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Answer :
Final answer:
The rate of change of f'(x) at (1,18) is found by evaluating the second derivative of f(x) at x=1, which yields a value of 32.
Explanation:
The rate of change of f'(x) at a point is given by its derivative, which in this case is the second derivative of f(x) since f'(x) represents the first derivative. To find this, we need to differentiate f(x) = 19x² - x³ twice.
First, find the first derivative:
f'(x) = d/dx(19x² - x³) = 38x - 3x²
Now, find the second derivative:
f''(x) = d/dx(38x - 3x²) = 38 - 6x
The rate of change of f'(x) at the point (1,18) is the value of f''(1):
f''(1) = 38 - 6(1) = 38 - 6 = 32
Therefore, the rate of change of f'(x) at (1,18) is 32.
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