The minimum value of f(8) - f(3) is 20.
The maximum value of f(8) - f(3) is 25.
In the question, we are given that, 4 ≤ f'(x) ≤ 5 for all values of x.
Taking the given inequality as (i).
We are asked to find the minimum and maximum possible values of f(8) - f(3).
We multiply (i) by dx throughout, to get:
4dx ≤ f'(x)dx ≤ 5dx.
To find this, we integrate (i) in the definite interval [8, 3] with respect to dx, to get:
![\int_{3}^{8}4dx \leq \int_{3}^{8}f'(x)dx \leq \int_{3}^{8}5dx\\\Rightarrow [4x]_{3}^{8} \leq [f(x)]_{3}^{8} \leq [5x]_{3}^{8}\\\Rightarrow 4*8 - 4*3 \leq f(8)-f(3) \leq 5*8 - 5*3\\\Rightarrow 20 \leq f(8) -f(3) \leq 25](https://tex.z-dn.net/?f=%5Cint_%7B3%7D%5E%7B8%7D4dx%20%5Cleq%20%5Cint_%7B3%7D%5E%7B8%7Df%27%28x%29dx%20%5Cleq%20%5Cint_%7B3%7D%5E%7B8%7D5dx%5C%5C%5CRightarrow%20%5B4x%5D_%7B3%7D%5E%7B8%7D%20%5Cleq%20%5Bf%28x%29%5D_%7B3%7D%5E%7B8%7D%20%5Cleq%20%5B5x%5D_%7B3%7D%5E%7B8%7D%5C%5C%5CRightarrow%204%2A8%20-%204%2A3%20%5Cleq%20f%288%29-f%283%29%20%5Cleq%205%2A8%20-%205%2A3%5C%5C%5CRightarrow%2020%20%5Cleq%20f%288%29%20-f%283%29%20%5Cleq%2025)
Thus, the minimum value of f(8) - f(3) is 20.
The maximum value of f(8) - f(3) is 25.
Learn more about definite integrals at
brainly.com/question/17074932
#SPJ4