Let f(x) = x^5 + 3x^2 − 5x − 7, find the interval(s) where f is increasing and decreasing.

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  1. f is increasing where f' > 0

    f' = 5x^4+6x-5
    f' = 0 at x = -1.2588 and 0.6677
    f' changes sign at those roots, so
    f' > 0 on (-∞,-1.2588)U(0.6677,∞)
    f' < 0 on (-1.2588,0.6677)

    a look at the graph shows that f is in/de-creasing on these intervals

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