Question

Suppose that 4 ≤ f '(x) ≤ 5 for all values of x. what are the minimum and maximum possible values of f(8) − f(3)?

Answer 1

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:

$\int_{3}^{8}4dx \leq \int_{3}^{8}f'(x)dx \leq \int_{3}^{8}5dx\\\Rightarrow [4x]_{3}^{8} \leq [f(x)]_{3}^{8} \leq [5x]_{3}^{8}\\\Rightarrow 4*8 - 4*3 \leq f(8)-f(3) \leq 5*8 - 5*3\\\Rightarrow 20 \leq f(8) -f(3) \leq 25$

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