Final Answer:
Using the definition of the derivative, find f'(x). Then find f'(1), f'(2), and f'(3) when the derivative exists.
If this question above needs to be answered, the answer is when the derivative exists, f'(1) = 3, f'(2) = 1, and f'(3) = -1.
Step-by-step explanation:
To find the derivative of f(x) = -x² + 5x - 4 using the definition of the derivative, we can use the following steps:
1. Start with the definition of the derivative:
f'(x) = lim(h->0) [f(x+h) - f(x)] / h
2. Substitute the given function into the definition of the derivative:
f'(x) = lim(h->0) [(-x-h)² + 5(x+h) - 4 - (-x² + 5x - 4)] / h
3. Expand and simplify the numerator:
f'(x) = lim(h->0) [(-x² - 2xh - h²) + 5x + 5h - 4 + x² - 5x + 4] / h
4. Combine like terms:
f'(x) = lim(h->0) [-2xh - h² + 5h] / h
5. Cancel out the common factor of h in the numerator and denominator:
f'(x) = lim(h->0) [-2x - h + 5]
6. Take the limit as h approaches 0:
f'(x) = -2x + 5
Now, we can find f'(1), f'(2), and f'(3) by substituting the respective values of x into the derivative equation:
- f'(1) = -2(1) + 5 = 3
- f'(2) = -2(2) + 5 = 1
- f'(3) = -2(3) + 5 = -1
Therefore, when the derivative exists:
- f'(1) = 3
- f'(2) = 1
- f'(3) = -1
Complete question:
- f(x) = -x² +5x-4. Using the definition of the derivative, find f'(x). Then find f'(1), f'(2), and f'(3) when the derivative exists.