Find the intervals on which f is increasing and decreasing. f(x)=-12x^5 + 135x^4

Question

Find the intervals on which f is increasing and decreasing. f(x)=-12x^5 + 135x^4 - 400x^3. Select one of the choices below filling in blank spaces. A) The function is increasing on ____. The function is never decreasing. (Type answers in interval notation using comma to separate answers as needed). B) The function is increasing on ____ and decreasing on ____. (Type answers in interval notation using comma to separate answers as needed). C) The function is decreasing on ____ . The function is never increasing. (Type answers in interval notation using comma to separate answers as needed). D) The function is never increasing nor decreasing.

Explanation / Answer

f'(x) = -60x^4 + 540 x^3 - 1200 x^2 = -60x^2 (x^2 - 9 x + 20) = -60x^2(x - 5)(x - 4) Thus, we need to consider four intervals for f'(x) (- infinity, 0); (0, 4); (4, 5); (5, infinity) On (- infinity, 0), -60x^2 is negative, (x - 5) is negative, and (x - 4) is negative, so the product of an odd number of negatives, 3, along with any number of positives is negative, and the function is decreasing; (I could consider this 5 negatives rather than 3 if I separated out the 2 x's in x^2 from - 60) On (0, 4), -60x^2 is negative, (x - 5) is negative, and (x - 4) is negative, so the product of an odd number of negatives, 3, along with any number of positives is negative, and the function is decreasing; (Note that we have an inflection point at x = 0, but the function is decreasing on both sides of 0) Then, as the function is continuous on (-infinity, 4) and decreasing on (- infinity, 0) and (0, 4), we can say that it is decreasing on(-infinity, 4). On (4, 5), -60x^2 is negative, (x - 5) is negative, and (x - 4) is positive; the product of an even number of negatives, 2, is positive, so the function is increasing. On (5, infinity), -60x^2 is negative, (x - 5) is positive, and (x - 4) is positive; the product of an odd number of negatives, 1, is negative, so the function is decreasing. The answer is B. The function is increasing on (4,5) and decreasing on (- infinity, 4), (5, infinity)

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