2].
Step 2: Evaluate the function at the endpoints of the interval.
Evaluate the function at both -1 and 2 since they are the endpoints of the interval [-1, 2]:
f(-1) = 2(3 - (-1)) = 2(3 + 1) = 2(4) = 8
f(2) = 2(3 - 2) = 2(1) = 2
Step 3: Compare the values obtained in step 2 and identify the absolute maximum and minimum.
From step 2, we got f(-1) = 8 and f(2) = 2.
So, the absolute maximum of the function is 8, which occurs at x = -1 on the given interval [-1, 2]. The absolute minimum of the function is 2, which occurs at x = 2 on the given interval [-1, 2].
Therefore, the absolute maximum is 8, and it occurs at x = -1, while the absolute minimum is 2, and it occurs at x = 2.