Final Answer:
Using the definition of the derivative for f(x) = -x² + 6x - 4, the derivative f'(x) is found by differentiating term by term: f'(x) = d/dx(-x²) + d/dx(6x) - d/dx(4) = -2x + 6. To find f'(1), f'(2), and f'(3), substitute x = 1, x = 2, and x = 3, respectively, into the derivative function f'(x) = -2x + 6, resulting in f'(1) = 4, f'(2) = 2, and f'(3) = 0.
Step-by-step explanation:
To determine the derivative of the function f(x) = -x² + 6x - 4 using the definition of the derivative, we apply the differentiation rules term by term. Deriving each term separately, we get d/dx(-x²) + d/dx(6x) - d/dx(4). The derivative of -x² is -2x, the derivative of 6x is 6, and the derivative of a constant term -4 is zero. Therefore, the derivative of f(x) is f'(x) = -2x + 6.
Upon obtaining the derivative function f'(x) = -2x + 6, we can evaluate specific values of x to find the corresponding values of the derivative. Substituting x = 1, x = 2, and x = 3 into f'(x) = -2x + 6, we get f'(1) = 4, f'(2) = 2, and f'(3) = 0. These values represent the instantaneous rate of change or slope of the function f(x) at x = 1, x = 2, and x = 3, respectively, where the derivative exists.
Understanding the derivative of a function allows us to explore its behavior at various points. In this case, the values of f'(1), f'(2), and f'(3) represent the slope of the tangent line to the curve of f(x) at x = 1, x = 2, and x = 3, indicating how the function changes at these specific x-values.