Determine the critical points of f(x) = x^4 - 8x^2.
Correct Answer: x = 0, ±2
- Step 1: Start with the function f(x) = x^4 - 8x^2.
- Step 2: Find the derivative of the function, which is f'(x).
- Step 3: Use the power rule to differentiate: f'(x) = 4x^3 - 16x.
- Step 4: Factor the derivative: f'(x) = 4x(x^2 - 4).
- Step 5: Set the derivative equal to zero to find critical points: 4x(x^2 - 4) = 0.
- Step 6: Solve for x by setting each factor to zero: 4x = 0 or x^2 - 4 = 0.
- Step 7: From 4x = 0, we get x = 0.
- Step 8: From x^2 - 4 = 0, we get x^2 = 4, which gives x = ±2.
- Step 9: The critical points are x = 0, x = 2, and x = -2.
- 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.
- Factoring Polynomials – Understanding how to factor polynomials is essential for solving equations derived from the derivative.
- Understanding Local Extrema – Critical points are used to determine where a function may have local maxima or minima.
