Find the second derivative of f(x) = x^3 - 6x^2 + 9x.

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Question: Find the second derivative of f(x) = x^3 - 6x^2 + 9x.

Options:

Correct Answer: 6

Solution:

f\'(x) = 3x^2 - 12x + 9; f\'\'(x) = 6x - 12. At x = 2, f\'\'(2) = 6(2) - 12 = 0.

Find the second derivative of f(x) = x^3 - 6x^2 + 9x.

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.