Suppose that 2 ≤ f '(x) ≤ 3 for all values of x. what are the minimum and maximum possible values of f(5) − f(1)?

The question is an illustration of mean value theorem.

The minimum and the maximum values are: 8 and 12, respectively.

The given parameters are:

Mean value theorem states that:

In this case: a = 1, b = 5

So, we have:

Multiply both sides by 4

Rewrite as:

Substitute x for c

When f'(x) = 2, we have:

When f'(x) = 3, we have:

So, the minimum and the maximum values are: 8 and 12, respectively.

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Assuming is differentiable over the interval (1, 5), by the mean value theorem we know that there is some such that

We're given that for all , so

that is, the minimum possible value is 8, and the maximum possible value is 12.

10+4√2 is the answer. i dont have the time to explain.