To calculate the mean value of the function f(x) = |7 - x| over the interval [5,9], we need to consider the behavior before and after the point x=7. The mean value is found to be 1 after averaging the function values on the two subintervals [5,7] and [7,9].
To find the mean value of the function f(x) = |7 - x| on the closed interval [5,9], we need to evaluate the function at the endpoints of the interval and consider any points within the interval where the absolute value expression may change. First, the function f(x) = |7 - x| equals 7 - x when x ≤ 7 and x - 7 when x > 7. Thus, on the interval [5,9], the function changes from 7 - x at x=7.
From x=5 to x=7, the function is 7 - x, and from x=7 to x=9, the function is x - 7. Therefore, we need to calculate the average of the two segments separately and then combine the results for the entire interval:
For x=5 to x=7, the mean is the average of f(5) and f(7), which is (|7-5| + |7-7|)/2 = (2 + 0)/2 = 1.
For x=7 to x=9, the mean is the average of f(7) and f(9), which is (|7-7| + |7-9|)/2 = (0 + 2)/2 = 1.
The overall mean value for the entire interval of f(x) is the average of these two means: (1 + 1)/2 = 1. Hence, the mean value is 1.