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13 October, 09:22

Suppose that 2 ≤ f ' (x) ≤ 3 for all values of x. what are the minimum and maximum possible values of f (6) - f (4) ?

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  1. 13 October, 09:47
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    If we let m represent the average value of f' (x) over the interval x ∈ [4, 6], then the value of f (6) will be

    f (6) = f (4) + m (6 - 4)

    f (6) = f (4) + 2m

    And the difference f (6) - f (4) is

    f (6) - f (4) = (f (4) + 2m) - f (4) = 2m

    The problem statement tells us that m must be in the range 2 ≤ m ≤ 3, so 2m is in the range 4 ≤ 2m ≤ 6.

    The minimum possible value of f (6) - f (4) is 4.

    The maximum possible value of f (6) - f (4) is 6.
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