If f(x) = x^3 - 3x^2 + 4, then the local maxima occurs at x = ?

If f(x) = x^3 - 3x^2 + 4, then the local maxima occurs at x = ?

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 slope is zero. The first derivative is f'(x) = 3x^2 - 6.
Step 3: Set the first derivative equal to zero to find critical points: 3x^2 - 6 = 0.
Step 4: Solve for x by simplifying the equation: x^2 - 2 = 0.
Step 5: Factor the equation to find the values of x: x = ±√2.
Step 6: Now, we need to determine if these points are local maxima or minima. We do this by finding the second derivative f''(x).
Step 7: Calculate the second derivative: f''(x) = 6x.
Step 8: Evaluate the second derivative at the critical points. First, check x = 1: f''(1) = 6(1) = 6, which is positive, indicating a local minimum.
Step 9: Check the other critical points x = √2 and x = -√2 to find the local maxima.
Step 10: Since f''(1) is positive, we conclude that the local maxima occurs at x = 1.

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