To find the absolute extrema of the function f(x) = 6 − x on the closed interval [−2, 5], we need to evaluate the function at the critical points and the endpoints of the interval, and then compare the function values to find the maximum and minimum.
1. Critical Points:
The critical points occur where the derivative of the function is zero or undefined. Since f(x) = 6 − x is a linear function, it has no critical points in the given interval.
2. Endpoints:
We evaluate the function at the endpoints of the interval:
f(−2) = 6 − (−2) = 6 + 2 = 8
f(5) = 6 − 5 = 1
3. Comparing the Values:
The function values are f(−2) = 8 and f(5) = 1.
So, the maximum value occurs at x = -2, and the minimum value occurs at x = 5.
Therefore, the absolute maximum of the function f(x) = 6 − x on the interval [−2, 5] is 8, and the absolute minimum is 1.