Find the second derivative of f(x) = x^3 - 3x^2 + 4. (2020)

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Question: Find the second derivative of f(x) = x^3 - 3x^2 + 4. (2020)

Options:

Correct Answer: 6x - 6

Exam Year: 2020

Solution:

First derivative f\'(x) = 3x^2 - 6x; second derivative f\'\'(x) = 6x - 6.

Find the second derivative of f(x) = x^3 - 3x^2 + 4. (2020)

Find the second derivative of f(x) = x^3 - 3x^2 + 4. (2020)

Step 1: Start with the function f(x) = x^3 - 3x^2 + 4.
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 3*x^(3-1) = 3x^2.
- For -3x^2, the derivative is -3*2*x^(2-1) = -6x.
- For the constant 4, the derivative is 0.
Step 4: Combine the derivatives from each term to get the first derivative: f'(x) = 3x^2 - 6x.
Step 5: Now, find the second derivative f''(x) by differentiating f'(x).
Step 6: Again, apply the power rule to f'(x):
- For 3x^2, the derivative is 3*2*x^(2-1) = 6x.
- For -6x, the derivative is -6.
Step 7: Combine these results to get the second derivative: f''(x) = 6x - 6.

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.