Final answer:
The percentage rate of change of the function f(x) = 4950 - 3x² at x = 25 is approximately -4.9%. This is calculated using the derivative of the function for the rate of change and expressing it as a percentage relative to the function's value at x = 25.
Step-by-step explanation:
To find the percentage rate of change of a function at a specific point, we can use the derivative of the function to determine the instant rate of change, and then calculate its percentage relative to the function's value at that point.
Given the function f(x) = 4950 - 3x², the first step is to find its derivative f'(x). The derivative of f(x) is f'(x) = -6x. Then, to find the rate of change at x = 25, we substitute x with 25 in f'(x), obtaining f'(25) = -6(25) = -150.
The next step is to find the value of f(x) at this point: f(25) = 4950 - 3(25)² = 4950 - 1875 = 3075. Now, using the percentage change formula:
Percentage rate of change = [(rate of change) / (value of the function at that point)] × 100 = (-150 / 3075) × 100
When calculated, this gives us a percentage rate of change of approximately -4.878%, which can be rounded to -4.9%.