Answer :

Final answer:

The function f(x) = 45x3 - 3x5 decreases in the interval (3,∞). The interval was found by calculating the derivative of the function, setting it equal to zero to identify critical points, and testing these points and the end values in the derivative to determine where it's negative.

Explanation:

To determine where the function f(x) = 45x3 − 3x5 decreases, we first need to find the derivative f'(x) of the function.

The derivative will give us the slope at each point. When the derivative is negative, that means the function is decreasing. So, let's compute the derivative:

f'(x) = 135x2 - 15x4.

Now, we want to find where this derivative is less than zero. Set it equal to zero and solve: 135x2 - 15x4 = 0, which simplifies to 9x2 - x4 = 0, x2(9 - x2) = 0, gives x = 0, √9 = ±3.

Now, test these critical numbers plus the end values in the derivative to find out where it's positive or negative: f'(−1) = 144, f'(0) = 0, f'(1) = 144, f'(3) = 0, and f'(4) = -192.

We find that the function is increasing in the intervals (-∞,0) U (0,3) and decreasing in the interval (3,∞).

Learn more about Decreasing Functions here:

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