Determine the local minima of f(x) = x^3 - 3x + 2. (2021)

Determine the local minima of f(x) = x^3 - 3x + 2. (2021)

Step 1: Write down the function f(x) = x^3 - 3x + 2.
Step 2: Find the derivative of the function, f'(x). The derivative is f'(x) = 3x^2 - 3.
Step 3: Set the derivative equal to zero to find critical points: 3x^2 - 3 = 0.
Step 4: Solve the equation 3x^2 - 3 = 0. This simplifies to x^2 = 1.
Step 5: Take the square root of both sides to find x. This gives x = 1 and x = -1.
Step 6: To determine if these points are local minima or maxima, we can use the second derivative test. Find the second derivative, f''(x) = 6x.
Step 7: Evaluate the second derivative at the critical points. For x = 1, f''(1) = 6(1) = 6 (which is positive, indicating a local minimum). For x = -1, f''(-1) = 6(-1) = -6 (which is negative, indicating a local maximum).
Step 8: Calculate the function value at the local minimum point x = 1. f(1) = 1^3 - 3(1) + 2 = 0.
Step 9: Conclude that the local minimum of the function occurs at the point (1, 0).

Finding Local Minima – The process of determining points where a function reaches a local minimum by analyzing its first derivative.
Critical Points – Identifying points where the first derivative is zero or undefined to find potential local extrema.
Second Derivative Test – Using the second derivative to confirm whether a critical point is a local minimum, maximum, or neither.