Find the second derivative of f(x) = x^3 - 6x^2 + 9x.
Correct Answer: f''(x) = 6x - 12
- Step 1: Start with the function f(x) = x^3 - 6x^2 + 9x.
- Step 2: Find the first derivative f'(x) by using the power rule. The power rule states that if f(x) = x^n, then f'(x) = n*x^(n-1).
- Step 3: Apply the power rule to each term in f(x):
- - For x^3, the derivative is 3x^2.
- - For -6x^2, the derivative is -12x.
- - For 9x, the derivative is 9.
- Step 4: Combine these results to get the first derivative: f'(x) = 3x^2 - 12x + 9.
- Step 5: Now, find the second derivative f''(x) by differentiating f'(x) again using the power rule.
- Step 6: Apply the power rule to each term in f'(x):
- - For 3x^2, the derivative is 6x.
- - For -12x, the derivative is -12.
- - The constant 9 has a derivative of 0.
- Step 7: Combine these results to get the second derivative: f''(x) = 6x - 12.
- Step 8: To find f''(2), substitute x = 2 into f''(x): f''(2) = 6(2) - 12.
- Step 9: Calculate f''(2): 6(2) = 12, and then 12 - 12 = 0.
- Differentiation – The process of finding the derivative of a function, which measures the rate of change.
- Second Derivative – The derivative of the first derivative, which provides information about the concavity of the function.
- Critical Points – Points where the second derivative is zero or undefined, indicating potential inflection points.
