At what point does the function f(x) = x^3 - 3x^2 + 4 have a local minimum? (202

At what point does the function f(x) = x^3 - 3x^2 + 4 have a local minimum? (2020)

At what point does the function f(x) = x^3 - 3x^2 + 4 have a local minimum? (2020)

Step 1: Write down 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 - 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. This gives us two solutions: x = 0 and x = 2.
Step 5: 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 6: Evaluate the second derivative at the critical points. First, check x = 0: f''(0) = 6(0) - 6 = -6 (which indicates a local maximum).
Step 7: Now check x = 2: f''(2) = 6(2) - 6 = 6 (which indicates a local minimum).
Step 8: The local minimum occurs at x = 2. To find the corresponding y-value, substitute x = 2 back into the original function: f(2) = 2^3 - 3(2^2) + 4 = 8 - 12 + 4 = 0.
Step 9: Therefore, the local minimum point is (2, 0).

Finding Local Minima – The process of determining points where a function reaches a local minimum by using first and second derivatives.
Critical Points – Points where the first derivative is zero or undefined, indicating potential local extrema.
Second Derivative Test – A method to classify critical points as local minima, local maxima, or points of inflection based on the sign of the second derivative.