Final answer:
The fourth derivative of the function f(x) = 4x^4 - 4x^3 + 5x^2 + 3x - 7 is 96, a constant, indicating no further changes in curvature after this derivative.
Step-by-step explanation:
To find the fourth derivative of the function f(x) = 4x⁴ - 4x³ + 5x² + 3x - 7, we must differentiate the function four times with respect to x.
First derivative:
f'(x) = 16x³ - 12x² + 10x + 3
Second derivative:
f''(x) = 48x² - 24x + 10
Third derivative:
f'''(x) = 96x - 24
Fourth derivative:
f''''(x) = 96
The fourth derivative is a constant, which shows that the rate of change of the curvature has stabilized and no further changes in curvature occur after the fourth derivative.