The minimum value of f(4) - f(1) is <u>6</u>.
The maximum value of f(4) - f(1) is <u>12</u>.
In the question, we are given that, 2 ≤ f'(x) ≤ 4 for all values of x.
Taking the given inequality as (i).
We are asked to find the minimum and maximum possible values of f(4) - f(1).
We multiply (i) by dx throughout, to get:
4dx ≤ f'(x)dx ≤ 5dx.
To find this, we integrate (i) in the definite interval [4, 1] with respect to dx, to get:
Thus, the minimum value of f(4) - f(1) is 6.
The maximum value of f(4) - f(1) is 12.
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