Calculus

Suppose the derivative of a function f is f′(x)=(x−8)^7(x−1)^4(x+19)^8.
Then the function f is increasing on the interval what?

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  1. f(x) is increasing if f'(x) is positive
    so (x−8)^7(x−1)^4(x+19)^8 > 0

    we know that (x−1)^4(x+19)^8 is always positive since we have even exponents.
    So all we have to do is look at
    (x-8)^7 > 0

    if x > 8 then (x-8)^7 > 0
    if x < 8 it is negative

    so the function is increasing for
    x > 8
    However, when x = 1 or -19, f'(x) = 0, so the function would not be increasing, but those two values would be excluded anyway by x > 8

    so it is increasing for all real values of x such that
    x > 8

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    posted by Reiny

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