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Given the function f'(x)=-x³-9x²+90x, determine all intervals on which ff is decreasing.

A) (−[infinity],−9)∪(0,6)
B) (−[infinity],−6)∪(0,3)
C) (−[infinity],−15)∪(0,6)
D) (−[infinity],−6)∪(0,9)

1 Answer

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Final answer:

To determine where a function f is decreasing, look where its derivative f'(x) is less than zero. By factoring the derivative provided (-x³-9x²+90x), we can identify the intervals where f'(x) is negative. The function f is decreasing on intervals (-∞, -15) and (0, 6), which corresponds to option C.

Step-by-step explanation:

The function given is the derivative of a function f, represented by f'(x) = -x³ - 9x² + 90x. To determine intervals on which f is decreasing, we examine where f'(x) is less than zero (meaning the slope of the function f is negative).

To find the intervals, we can set f'(x) < 0 and solve the inequality:

-x³ - 9x² + 90x < 0

To solve this cubic inequality, we try to factor the expression:

f'(x) = -x(x² + 9x - 90)

The quadratic can be factored further:

f'(x) = -x(x + 15)(x - 6)

We have three roots: x = 0, x = -15, and x = 6. To determine the sign of f'(x) we check intervals around these roots:

Less than -15

Between -15 and 0

Between 0 and 6

Greater than 6

We find that f'(x) > 0 for x in (-15, 0) and (6, ∞), and f'(x) < 0 for x in (-∞, -15) and (0, 6). Therefore, the function f is decreasing on the intervals (-∞, -15) and (0, 6).

The correct option answer in the final answer is: C) (-∞,-15)∪(0,6).

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