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).