find the absolute extrema for f(x)= (5-x) with a closed interval of -1,2
since f'(x) = -1 which is never zero, there are no relative min or max points.
So, the extrema will just be the values at the endpoints of the interval. So, just find f(-1) and f(2).
I mean, don't forget your Algebra I just 'cause you're in calculus now. f(x) is just a straight line, doncha know...
To find the absolute extrema of the function f(x) = 5 - x on the closed interval [-1, 2], we need to consider both the critical points and the endpoints of the interval.
Step 1: Find the critical points by taking the derivative of f(x) and setting it equal to zero.
f'(x) = -1
Setting f'(x) = 0, we have -1 = 0. However, this equation has no solution. Hence, there are no critical points.
Step 2: Evaluate the function at the endpoints of the interval.
For x = -1:
f(-1) = 5 - (-1) = 6
For x = 2:
f(2) = 5 - 2 = 3
Step 3: Compare the values obtained from the critical points and the endpoints to determine the absolute extrema.
In this case, since there are no critical points, we only need to compare the values at the endpoints.
The function values at the endpoints are:
f(-1) = 6
f(2) = 3
Since 6 is the larger value, the absolute maximum occurs at x = -1 with a value of 6, and the absolute minimum occurs at x = 2 with a value of 3.
Therefore, the absolute extrema of f(x) = 5 - x on the closed interval [-1, 2] are:
Absolute maximum: f(-1) = 6
Absolute minimum: f(2) = 3
To find the absolute extrema of a function on a closed interval, you need to evaluate the function at its critical points within the interval, as well as at the endpoints of the interval.
1. First, let's find the critical points by taking the derivative of the function f(x) = 5 - x. The critical points occur where the derivative is zero or undefined.
f'(x) = -1
Since f'(x) is a constant (-1), it is never equal to zero or undefined. Therefore, there are no critical points within the interval (-1,2).
2. Next, we evaluate the function at the endpoints of the interval: f(-1) and f(2).
f(-1) = 5 - (-1) = 6
f(2) = 5 - 2 = 3
3. Now, compare the function values at the critical points and the endpoints.
The function values at the critical points are not applicable since there are no critical points within the interval.
The function value at the endpoint -1 is 6, and at the endpoint 2, it is 3.
4. Finally, determine the absolute extrema. The absolute maximum is the highest function value, and the absolute minimum is the lowest function value.
In this case, the function value of 6 at x = -1 is the absolute maximum, and the function value of 3 at x = 2 is the absolute minimum.
Therefore, the absolute extrema for the function f(x) = 5 - x on the closed interval [-1,2] are:
Absolute Maximum: f(-1) = 6
Absolute Minimum: f(2) = 3