Determine the local maxima and minima of f(x) = x^4 - 8x^2 + 16. (2023)

Determine the local maxima and minima of f(x) = x^4 - 8x^2 + 16. (2023)

Step 1: Write down the function f(x) = x^4 - 8x^2 + 16.
Step 2: Find the first derivative f'(x) to determine where the function's slope is zero. The first derivative is f'(x) = 4x^3 - 16x.
Step 3: Set the first derivative equal to zero to find critical points: 4x^3 - 16x = 0.
Step 4: Factor the equation: 4x(x^2 - 4) = 0. This gives us x = 0, x = 2, and x = -2.
Step 5: Find the second derivative f''(x) to determine the nature of the critical points. The second derivative is f''(x) = 12x^2 - 16.
Step 6: Evaluate the second derivative at each critical point: f''(0) = 12(0)^2 - 16 = -16 (which is less than 0, indicating a local maximum), f''(2) = 12(2)^2 - 16 = 32 (which is greater than 0, indicating a local minimum), and f''(-2) = 12(-2)^2 - 16 = 32 (which is also greater than 0, indicating a local minimum).
Step 7: Conclude that there is a local maximum at x = 0 and local minima at x = 2 and x = -2.

Critical Points – Finding where the first derivative is zero or undefined to identify potential local maxima and minima.
Second Derivative Test – Using the second derivative to determine the concavity at critical points to classify them as local maxima or minima.
Polynomial Functions – Understanding the behavior of polynomial functions and their derivatives.