Determine the critical points of f(x) = x^3 - 3x + 2.
Correct Answer: x = -1 and x = 1
- Step 1: Start with the function f(x) = x^3 - 3x + 2.
- Step 2: Find the derivative of the function, which is f'(x). The derivative of f(x) is f'(x) = 3x^2 - 3.
- Step 3: Set the derivative equal to zero to find critical points: 3x^2 - 3 = 0.
- Step 4: Simplify the equation: 3x^2 = 3.
- Step 5: Divide both sides by 3: x^2 = 1.
- Step 6: Solve for x by taking the square root of both sides: x = ±1.
- Step 7: The critical points are x = -1 and x = 1.
- Finding Critical Points – This involves taking the derivative of a function and setting it to zero to find points where the function's slope is zero.
- Derivative Calculation – Understanding how to differentiate polynomial functions correctly.
- Identifying Local Extrema – Recognizing that critical points can indicate local maxima, minima, or points of inflection.
