If f(x) = x^3 - 3x^2 + 4, find the point where the function has a local minimum.

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Question: If f(x) = x^3 - 3x^2 + 4, find the point where the function has a local minimum.

Options:

Correct Answer: (1, 2)

Solution:

f\'(x) = 3x^2 - 6x. Setting f\'(x) = 0 gives x(3x - 6) = 0, so x = 0 or x = 2. f\'\'(2) = 6 > 0, so (2, f(2)) = (2, 1) is a local minimum.

If f(x) = x^3 - 3x^2 + 4, find the point where the function has a local minimum.

If f(x) = x^3 - 3x^2 + 4, find the point where the function has a local minimum.

Correct Answer: (2, 1)

Step 1: Start with the function f(x) = x^3 - 3x^2 + 4.
Step 2: Find the first derivative f'(x) to determine where the function's slope is zero. The first derivative is f'(x) = 3x^2 - 6x.
Step 3: Set the first derivative equal to zero to find critical points: 3x^2 - 6x = 0.
Step 4: Factor the equation: 3x(x - 2) = 0.
Step 5: Solve for x: This gives us two solutions, x = 0 and x = 2.
Step 6: To determine if these points are local minima or maxima, we need to find the second derivative f''(x). The second derivative is f''(x) = 6x - 6.
Step 7: Evaluate the second derivative at the critical points. First, check x = 0: f''(0) = 6(0) - 6 = -6 (which indicates a local maximum).
Step 8: Now check x = 2: f''(2) = 6(2) - 6 = 6 (which indicates a local minimum).
Step 9: To find the local minimum point, calculate f(2): f(2) = (2)^3 - 3(2)^2 + 4 = 8 - 12 + 4 = 0.
Step 10: Therefore, the point where the function has a local minimum is (2, 0).

Critical Points – Identifying where the first derivative is zero to find potential local extrema.
Second Derivative Test – Using the second derivative to determine the nature of the critical points (minimum or maximum).
Function Evaluation – Calculating the function value at the critical points to find the corresponding local minimum or maximum.