If f(x) = x^3 - 3x^2 + 4, what is f'(2)? (2020)

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Question: If f(x) = x^3 - 3x^2 + 4, what is f\'(2)? (2020)

Options:

Correct Answer: 0

Exam Year: 2020

Solution:

First, find f\'(x) = 3x^2 - 6x. Then, f\'(2) = 3(2^2) - 6(2) = 12 - 12 = 0.

If f(x) = x^3 - 3x^2 + 4, what is f'(2)? (2020)

If f(x) = x^3 - 3x^2 + 4, what is f'(2)? (2020)

Step 1: Identify the function f(x) = x^3 - 3x^2 + 4.
Step 2: Find the derivative of the function, f'(x). The derivative of x^3 is 3x^2, and the derivative of -3x^2 is -6x. So, f'(x) = 3x^2 - 6x.
Step 3: Substitute x = 2 into the derivative f'(x). This means we need to calculate f'(2) = 3(2^2) - 6(2).
Step 4: Calculate 2^2, which is 4. Then multiply by 3: 3 * 4 = 12.
Step 5: Calculate 6 * 2, which is 12.
Step 6: Subtract the two results: 12 - 12 = 0.
Step 7: Therefore, f'(2) = 0.

Differentiation – The process of finding the derivative of a function to determine its rate of change.
Evaluation of Derivatives – Substituting a specific value into the derivative to find the slope of the function at that point.