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Answer :
Final answer:
The interval(s) for which f(x) decreases for the function f(x) = 45x³−3x⁵ are (-∞, -3) ∪ (3, ∞), which corresponds to option d) (-3,0)∪(3,[infinity]).
Explanation:
To determine the interval(s) for which f(x) decreases for the function f(x) = 45x³−3x⁵, we need to find the values of x where the derivative of f(x) is negative.
- Find the derivative of f(x): f'(x) = 135x² - 15x⁴
- Set f'(x) less than zero and solve for x. 135x² - 15x⁴ < 0
- Factor out x² from the equation: x²(135 - 15x²) < 0
- Solve for x. We have two critical points: x = -√9 = -3 and x = √9 = 3
Therefore, the interval(s) for which f(x) decreases are (−∞, -3) ∪ (3, ∞), which corresponds to option d) (−3,0)∪(3,[infinity]).
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